UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
POSGRADO EN CIENCIAS FÍSICAS
Study of Modifications of Gravity in Three
Dimensions
TESIS
QUE PARA OPTAR POR EL GRADO DE
DOCTORA EN CIENCIAS (FÍSICA)
PRESENTA
LORENA PARRA RODRÍGUEZ
TUTOR PRINCIPAL
DR. JOSÉ DAVID VERGARA OLIVER
INSTITUTO DE CIENCIAS NUCLEARES
MIEMBROS DEL COMITÉ TUTOR
DRA. ROCÍO JAÚREGUI RENAUD
INSTITUTO DE FÍSICA
DR. JOSÉ ANTONIO GARCÍA ZENTENO
INSTITUTO DE CIENCIAS NUCLEARES
 
UNAM – Dirección General de Bibliotecas 
Tesis Digitales 
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The work described in this thesis was performed on a double degree program
between the Van Swinderen Institute for Particle Physics and Gravity of the
University of Groningen under the supervision of Prof. Eric A. Bergshoeff and
the Instituto de Ciencias Nucleares of the Universidad Nacional Autónoma de
México under the supervision of Prof. J. David Vergara. It was supported by the
University of Groningen, the Universidad Nacional Autónoma de México, and
the Consejo Nacional de Ciencia y Tecnología (CONACyT).
Chapters about the Supersymmetric Massive Extension, the Non Relativistic and
the Carroll limits, are published under the title "Extensions and Limits of Gravity
in Three Dimensions" with:
ISBN: 978-90-367-8124-4 (printed version)
ISBN: 978-90-367-8123-7 (electronic version)
© 2016 Lorena Parra Rodríguez
Contents
1 Introduction 1
1.1 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Preliminaries 11
2.1 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Massless multiplets . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 Massive supermultiplets . . . . . . . . . . . . . . . . . . . . 16
2.2 Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Kinematical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Non-Linear Realizations . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4.1 Limits vs Non Linear Realizations . . . . . . . . . . . . . . 31
2.5 Group averaging method . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.1 The relativistic free particle . . . . . . . . . . . . . . . . . . 33
2.6 Killing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6.1 Massive Relativistic Particle . . . . . . . . . . . . . . . . . . 35
2.6.2 Massive Relativistic Superparticle . . . . . . . . . . . . . . 37
3 Supersymmetric Massive Extension 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Constructing Massive Actions . . . . . . . . . . . . . . . . . . . . . 43
3.2.1 Proca Action . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2.2 Fierz-Pauli action . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 New Massive Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 Supersymmetric Proca . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.1 Kaluza–Klein reduction . . . . . . . . . . . . . . . . . . . . 52
3.4.2 Truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.3 Massless limit . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5 Supersymmetric Fierz–Pauli . . . . . . . . . . . . . . . . . . . . . . 58
3.5.1 Kaluza–Klein reduction and truncation . . . . . . . . . . . 58
I
II CONTENTS
3.5.2 Massless limit . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6 Linearized SNMG without Higher Derivatives . . . . . . . . . . . . 66
3.7 The non-linear case . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.A General Multiplets and Degrees of Freedom . . . . . . . . . . . . . 74
3.B Off-shell N = 1 Massless Multiplets . . . . . . . . . . . . . . . . . . 75
3.C Boosting up the Derivatives in Spin-3/2 FP . . . . . . . . . . . . . 77
4 Symmetry Analysis for Anisotropic Field Theories 79
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.1 Scalar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.3.2 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3.3 Higher Derivative Chern-Simons Extensions . . . . . . . . . 85
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Path Integrals and Polymer Quantum Mechanics 87
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Polymer Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . 88
5.3 Path Integrals for Polymer Quantization . . . . . . . . . . . . . . . 90
5.3.1 Polymer non-relativistic free particle . . . . . . . . . . . . . 91
5.3.2 Polymer relativistic particle . . . . . . . . . . . . . . . . . . 95
5.3.3 Polymer Harmonic Oscillator . . . . . . . . . . . . . . . . . 98
5.3.4 Polymer Cosmology . . . . . . . . . . . . . . . . . . . . . . 104
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6 Non Relativistic Limit 109
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 The Free Newton-Hooke Particle . . . . . . . . . . . . . . . . . . . 112
6.2.1 Non-linear Realizations . . . . . . . . . . . . . . . . . . . . 113
6.2.2 The Killing Equations of the Newton-Hooke Particle . . . . 114
6.2.3 The Flat Limit . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.3 The Newton–Hooke Superalgebra . . . . . . . . . . . . . . . . . . . 115
6.3.1 Non-linear Realizations . . . . . . . . . . . . . . . . . . . . 117
6.3.2 The Killing Equations of the Newton-Hooke Superparticle . 120
6.3.3 The Kappa-symmetric Newton Hooke Superparticle . . . . 121
6.3.4 The Flat Limit . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.4 Gauging Procedure: Bosonic Case . . . . . . . . . . . . . . . . . . 125
6.4.1 The Galilean Particle . . . . . . . . . . . . . . . . . . . . . 125
CONTENTS III
6.4.2 The Curved Galilean Particle . . . . . . . . . . . . . . . . . 126
6.4.3 The Newton-Cartan Particle . . . . . . . . . . . . . . . . . 127
6.5 Gauging Procedure: Supersymmetric Case . . . . . . . . . . . . . . 128
6.5.1 The Galilean Superparticle . . . . . . . . . . . . . . . . . . 128
6.5.2 The Curved Galilean Superparticle . . . . . . . . . . . . . . 129
6.5.3 The Newton–Cartan Superparticle . . . . . . . . . . . . . . 132
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7 Carroll Limit 135
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 The Free AdS Carroll Particle . . . . . . . . . . . . . . . . . . . . . 139
7.2.1 The AdS Carroll Algebra . . . . . . . . . . . . . . . . . . . 139
7.2.2 Non-Linear Realizations . . . . . . . . . . . . . . . . . . . . 140
7.2.3 The Killing Equations of the AdS Carroll Particle . . . . . 143
7.2.4 The Carroll action as a Limit of the AdS action . . . . . . . 146
7.3 The N = 1 AdS Carroll Superalgebra . . . . . . . . . . . . . . . . 147
7.3.1 Superparticle Action . . . . . . . . . . . . . . . . . . . . . . 148
7.3.2 The Super Killing Equations . . . . . . . . . . . . . . . . . 150
7.3.3 The Flat Limit . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.3.4 The super-AdS Carroll action as a limit of the super-AdS
action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.4 The N = 2 Flat Carroll Superparticle . . . . . . . . . . . . . . . . 154
7.4.1 The N = 2 Carroll Superalgebra . . . . . . . . . . . . . . . 155
7.4.2 Superparticle Action and Kappa Symmetry . . . . . . . . . 156
7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
7.A The 3D N = 2 AdS Carroll Superparticle . . . . . . . . . . . . . . 159
7.A.1 The N = (2, 0) AdS Carroll Superalgebra . . . . . . . . . . 160
7.A.2 Superparticle action . . . . . . . . . . . . . . . . . . . . . . 161
7.A.3 Global Symmetries and Kappa symmetry . . . . . . . . . . 162
7.A.4 The N = (1, 1) AdS Carroll Superalgebra . . . . . . . . . . 165
7.A.5 Superparticle Action . . . . . . . . . . . . . . . . . . . . . . 166
7.A.6 Global Symmetries . . . . . . . . . . . . . . . . . . . . . . . 167
8 Conclusions and Outlook 169
Bibliography 177
Resumen 191
Agradecimientos 197
1
Introduction
General Relativity (GR) has resisted the test of time even a hundred years
after its conception. From an experimental point of view it has been tested to
incredible accuracy by solar system experiments and pulsar timing measurements,
and it has predicted with precision, such phenomena as the deflection of light, the
perihelion advance of Mercury, and the gravitational frequency shift of light. GR
also predicts that over time a binary system’s orbital energy will be converted to
gravitational radiation and the measured rate of orbital period decay turns out to
be almost precisely as predicted. From a theoretical point of view GR provides
a comprehensive, consistent and elegant description of gravity, spacetime and
matter on the macroscopic level.
However, GR loses precision and predictability in strong gravity regimes and
it fails to explain some phenomena in weak gravity regimes such as the galaxy
rotation problem and the missing mass problem in clusters of galaxies. Further-
more, there is not a satisfactory model that explains dark energy and the accel-
erated expansion of the universe. The theory also has some shortcomings such
as its failure to unify gravity with strong and electroweak forces, its prediction of
spacetime singularities, and its incompatibility with quantum mechanics.
Due to GR’s experimental and theoretical importance, many efforts have been
made to study its possible deformations. Extending GR is technically very chal-
lenging because any new theory would have to resolve the above-mentioned large
scale problems while taking into account the behavior that arises in quantum
theory. In other words, we are searching for a mathematically consistent theory
(free of instabilities and ghosts), that can reproduce GR at cosmological scales,
provide an explanation of today’s acceleration of the universe and at the same
time, capable of giving a description of gravity following the principles of quantum
mechanics.
1
2 Introduction
In view of the complexity of this task, this work aims to take a first step
towards dissecting GR and studying a number of different extensions and limits
of the theory in order to have a better understanding of it. Modifying a known
theory is one of the best ways to discover new structures, which may have un-
foreseen applications. In this thesis we therefore address the following question:
What are the possible ways of modifying and studying different limits of gravity?
Table 1.1 lists the extensions and limits studied in this thesis.
Extensions
a) Add new symmetries: Supersymmetry.
b) Add new parameters: Mass parameter.
c) Modify symmetries: Lorentz violation.
d) Modify quantization scheme: Polymer quantization.
Limits
e) Study the non-relativistic limit.
f) Study the ultra-relativistic limit.
Table 1.1
Extensions and Limits of Gravity as described in this thesis.
In order to explain the usefulness of supersymmetry, case (a) in Table 1.1,
we should keep in mind the current status of the Standard Model of particle
physics. The Standard Model was developed in the early 1970s and has since
then successfully explained observations from particle colliders. The Standard
Model has the symmetry group
SU(3)C × SU(2)L × U(1)Y . (1.1)
Roughly speaking, the three factors give rise to the fundamental interactions.
The first SU(3)C part is related to the chromodynamics sector. The gauge group
SU(2)L×U(1)Y is related to the electroweak theory which describes two different
forces below the breaking scale: the weak force and the electromagnetic force.
Despite the huge success of the Standard Model in describing high energy
phenomena, it does leave some features unexplained. For instance, the Standard
Model Higgs sector is not natural, the theory does not provide a dark matter
candidate, fails to explain the origin of the three gauge interactions and cannot
provide an explanation for the inflation period in the early universe [1]. There
are several possible approaches to address these issues, such as adding more
3
particles or interactions to the model, including more general internal symmetries
(in other words, the symmetries of the Standard Model are obtained from the
symmetry breaking of a larger symmetry group) or adding more general spacetime
symmetries (such as extra spacetime dimensions or supersymmetry).
With the development of quantum mechanics, symmetry principles and con-
servation laws came to play a fundamental role in physics. Particle physics sym-
metries include the Poincaré invariance (translations and rotations), ‘internal’
symmetries which can be further divided into global (related to conserved quan-
tum numbers such as the electric charge) and local symmetries (which form the
basis for gauge theories and are therefore related to forces) and discrete sym-
metries (such as charge conjugation, parity transformation and time reversal
invariance).
In 1967 Coleman and Mandula [2] proved that, given certain assumptions,
these are the only possible symmetries for a physical system. The Coleman-
Mandula theorem states that for quantum field theories, spacetime and internal
symmetries can only be combined in a direct product symmetry group (in other
words, Poincaré and internal symmetries do not mix) [3]. This theorem is based
on certain assumptions: that the Poincaré, internal and discrete symmetries are
the only symmetries of the S matrix, that there is a mass gap and that internal
symmetries form a so called Lie group.
It is possible to bypass this theorem by relaxing some of these assumptions.
For example, Poincaré and internal symmetries do in fact mix in the case of
spontaneous broken symmetries. Another option is to consider conformal field
theories, in which there is no mass gap. If the theory only presupposes massless
particles, the Poincaré algebra is extended to the conformal algebra. Yet an-
other alternative is to relax the Lie group structure of the Poincaré and internal
symmetries and to unify them into a superalgebra. In particular, the theorem
assumes that the symmetry generators of the algebra only involve commutators.
Weakening this assumption to allow both anticommuting and commuting gen-
erators opens the possibility of supersymmetry, in other words, an algebra that
includes fermionic and bosonic symmetry generators.
Performing a supersymmetry transformation on a field results in an inter-
change of bosonic and fermionic fields. Exact invariance under supersymmetry
implies that every particle in nature should have a partner with the same mass
and same quantum numbers but with a spin differing by one-half. Performing
two successive supersymmetry transformations on a field leads to the same field
but with a different dependence of spacetime coordinates. It turns out that su-
persymmetry is the only possible extension of the known spacetime symmetries
of particle physics, so it is natural to include it in the study of gravity which is
4 Introduction
the gauge field of the Poincaré spacetime symmetries.
Supergravity is an extension of Einstein’s general relativity that treats super-
symmetry as a gauge symmetry. In other words, supergravity theories have local
supersymmetry and corresponding gauge fields, the gravitini. The gravitini are
the fermionic partners of the graviton. Including supersymmetry in a theory of
gravity turns out to be a valuable and useful technical tool. Moreover, supergrav-
ity theories arise naturally from string theory which is a major candidate for a
theory of quantum gravity. Supergravity is also an essential ingredient for super-
symmetric phenomenology and for obtaining precise predictions, often relating
strong- to weak-coupling regimes leading to the AdS/CFT correspondence.
In addition to adding new symmetries, such as supersymmetry, one could
also add new parameters to gravity. For instance, adding a mass parameter to
GR leads to a theory of massive gravitons, corresponding to case (b) in Table
1.1. Recent years have witnessed a growing interest in massive gravity as a
result of the cosmological constant problem and the discovery of the acceleration
of the universe [4, 5], which could be better explained in terms of an infrared
modification of GR [6, 7] that gives the graviton a mass than by using a dark
energy component. Further motivation arises from the conjecture that some
theories involving massive gravitons could be the low energy limit of a non-critical
string-theory underlying quantum chromodynamics [8].
There are number of ways to acquire a theory of massive gravitons, one
of which is to include explicit mass terms in the Einstein-Hilbert action as in
the so-called Fierz–Pauli theory [9]. A second alternative is to consider higher-
dimensional scenarios, as in the Dvali-Gabadadze-Porrati model [10]. Other theo-
ries, referred to as bi-metric theories, consider two dynamical spin-2 fields [11–13].
Finally, one could introduce higher-derivative terms in the action as in the New
Massive Gravity (NMG) theory [14].
Massive gravity theories pose two main problems. First of all, they usually
make it necessary to work with non-linear theories because linear theories would
not yield the predictions of GR when taking the massless limit. The second
problem that we must take into account is that most massive gravity theories
are plagued with ghost instabilities (fields with negative kinetic energy). In an
attempt to resolve these problems, researchers have tried to construct massive
gravity theories in three dimensions.
Although we live in a four dimensional spacetime it is often convenient to
study gravity in different dimensions. Dimensionality is one of the main features
used to describe physical systems. For example, it is possible to build theoretical
models in dimensions higher than four, or to study systems that are in lower
spatial dimensions, such as one dimensional wires, or two dimensional interfaces.
5
Recently, the study of GR extensions in more than four dimensions has gained
a lot of attention, in particular thanks to the black hole solutions this offers [15].
Another motivation for studying gravity in higher spacetime dimensions is that
the AdS/CFT correspondence relates the properties of aD-dimensional black hole
with those of a quantum field theory in D − 1 dimensions [16], thus relating an
AdS black hole in the gravitational theory with a strongly coupled CFT theory at
finite temperature [17]. Additionally, many studies have focused on this approach
because string theory is formulated in higher dimensions and it includes gravity
[18].
The quantum theory of gravitons is non-renormalizable in four dimensions.
Therefore, theorists have considered GR and its variants in three dimensions
because one expects less severe short-distance behavior in lower dimensions. This
makes it interesting to use three-dimensional gravity models to study problems
in quantum gravity.
Three dimensions are special because a massless graviton in three dimensions
has no propagating degrees of freedom (we discuss in detail the counting of de-
grees of freedom in section (3.1)). In three dimensional gravity, the Weyl tensor
vanishes identically and the Riemann tensor is therefore completely determined
by the Ricci tensor. Thus, all the components of the Riemann tensor are fixed
by Einstein’s equations and there are no free degrees of freedom. Despite this,
three dimensional gravity allows for solutions with non-trivial topology. A mas-
sive graviton in three dimensions has two degrees of freedom which is the same
number as a massless graviton in four dimensions. Therefore, in three dimensions
it should be possible to construct a theory of massive gravity that is invariant
under diffeomorphisms.
New Massive Gravity (NMG) is a higher-derivative extension of three-dimen-
sional (3D) Einstein–Hilbert gravity, with a particular set of terms quadratic
in the 3D Ricci tensor and Ricci scalar [14]. The NMG model is interesting
because although the theory contains higher derivatives, it nevertheless describes,
unitarily, two massive degrees of freedom of helicity +2 and −2. Furthermore, it
has been shown that even at the non-linear level ghosts are absent [19]. The 3D
NMG model is an interesting laboratory to study the validity of the AdS/CFT
correspondence in the presence of higher derivatives. A supersymmetric version of
NMG was constructed in [20]. In addition to the fourth-order-derivative terms of
the metric tensor, this model also contains third-order-derivative terms involving
the gravitino.
For many purposes, it is convenient to work with a formulation of this model
without higher derivatives, see, e.g. [21]. In the linearized NMG model, this can
be achieved by introducing an auxiliary symmetric tensor that couples to (the
6 Introduction
Einstein tensor of) the 3D metric tensor and has an explicit mass term [14].
The first GR modification that we consider in this work combines extensions
(a) and (b) from Table 1.1, we modify GR by adding an additional symmetry
(supersymmetry) and an additional parameter (a mass parameter). We work
with a supersymmetric reformulation of the NMG model (SNMG) without higher
derivatives. This requires us to introduce both an auxiliary symmetric tensor and
further auxiliary fermionic fields that effectively lower the number of derivatives
of the gravitino kinetic terms.
As previously mentioned, the rules of quantum mechanics have not been ap-
plied to gravity in a satisfactory manner up to now, although recently, many
theories have emerged as possible avenues towards the quantization of GR such
as string theory, supergravity, loop quantum gravity, Hořava-Lifshitz gravity. . .
Despite these efforts, the candidate models still need to overcome major formal
and conceptual problems besides, with our current technology development on
particle physics and cosmological observations, there is no way to validate their
predictions through experimental tests.
Even though there are no experimental accessible situations where one can
check the quantum gravity framework, there are many physical scenarios that
one can imagine that require a quantum theory of gravity for their description
where we can study their properties and outcomes with our current knowledge.
Recently, Hořava proposed a new theory of gravity [22–24] based on an
anisotropic scaling of the spatial and time coordinates, which leads to the break-
ing of the relativistic invariance at short distances, with GR being recovered for
low-energy limits. The main ingredient of this theory is the anomalous scal-
ing relation between the Minkowskian space and time coordinates, breaking the
spacetime diffeomorphism invariance.
The second modification that we study in this thesis (case (c) on Table 1.1)
was inspired by the Lorentz violation in Hořava’s theory. We use Noether’s
theorem to study the symmetries of anisotropic actions for arbitrary fields which
generally depend on higher order spatial derivatives, and find the corresponding
current densities and charges. In particular, we consider scale invariance on higher
derivative four dimensional extensions of the scalar field and electrodynamics, and
a three dimensional Chern-Simons theory.
On the other hand, in the mid 1980s, Ashtekar pointed out that it is possible
to rewrite the equations of gravity in terms of variables that made the theory re-
semble the theories of particle physics in such a way that techniques from particle
physics could be imported to the quantization of gravity. The resulting approach
is called loop quantum gravity (see for example [25] for a brief introduction).
Loop Quantum Gravity considers GR’s insight that spacetime is a dynamical
7
field and is therefore a quantum object so there should be no background metric,
implying that geometry and matter should both be ‘born quantum mechani-
cally’ [25]. A second consideration in loop quantum gravity is that the quantum
discreteness that determines the particle-like behavior of other field theories also
affects the structure of space. One of the main results of loop quantum gravity
is the derivation of a granular structure of space at the Planck length.
Polymer Quantum Mechanics is a quantization that replicates some features
of the quantization in Loop Quantum Gravity. It was introduced by Ashtekar
in [26] and later used in [27–30] as a novel quantization for scalar field theories,
to construct a quantization scheme not equivalent to Schrödinger Quantum Me-
chanics. This scheme has been successfully applied to the quantization of several
cosmologies, avoiding the big bang singularity.
In order to gain a better insight on this quantization, in our third study
case (case (d) on Table 1.1) we analyze the polymer quantization via the path
integral procedure of some toy models such as the non-relativistic free particle,
the relativistic free particle, the harmonic oscillator and the simplest Friedman-
Robertson-Walker model. This scheme provides us with an effective action that
we can use to understand more deeply the non-perturbative aspects of the dy-
namical system at classical and quantum levels.
Apart from these gravity extensions, another approach to study gravity is to
test its properties in limiting cases [31–34]. Here we consider the supersymmetric
non-relativistic and ultra-relativistic limits of GR: the limits when the speed
of light goes to infinity (we combine extension (a) and limit (c) of Table 1.1)
and to zero (combining cases (a) and (d)) respectively. Geometrically, the non-
relativistic transition can be viewed as the opening of the light cones (the cones
become space-like hypersurfaces) while the opposite ultra-relativistic transition
can be understood as the shrinking of the light cones (the cones become time-
like hypersurfaces). While the non-relativistic limit is governed by the Galilei
algebra, the corresponding algebra in the ultra-relativistic limit is the Carroll
algebra. Both invariance groups can be obtained from adequate contractions of
the Poincaré group. When starting from the AdS group, different contractions
will lead to the non-relativistic Newton-Hooke and the ultra-relativistic AdS-
Carroll groups.
There are many non-relativistic conformal field theories which describe physi-
cal systems within condensed matter physics [35], atomic physics [36] and nuclear
physics [37]. Non-relativistic versions of the AdS/CFT correspondence have re-
cently been investigated because they open the way for possible applications of
gauge-gravity duality to a variety of real-life strongly interacting systems. Non-
relativistic symmetry groups, such as the Schrödinger or the Galilean conformal
8 Introduction
symmetry groups, are relevant for the study of cold atoms, which have a gravity
dual possessing these symmetries [38, 39]. Furthermore, in the case of strings,
non-relativistic limits may have applications in the context of non-relativistic
versions of AdS/CFT [40].
In addition, the so-called Carroll symmetries that arise in the ultra-relativistic
limit, have played an important role in recent investigations [41], for example in
studies of tachyon condensation [42]. More recently, they have also appeared in
the study of warped conformal field theories [43].
There are essentially two procedures for constructing non-relativistic and
ultra-relativistic gravity/supergravity theories. The first involves using the limits
from vielbein formulations of relativistic gravity/supergravity. As shown in [44],
such a limit can be defined and implemented in a consistent manner in the non-
relativistic case. The second procedure consists of gauging the algebras with
non-relativistic or ultra-relativistic symmetries (the gauging procedure in the
non-relativistic case was studied in [45–47]). In other words, the first proce-
dure considers the non-relativistic and ultra-relativistic limits of Einstein gravity
while the second procedure involves performing a gauging of the limit versions
of the Poincaré algebra. It turns out that the relevant non-relativistic and ultra-
relativistic versions of the Poincaré algebra represent a particular contraction of
this algebra.
Using the non relativistic limit of GR leads to the well-known non-relativistic
Galilean gravity (valid in frames with constant acceleration), Curved Galilean
gravity (valid in frames with time-dependent acceleration) and, Newton–Cartan
gravity (valid in arbitrary frames). So far, no satisfactory ultra-relativistic limit
of GR has been found.
1.1. Outline of the thesis
This thesis is organized as follows. The first part of Chapter 2 contains a
review of supersymmetry and supergravity, as these topics play an important
role in this thesis. Since we discuss different types of limits and contractions of
the AdS groups, we dedicate a section of this chapter to reviewing some features
of the different kinematical groups. We also explain the non-linear realizations
method which allows us to construct actions invariant under the symmetry group
under consideration.
In Chapter 3 we make the case for constructing a supersymmetric model of
New Massive Gravity without higher derivatives. We show how to explicitly
construct the linearized, massive, off-shell, spin 1 three dimensional N = 1 su-
permultiplet and we comment about its massless limit as a warming-up example
1.1 Outline of the thesis 9
before moving on to the spin-2 case. We then discuss how to obtain a linearized
supersymmetric new massive gravity theory without higher derivatives and we
offer some comments on the non-linear case.
At the linearized level, the NMG model decomposes into the sum of a massless
spin-2 Einstein–Hilbert theory and a massive spin-2 Fierz–Pauli (FP) model [14].
In the supersymmetric case we therefore need a 3D massless and a 3D massive
spin-2 supermultiplet.
Chapters 4 and 5 are devoted to the symmetry analysis of higher order spa-
tial derivatives systems with the anisotropic scaling between the space and time
coordinates introduced by Hořava and the construction of the path integral for
some physical systems in the framework of polymer quantization respectively. In
Chapter 5 we also study the measure of the integral in detail in order to obtain
the form of the exact propagators. Afterwards we analyze the effective action of
several systems and their dynamics.
Chapter 6 addresses the problem of how to describe a non-relativistic super-
particle in a curved background. Describing a non-relativistic superparticle in a
curved background requires first a supersymmetric extension of the gravity back-
grounds. Since non-relativistic supergravity multiplets have to our knowledge
only been explicitly constructed in three dimensions, we will only consider su-
perparticles in a three-dimensional (3D) background. A supersymmetric version
of the 3D Galilean background was recently constructed by gauging the Galilei,
or more accurately the Bargmann, superalgebra [48]. Our aim is to investigate
the action of a 3D superparticle first in a flat background and then in a Galilean
supergravity background without a cosmological constant.
In Chapter 7 we investigate particles whose dynamics are invariant under
the Carroll superalgebra. We investigate the geometry of flat and curved (AdS)
Carroll space and the symmetries of a particle moving in such a space both in
the bosonic as well as in the supersymmetric case. In the bosonic case we find
that the Carroll particle possesses an infinite-dimensional symmetry which only
includes dilatations in the flat case. The duality between the Bargmann and
Carroll algebra that is relevant for the flat case does not extend to the curved
case.
In the supersymmetric case we study the dynamics of the N = 1 AdS Carroll
superparticle. Only in the flat limit do we find that the action is invariant under
an infinite-dimensional symmetry that includes a supersymmetric extension of
the Lifshitz Carroll algebra with dynamical exponent z = 0. We also discuss the
extension to N = 2 supersymmetry in the flat case and show that the flat N = 2
superparticle is equivalent to the (non-moving) N = 1 superparticle making it
non-BPS, unlike its Galilei counterpart. This is due to the fact that in this case
10 Introduction
the so-called kappa-symmetry eliminates the linearized supersymmetry.
In the last chapter, Chapter 8, we conclude and offer some possible directions
for future research.
2
Preliminaries
In the first part of this chapter we review some general theoretical aspects
that we will use through this thesis. We start with supersymmetry and super-
gravity and we discuss the construction of massless and massive representations
of supersymmetry algebras in four and three dimensions respectively in order to
have a better understanding of the supermultiplets involved in the New Massive
Supergravity section.
We also make a review of the kinematical groups obtained from the differ-
ent contractions of the AdS group that we use in the non-relativistic and in the
Carroll sections. We give the generalities of the non-linear realizations proce-
dure to obtain particle actions and we discuss how to study the symmetries of
Hamiltonian systems.
2.1. Supersymmetry
The Standard Model has a huge and continued success in providing experi-
mental predictions, however it does leave some unexplained phenomena like the
hierarchy problem or the cosmological constant problem. Besides, it describes
three of the four fundamental interactions at the quantum level, and we need
a quantum theory of gravity because at energies higher than the Planck scale,
gravity becomes comparable with other forces and cannot be neglected. At this
scales quantum effects of gravity have to be included but Einstein’s theory is
non-renormalizable and therefore it can not provide proper answers to observ-
ables beyond this scale.
Supersymmetry solves the naturalness issue of the hierarchy problem. Com-
bined with string theory it solves the quantum gravity issue. It also provides
11
12 Preliminaries
the best example for dark matter candidates and it also allows one to evade the
Coleman-Mandula theorem which asserted the impossibility of putting together
spacetime symmetries and internal symmetries in a non-trivial way. Moreover,
Haag, Lopuszanski and Sohnius showed that supersymmetry is the only possi-
ble option of non-trivial interactions between internal symmetries and spacetime
symmetries. There is a lot of literature devoted to these topics, throughout
this thesis we will mainly follow the notation from [49]. Other useful references
are [50–55].
Supersymmetry is a spacetime symmetry mapping bosonic states (integer
spin) into fermionic states (half-integer spin) and vice versa. The operator Q
that generates such transformation must be a spinor with
Q|boson〉 = |fermion〉 and Q|fermion〉 = |boson〉 . (2.1)
Supersymmetry is not an internal symmetry but a spacetime symmetry because it
changes the spin of a particle and the behaviour of bosons and fermions is different
under rotations. This can be seen trough the commutation relations between the
supersymmetry generators and the other ones. The operator Q commutes with
time and spatial translations and also with internal quantum numbers (like gauge
and global symmetries). However, it does not commute with Lorentz generators
[Q,Pµ] = 0 , [Q,Mµν ] 6= 0 . (2.2)
We should remark here that the full supersymmetry algebra contains the Poincaré
algebra as a subalgebra, and since each irreducible representation of the Poincaré
algebra corresponds to a particle, an irreducible representation of the super-
symmetry algebra in general corresponds to several particles (because any rep-
resentation of the full supersymmetry algebra also gives in general a reducible
representation of the Poincaré algebra). The resulting states are related between
them by the Q generators and have spins differing by units of one half. They form
a supermultiplet. A supermultiplet always contains an equal number of bosonic
and fermionic degrees of freedom (here it is necessary to make a distinction be-
tween on- and off-shell multiplets. In the off-shell case, it is necessary to add
auxiliary fields to the theory in order to close the algebra off-shell and then we
will have equality of bosonic and fermionic degrees of freedom). In other words,
in all supersymmetric models each one-particle state has at least one superpart-
ner, therefore instead of having single particle states we have supermultiplets of
particle states. From the commutation relations we can also see that the particles
belonging to the same supermultiplet have different spin but same mass and same
quantum numbers.
2.1 Supersymmetry 13
It is possible to write down theories with any number N of supersymmetry
generators but this number cannot be arbitrarily large, because as we will show
later on, N is related with particle states of a given spin, so by increasing N
the spin will also increase and it is very difficult to built higher spin consistent
theories. For example in 4 dimensions to describe local and interacting massless
theories with global supersymmetry, N can be at most as large as 4 for theories
with maximal spin 1 (gauge theories) and as large as 8 for theories with maximal
spin 2 (supergravity theories). For N ≥ 9 the representations contain some
particles with spin s > 2. It turns out, that there is a one-to-one correspondence
between the massless on shell supermultiplets of N -extended four dimensional
supersymmetry and the massive on shell-supermultiplets of N extended three-
dimensional supersymmetry [56], we discuss both cases in detail in the following
sections.
2.1.1. Massless multiplets
The maximum allowed number of supersymmetry generators N , depends on
the dimension of the spinor representation of the Lorentz group which at the same
time depends on the spacetime dimensions where one is working (for example, in
ten dimensions, which is the dimension where superstring theory lives, only two
supersymmetry generators are allowed). In this section we will focus in the case
of four dimensions.
Lets consider superalgebras containing N ≥ 1 Majorana spinor charges Qiα,
where α is the spinor index and i is the index that labels the supercharges i =
1, . . .N . We define the supercharges Qiα as the left-handed chiral components
and use the Weyl representation, then Qiα = (Qi1, Qi2, 0, 0), where the i index
stands up for their hermitian conjugates Q†iα = ((Qi1)∗, (Qi2)∗, 0, 0) so we can
use the index range α = 1, 2. This gives the algebra
[Pµ, Qiα] = 0 , [Mµν , Qiα] = −1
2
[γµν ]βαQiβ , {Qiα, Q†jβ} =
1
2
δji [γµγ
0]βαP
µ .
(2.3)
Since the momentum operators Pµ commute with the supercharges, we may
consider the states at arbitrary but fixed momentum Pµ which for massless rep-
resentations, satisfies P 2 = 0, so it will be enough to consider the action of the
supersymmetry generators on a set of particle states |pµ, s, λ〉 where s is the spin
and λ is the helicity of the particle (the component of angular momentum in the
direction of motion of the particle). Going to the rest frame Pµ = (E, 0, 0, E)
and
{Qiα, Q†jβ} =
(
0 0
0 E
)
δji , (2.4)
14 Preliminaries
hence half of the spinors must vanish on physical states while the other half will
generate a Clifford algebra. From the non-trivial generators we can choose Qi2
as the annihilation and Q†i2 as the creation operators. In the present frame
[J3, Q†i2] = −1
2
Q†i2 , (2.5)
which means that Q†i2 lowers the helicity of a state by 1/2 and Qi2 arises it by the
same amount. Now, it is possible to choose a vacuum state (a state annihilated
by all the annihilation operators). This vacuum state will carry some irreducible
representation of the Poincaré algebra. We denote this state as |λ0〉. We can
construct the supermultiplet by applying the creation operator over |λ0〉 creating
a tower of helicity states of maximum helicity λ0 and minimum helicity λ0 − 1
2N
as
|λ0〉 , |λ0 − 1
2〉i , |λ0 − 1〉[ij] . . . |λ0 − N
2 〉[i...N ] , (2.6)
where i = 1, . . .N . Note that the particle spin s = |λ| is a redundant label
and pµ is fixed so we omitted them when writing the states. Since the action
of different creation operators is antisymmetric, the interchange of the indices
i, j, . . . is antisymmetric too and states with helicity λ = λ0 − k
2 have multiplicity
(N
k
)
with k = 0, 1, . . .N , see Table 2.1. The sequence stops at the multiplicity
1 state of lowest helicity λ0 − 1
2N and it is easy to see that in every multiplet
λmax − λmin = N
2 . Summing the binomial coefficients gives a total of 2N states
State Number of states
|λ0〉 1 =
(
N
0
)
|λ0 − 1
2 〉 N =
(
N
1
)
|λ0 − 1〉 1
2! N (N − 1) =
(
N
2
)
...
...
|λ0 − N
2 〉 1 =
(
N
N
)
Table 2.1
This table gives the number of states that we will get in 4 dimensions depending on the
number of supersymmetry generators N that we are considering.
with 2N −1 having integer helicity (bosons) and 2N −1 having half-integer helicity
2.1 Supersymmetry 15
(fermions).
N
∑
k=0
(
N
k
)
= 2N = (2N −1)bosons + (2N −1)fermions , (2.7)
For unextended supersymmetry N = 1 each massless supermultiplet only
contains two states |λ0〉 and |λ0 − 1
2〉. We denote these multiplets by (λ0, λ0 − 1
2).
To satisfy CPT invariance and since they can never be CPT self-conjugate and one
needs to double these multiplets by adding their CPT conjugate with opposite
helicities and opposite quantum numbers. Thus one arrives at the following
massless N = 1 multiplets.
The chiral multiplet (λ0 = 0): contains (0,−1
2) and its CPT conjugate
(+1
2 , 0). The degrees of freedom correspond to one Weyl fermion and one
complex scalar. This is the representation to describe a matter multiplet.
The vector multiplet (λ0 = −1
2): consists of (−1
2 ,−1) plus (+1,+1
2), corre-
sponding to a gauge boson (massless vector) and one Weyl fermion (gaug-
ino). This representation describes gauge fields in a supersymmetric theory.
The gravitino multiplet (λ0 = −1): contains (−1,−3
2) and (+3
2 ,+1), which
describe a gravitino and a gauge boson. Notice that there is a corresponding
free field theory for a massless spin 3/2 fermion but no interacting field
theory is know for this multiplet without supergravity.
The graviton multiplet (λ0 = −3
2): describes a (−3
2 ,−2) and (+2,+3
2)
supermultiplets corresponding to the graviton and the gravitino. This is
the supergravity multiplet.
For N = 2 the supermultiplets will contain 22 = 4 states and again it is
necessary to add their CPT conjugates so we will get
The matter multiplet or hypermultiplet (λ0 = 1
2): contains 2×(1
2 , 0, 0,−1
2)
corresponding to two Weyl fermions and two complex scalars. Note that
although this multiplet is self-conjugate it requires to be doubled because
of hermiticity. This can be decomposed in terms of two N = 1 chiral
multiplets.
The gauge multiplet (λ0 = 0): consisting of (0,−1
2 ,−1
2 ,−1) and (+1,+1
2 ,
+1
2 , 0), so the degrees of freedom are those of one vector, two Weyl fermions
and one complex scalar. This supermultiplet can be decomposed in terms
of one N = 1 vector and one N = 1 chiral multiplets.
16 Preliminaries
The gravitino multiplet (λ0 = 3
2): describing a (+3
2 ,+1,+1,+1
2) and (−1
2 ,
−1, −1,−3
2) corresponding to a spin 3/2 fermion, two vectors and one Weyl
fermion.
The graviton multiplet (λ0 = 2): containing (+2,+3
2 ,+
3
2 ,+1) and (−1,−3
2 ,
−3
2 ,−2) a graviton, two gravitini and a vector.
Consider now the N = 4 case
The vector multiplet (λ0 = +1): consisting of (1 × (+1), 4 × (+1
2), 6 ×
(0), 4 × (−1
2), 1 × (−1)). This is the only multiplet in N = 4 with states
with helicity λ < 2. It consist of three N = 1 chiral multiplets plus their
CPT conjugates and one N = 1 vector. Or one N = 2 vector multiplet and
two N = 2 hypermultiplets plus their CPT conjugates.
The gravitino multiplet (λ0 = +3
2): contains (1×(+3
2), 4×(+1), 6×(+1
2), 4×
(0), 1 × (−1
2)) plus its CPT conjugate.
The graviton multiplet (λ0 = +2): consisting of (1 × (+2), 4 × (+3
2), 6 ×
(+1), 4 × (+1
2), 1 × (0)) and its CPT conjugate.
Finally, for N = 8 generators we obtain
The maximum multiplet: (1×(±2), 8×(±3
2), 28×(±1), 56×(±1
2), 70×(0)).
It is important to note that renormalizable theories have spin s ≤ 1 implying N ≤
4. Therefore N = 4 is the largest supersymmetry for renormalizable theories with
global supersymmetry. Besides this, the maximum number of supersymmetries
is N = 8 otherwise we will have particles with spin higher than two and theories
with more than one graviton. Supergravity theories will be those involving one
spin-2 graviton, N spin-3/2 gravitini, and, for N ≥ 2 lower spin particles.
2.1.2. Massive supermultiplets
Having discussed the algebra and representations of extended massless super-
symmetry, we will turn now to the case of massive supersymmetry N ≥ 1 in three
dimensions. In this section we follow the discussion given in [56]. The algebra in
this case is
[Pµ, Qiα] = 0 , [Mµν , Qiα] = −1
2
[γµν ]βαQiβ , {Qiα, Q†jβ} =
1
2
δji [γµγ
0]βαP
µ ,
{Qiα, Qjβ} = ǫijZαβ , {Q†iα, Q†jβ} = ǫijZ̄αβ ,
(2.8)
2.1 Supersymmetry 17
these are superalgebras with central charges. The generator Z and its complex
conjugate Z̄ commute with all generators of the full algebra. Because of the
presence of ǫij , there is no possibility of central charges for the simplest algebra
N = 1.
In general, massive multiplets are bigger than massless ones because the num-
ber of non-trivial generators is not reduced, unlike for the massless case where
one-half of the generators vanishes. However, it is possible to have a shortening of
the massive representation in the presence of special values of the central charges.
The shortened supermultiplets are known as BPS multiplets.
In the rest frame where Pµ = (m, 0, 0) and P 2 = m2 > 0, the supersymmetry
algebra becomes
{Qiα, Q†jβ} = mδji δ
β
α , {Qiα, Qjβ} = ǫijZαβ , {Q†iα, Q†jβ} = ǫijZ̄αβ . (2.9)
Since the central charges can or can not be non-vanishing, we will just focus in
the vanishing central charges Z = 0 case.
There are 2N creation and annihilation operators Q†iA and QiA respectively
(with A = 1, 2), in this case Q†iA lowers the helicity by 1
2 and QiA rises it by
the same amount. In 3D the helicity λ is equal to the Pauli-Lubanski pseudo
scalar divided by the mass. We can define a vacuum state with mass m and spin
s as |Ω〉 which is annihilated by both QiA and act with the creation operators to
construct the corresponding massive representation
|Ω〉, Q†iA|Ω〉, Q†jBQ†iA|Ω〉, Q†kCQ†jBQ†iA|Ω〉, . . . , Q†NB . . . Q†iA|Ω〉, (2.10)
leading to 2N states with helicities ranging from λ to λ+ N/2. It turns out that
formally these are the same massless particle multiplets in four dimensions for
N -extended supersymmetry.
Helicity +2 +3/2 +1 +1/2 0 -1/2 -1 -3/2 -2
N = 1 1 1
N = 2 1 2 1
N = 4 1 4 6 4 1
N = 8 1 8 28 56 70 56 28 8 1
Table 2.2
This table gives the multiplets containing the +2 helicities for N = 1, 2, 4 and 8
supersymmetries.
18 Preliminaries
One important difference between the 4D and 3D theories is that the last
one has half the total number of supersymmetries. The correspondence between
4D and 3D theories is for supermultiplets, and not for free field theories as it is
explained in [56].
2.2. Supergravity
Supergravity is a field theory that combines general relativity and supersym-
metry, therefore, supersymmetry holds locally in a supergravity theory (in other
words, supergravity is the supersymmetric theory of gravity, or equivalently, a
theory of local supersymmetry). The theory will be invariant under supersym-
metry transformations in which the spinor parameters are arbitrary functions of
the spacetime coordinates. The gauge field of supersymmetry is a spin 3/2 field
ψαµ called gravitino and it is the supersymmetric partner of the graviton.
The supersymmetry algebra (see (2.8)) will then involve local translation pa-
rameters which must be viewed as diffeomorphisms. Take the anticommutation
relation
{Qiα, Q†jβ} = 1
2δ
j
i [γµγ
0] βα P
µ , (2.11)
this indicates that having general coordinate transformations is equivalent to have
local supersymmetry. Thus local supersymmetry requires gravity. The converse is
also true. In any supersymmetric theory which includes gravity, supersymmetry
has to be realized locally. The reason is that a constant spinor is not compatible
with the symmetries required in a theory of gravity with fermions. One must
extend the constant parameter to a local one.
Since Einstein’s gravity is a gauge theory of local symmetry, then the genera-
tors of rotations Mab and the translation generators Pa have corresponding gauge
fields ωabµ , the spin connection, and eaµ, the vierbein (where a, b = 0, . . . , D − 1
are Lorentz gauge group indices and µ, ν = 0, . . . , D − 1 are spacetime indices).
Thus, the vierbein and the spin connection transform under general coordinate
transformations as collections of vectors. A vierbein formulation of gravity is
necessary since supersymmetry will require the presence of spinor fields. The
gauge fields have corresponding field strengths Rabµν , the Riemann curvature ten-
sor, and Caµν the torsion. Setting the torsion to zero allows us to solve for the
spin connection in terms of the vierbein, which leaves a theory with two degrees
of freedom: the graviton.
A supergravity theory is an interacting field theory that contains the gravity
multiplet plus other matter multiplets of the underlying global supersymmetry
algebra. The gauge multiplet consists of the vierbein eaµ(x) describing the graviton
2.3 Kinematical Groups 19
plus the supersymmetric partners of it, a specific number N of vector–spinor fields
ψiµ(x), i = 1, ..., N , the gravitini. In the basic case of N = 1 supergravity in four
spacetime dimensions, the gauge multiplet consists entirely of the graviton and
one Majorana spinor gravitino.
2.3. Kinematical Groups
Spacetime symmetries have played a central role in the understanding of var-
ious physical theories such as Newtonian Gravity, Maxwell’s Electromagnetism,
Special Relativity, General Relativity, Strings and Supergravity. Most of these
models are based on relativistic symmetries. An example of a model with non-
relativistic symmetries is Newtonian Gravity which is based on the Galilei sym-
metries. Such non-relativistic symmetries arise when the velocity of light is sent
to infinity.
A less well known example of a non-relativistic symmetry are the Carroll
symmetries which arise when the velocity of light is sent to zero [32]. In this sense
the Carroll symmetries are the opposite to the Galilei symmetries. This can also
be seen by looking at the light cone which in the Carroll case, at each point of
spacetime, collapses to the time axis whereas in the Galilei case it coincides with
the space axis, see Table 2.3.
c → ∞ c → 0
Table 2.3
The diagram at the left shows the transition Einstein→Newton that occurs in the limit
c → ∞, geometrically it can be viewed as the opening of the light cones. The diagram at the
right shows the transition Einstein→Carroll that occurs in the limit c → 0 which can be
interpreted as the shrinking of the light cones.
A systematic investigation of the possible relativity groups1 was initiated by
Bacry and Lévy-Leblond [31]. They showed that all these groups can be obtained
by a contraction of the anti-de Sitter (AdS) and de Sitter (dS) groups. There are
eleven kinematical Lie algebras:
1A relativity group is an invariance group of a physical theory that contains the generators
of special relativity: time translations, spatial translations, boosts and spatial rotations.
20 Preliminaries
Two simple Lie algebras: the dS and AdS algebras.
Five solvable Lie algebras: the Poincaré (P) algebra, two Newton-Hook
(NH±) and two AdS-Carroll (AC±) algebras.
Three nilpotent Lie algebras: the Galilei (G), the Carroll (C) and two Para-
Galilei (PG) algebras.
AdS
AC P NH
C G
c→0
R→∞
c→∞
R→∞
c→0 c→∞
R→∞
Table 2.4
The figure displays the different contractions of the AdS group. The different abbreviations are
explained in the text.
As Table 2.4 shows there are three different types of contractions2: the non-
relativistic limit c → ∞ of the AdS group leads to the Newton-Hooke (NH)
group. The flat limit R → ∞ leads to the Poincaré (P) group and the ultra-
relativistic limit c → 0 leads to the AdS-Carroll (AC) group [32]3. In a second
stage, the flat limit of the AdS-Carroll group and the ultra-relativistic limit of
the Poincaré group leads to the Carroll (C) group while the non-relativistic limit
of the Poincaré group and the flat limit of the Newton-Hooke group leads to the
Galilean (G) group.
All the algebras corresponding to the kinematical groups given in Table 2.4
contain the same commutators involving spatial rotations. These commutators
are given by
[Mab,Mcd] = 2ηa[cMd]b − 2ηb[cMd]a , (2.12)
[Mab, Pc] = 2δc[bPa] , [Mab,Kc] = 2δc[bKa] , (2.13)
2In this thesis we will only consider the AdS case.
3Bacry and Lévy-Leblond [31] call this algebra the para-Poincaré algebra.
2.3 Kinematical Groups 21
where a = 1, . . . , D−1, for a D-dimensional space-time. The Galilean algebra can
be extended with a central charge generator Z to the so-called Bargmann algebra
[57]. It has been recently shown that there is a duality between this Bargmann
algebra and the Carroll algebra by the exchange of Z and the generator of time
translations H [33]. Note that this duality does not extend to a duality between
the Newton-Hooke and AdS-Carroll algebras. This is due to the expression for
the commutator [Pa, Pb], see Table 2.54.
[Pa,Kb] [H,Ka] [H,Pa] [Pa, Pb] [Ka,Kb]
AdS δabH Pa − 1
R2Ka
1
R2Mab Mab
Poincaré δabH Pa 0 0 Mab
Newton-Hooke δabZ Pa − 1
R2Ka 0 0
AdS-Carroll δabH 0 − 1
R2Ka
1
R2Mab 0
Galilei δabZ Pa 0 0 0
Carroll δabH 0 0 0 0
Table 2.5
This table gives an overview of the algebras of the relativity groups that we consider.
The next thing to do is to consider a supersymmetric extension of the bosonic
relativity groups given in 2.4. We construct the N = 1 superalgebras in any
dimension (see Tables 2.5 and 2.6, where Q stands for the generator of supersym-
metry).
All these algebras contain the same commutator between spatial rotations and
supersymmetry generators
[Mab, Q] = −1
2
γabQ . (2.14)
Note that an adequate Newton-Hooke superalgebra should contain commutator
relations that yield {Q,Q} ∼ H and {Q,Q} ∼ /P , that is, the commutator of two
supersymmetries gives time- and space-translations. For N = 1 this cannot be
achieved, in order to obtain a true supersymmetric extension of the Bargmann
algebra in which the anti-commutator of two supersymmetry generators gives
4Note that the AdS Carroll algebra is formally isomorphic to the Poincaré algebra by replacing
the AC generators of time/spatial translations and boosts according to H → 1
R
H, Pa → 1
R
Ka
and Ka → Pa. We thank Sergey Krivonos for pointing this out to us.
22 Preliminaries
N = 1 [Ka, Q] [H,Q] [Pa, Q] {Qα, Qβ}
AdS − γa0
2 Q γ0
2RQ
γa
2RQ 2[γAC−1]αβPA + 1
R [γABC−1]αβMAB
Poincaré − γa0
2 Q 0 0 2[γAC−1]αβPA
Newton-Hooke 0 0 0 2δαβZ
AdS-Carroll 0 0 1
2RγaQ [γ0C−1]αβH + 2
R [γa0C−1]αβKa
Galilei 0 0 0 2δαβZ
Carroll 0 0 0 [γ0C−1]αβH
Table 2.6
In this table we give the (anti-)commutators of the N = 1 superalgebras that involve the
generators Q of supersymmetry.
both a time and a space translation we need at least two supersymmetries. On
the other hand the AdS Carroll superalgebra has a conventional supersymmetry
algebra, where the energy and boost generators appear in the anti-commutator
of the supersymmetries.
One way to obtain N = 2 locally supersymmetric theories in three space-
time dimensions is through dimensional reduction from four dimensional N = 1
supersymmetric systems, however, there are theories which cannot be obtained
from such procedure. Such theories contain Chern-Simons couplings, in [58] is
showed that in three dimensions N -extended AdS supergravity exists in several
incarnations [59,60]. These were called the (p, q) AdS supergravity theories where
the non-negative integers p ≥ q are such that N = p + q. The N = (p, q) no-
tation refers to the associated 3D AdS supergroups OSp(2, p) ⊕ OSp(2, q) with
R-symmetry group SO(p) × SO(q).
There are two different independent versions of the N = 2 supergravities in
three dimensions, the so-called N = (1, 1) and N = (2, 0) representations.
The basic commutators of the 3D N = (2, 0) AdS algebra are given by those
on Table 2.5 plus
[
MAB, Q
i
]
= −1
2
γABQ
i , [PA, Qi] = xγAQ
i , [R, Qi] = 2xǫijQj ,
{Qiα, Qjβ} = 2[γAC−1]αβPAδij + 2x[γABC−1]αβMABδ
ij + 2[C−1]αβǫijR ,
(2.15)
where (A = 0, 1, 2; i = 1, 2). Here PA,MAB,R andQiα are the generators of space-
time translations, Lorentz rotations, SO(2) R-symmetry transformations and su-
2.3 Kinematical Groups 23
persymmetry transformations, respectively. The bosonic generators PA,MAB
and R are anti-hermitian while the fermionic generators Qiα are hermitian. The
parameter x = 1/(2R), with R being the AdS radius. Note that the generator of
the SO(2) R-symmetry becomes the central element of the Poincaré algebra in
the flat limit x → 0.
To show that the above algebra corresponds to the N = (2, 0) AdS algebra it
is convenient to define the new generators
MC = ǫCABM
AB , J±
A = PA ± xMA . (2.16)
In terms of these new generators we obtain the following (anti-)commutation
relations:
[
J+
A , Q
i] = 2xγAQi ,
{
Qiα, Q
j
β
}
= 2[γAC−1]αβJ
+
A δ
ij , (2.17)
while the charges Qi do not transform under J−
A . This identifies the algebra as
the N = (2, 0) AdS algebra.
We now consider the 3D N = (1, 1) anti-de Sitter algebra which is given by
the commutators on Table 2.5 and the following (anti-)commutators:
[
MAB, Q
±] = −1
2
γABQ
± , [PA, Q±] = ±xγAQ± ,
{Q±
α , Q
±
β } = 4[γAC−1]αβPA ± 4x[γABC−1]αβMAB ,
(2.18)
Here PA,MAB and Q±
α are the generators of space-time translations, Lorentz
rotations and supersymmetry transformations, respectively. The bosonic gen-
erators PA and MAB are anti-hermitian while the fermionic generators Q±
α are
hermitian. The parameter x = 1/(2R) is a contraction parameter.
If we consider the AdS superalgebra (2.18) it is easy to show that this algebra
corresponds to the N = (1, 1) representation by redefining
MC = ǫCABM
AB , J± = PA ± xMA . (2.19)
This gives the (anti-)commutation relations
[J+, Q
+] = 2xγAQ+ , [J−, Q
−] = −2xγAQ− ,
{
Q±
α , Q
±
β
}
= 4[γAC−1]αβJ± ,
(2.20)
where the N = (1, 1) character is manifest.
Together with the bosonic commutators in Table 2.5 the algebras of the kine-
matical groups that we study in this thesis are given in Tables 2.7 and 2.8 for
24 Preliminaries
N = (2, 0) [Ka, Q
+] [Ka, Q
−] [H,Q+] [H,Q−] [Pa, Q+] [Pa, Q−] [Z,Q+] [Z,Q−]
AdS −γa0
2 Q− −γa0
2 Q+ − γ0
2RQ
+ 3γ0
2RQ
− γa
2RQ
− γa
2RQ
+ −3γ0
4RQ
+ − γ0
4RQ
−
P −γab
2 Q
− −γab
2 Q
+ 0 0 0 0 0 0
NH −γa0
2 Q− 0 − γ0
2RQ
+ 3γ0
2RQ
− γa
2RQ
− 0 0 0
AC 0 0 0 0 γa
2RQ
− γa
2RQ
+ 0 0
G −γa0
2 Q− 0 0 0 0 0 0 0
C 0 0 0 0 0 0 0 0
Table 2.7
This table gives an overview of the commutators of the N = (2, 0) superalgebras that involve
the supersymmetry generators Q+ and Q−
N = (2, 0) {Q+
α , Q
+
β } {Q−
α , Q
−
β } {Q+
α , Q
−
β }
AdS
[γ0C−1]αβH 2[γ0C−1]αβZ
[γaC−1]αβPa
+ 1
2R [γabC−1]αβMab + 1
2R [γABC−1]αβMAB
Poincaré [γ0C−1]αβH 2[γ0C−1]αβZ [γaC−1]αβPa
Newton-Hooke
[γ0C−1]αβH 2[γ0C−1]αβZ
[γaC−1]αβPa
+ 1
2R [γabC−1]αβMab + 1
2R [γa0C−1]αβKa
AdS-Carroll [γ0C−1]αβ(H + 2Z) [γ0C−1]αβ(H − 2Z) 1
2R [γa0C−1]αβKa
Galilei [γ0C−1]αβH 2[γ0C−1]αβZ [γaC−1]αβPa
Carroll [γ0C−1]αβ(H + 2Z) [γ0C−1]αβ(H − 2Z) 0
Table 2.8
This table gives the anticommutators of the N = (2, 0) superalgebras that we studied.
the N = (2, 0) case and in Tables 2.9 and 2.10 for the N = (1, 1) case. In order
to compare the (anti-)commutators of all the kinematical groups we rewrote the
generators of the AdS algebras P0, Ma0 and Q1,2 in terms of the generators H,
Ka and Q±.
The Newton–Hooke superalgebra that we will use is obtained by contracting
the N = (2, 0) AdS superalgebra. The reason why we do not use the N = (1, 1)
AdS algebra for the contraction in this case is essentially the same reason as
why we are interested in N = 2 rather than N = 1 algebras. Taking the non-
2.4 Non-Linear Realizations 25
relativistic contraction thereof amounts to taking the simultaneous contractions
of two independent N = 1 algebras. However, we already argued that this
cannot lead to a superalgebra of the desired form. On the other hand, for the
ultra-relativistic limit, the N = (2, 0) and N = (1, 1) AdS Carroll algebras are
not isomorphic. We will see in Chapter 7 that the associated particle actions are
rather different.
N = (1, 1) [Ka, Q
−] [H,Q+] [H,Q−] [Pa, Q+] [Pa, Q−]
AdS 0 γ0
2RQ
+ − γ0
2RQ
− γa
2RQ
+ − γa
2RQ
−
P 0 0 0 0 0
AC 0 0 0 γa
2RQ
+ − γa
2RQ
−
C 0 0 0 0 0
Table 2.9
This table gives the commutators of the N = (1, 1) superalgebras that involve the
supersymmetry generators Q+ and Q−.
N = (1, 1) {Q+
α , Q
+
β } {Q−
α , Q
−
β }
AdS
[γAC−1]αβPA [γAC−1]αβPA
+ 1
2R [γABC−1]αβMAB − 1
2R [γABC−1]αβMAB
Poincaré [γAC−1]αβPA [γAC−1]αβPA
AdS-Carroll
2[γ0C−1]αβH 2[γ0C−1]αβH
+ 4
R [γa0C−1]αβKa − 4
R [γa0C−1]αβKa
Carroll 2[γ0C−1]αβH 2[γ0C−1]αβH
Table 2.10
This table gives the anticommutators of the N = (1, 1) superalgebras of the kinematical
groups that we are considering.
2.4. Non-Linear Realizations
Non-linear realizations have played a very important role in our understand-
ing of symmetries in quantum field theories and in particle physics in general.
26 Preliminaries
Given a quantum field theory with a rigid symmetry group G which is spon-
taneously broken to a subgroup H, in many circumstances, its effective theory
is described by the non-linear realization of G with local subgroup H. Having
deduced the non-linear realization, one has discovered the symmetry G of the
underlying theory and possessing the symmetry one may then use it to try to
understand the underlying dynamics that it has. In this section we review this
procedure and later on we will use it when we discuss the non relativistic and the
Carroll superparticle sections.
In the original approach [61,62] Callan, Coleman, Wess and Zumino follow a
method in terms of coset representatives5. We use a reformulation of the theory
of non-linear realizations in terms of the group elements of G themselves rather
than those of the coset G/H [65, 66]. Let us consider a Lie supergroup G, a
subgroup H, and the coset G/H. We will use the following notation for the Lie
generators:
GA ∈ G , Hi ∈ H , GI ∈ G/H . (2.21)
The generators of the supergroup satisfy the graded commutation relations
[GA, GB} = fCABGC . (2.22)
The bracket structure { , ] signifies either commutator or anticommutator, ac-
cording to the even or odd character of the generators. The structure constants
are graded anti-symmetric
fCAB = −(−1)ABfCBA , where
(−)AB = −1 when A and B refer to odd
generators ,
(−)AB = +1 otherwise .
(2.23)
We will use the conventions corresponding to independent parity for forms and
Grassmann variables. For instance, the rule for a 0-form as GA and for a 1-form
as LB would be
LAGB = (−)ABGBLA , (2.24)
whereas for two 1-forms
LALB = −(−)ABLBLA . (2.25)
We consider group elements g(z) of G and take the symmetries of the non-linearly
realized theory given by
g(z) → g0g(z) and g(z) → g(z)h(z) , (2.26)
5For an early application of this method in a different situation than the one considered
in this work, namely to the construction of worldline actions of conformal and superconformal
particles, see [63, 64].
2.4 Non-Linear Realizations 27
where h ∈ H is a local (that is, spacetime dependent) transformation that belongs
to H and g0 ∈ G is a rigid transformation (that is, does not depend on spacetime)
that belongs to G. We note that the rigid and local transformations may be
carried out completely independently. This defines what the non-linear realization
is and one only has to find the dynamics that is invariant under this symmetry.
This theory contains gauge degrees of freedom that can be fixed using the
local H transformations. This formulation is a non-linear realization of a group
G with local subgroup H. The global G transformations of the coordinates
z’s are determined up to the local H transformations and are in general non-
linear. This general approach can be used for spacetime symmetries as well as
internal symmetries of fields. One can recover the original approach using from
the beginning these local H transformations to set to zero some fields (fixing
the gauge freedom) and so work only with coset representatives, it would also
require the H-compensating transformations for the global g0 transformations.
In this section we will review this more general way of constructing non-linear
realizations and in the non-relativistic and Carroll sections will choose specific
examples of the coset.
Let us now consider the structure equations for the forms LA. The usual
method is to consider the Maurer-Cartan (MC) one form which belongs to the
Lie algebra and so can be written in the form
Ω = g−1dg = GAL
A . (2.27)
This equation defines the set of 1-forms LA and satisfies the MC equation
dΩ = GAdL
A = d(g−1dg) = dg−1dg = −g−1dgg−1dg = −Ω ∧ Ω , (2.28)
and
Ω∧Ω = (−)ABGAGBLA∧LB =
1
2
(−)AB[GA, GB}LA∧LB = −1
2
fCABGCL
B∧LA ,
(2.29)
therefore we get the structure equations for the LA’s
dLC − 1
2
fCABL
B ∧ LA = 0 . (2.30)
When the forms LA are considered as depending on the parameters zA, we have
LA = dzBLAB. In order to evaluate the properties of transformations of the MC
form with respect to an infinitesimal variation of the group element g, it is very
useful to introduce the following quantity
Ωδ = g−1δg . (2.31)
28 Preliminaries
Notice that formally Ωδ → Ω when δg → dg. We may now evaluate the variation
of Ω to get
δΩ = dΩδ − [Ωδ,Ω] , (2.32)
which does not depend on the grading. Let us now consider a generic variation
of the parameters zA and define
Ωδ = GA[δzA] = (−)A[δzA]GA, [δzA] = δzBLAB , (2.33)
we get
δ(GALA) = d(GA[δzA]) − [GB[δzB], GCLC ] . (2.34)
Using the identity
[O1c1, O2c2] = [O1, O2}c2c1 , (2.35)
valid for graded c-numbers ci’s and graded operators Oi’s, i = 1, 2, with deg[ci] =
deg[Oi], we get
[GB[δzB], GCLC ] = fABCGAL
C [δzB] , (2.36)
and therefore
δLA = d([δzA]) − fABCL
C [δzB] . (2.37)
Now, the central problem in the theory of non-linear realizations is to con-
struct an action. Let us consider {Bi} as the set of generators of broken (su-
per)translations (it can also include central charge extensions), {Ti} the unbroken
translation generators, {Hi} the generators of the stability group H and {Ki} the
remaining broken transformations. We now consider the coset element g = g0 U,
where g0 is the coset representing the group (super)space of broken and unbroken
translations and U is in terms of the generators {Ka}. We may write g0 and U
in terms of an exponential parametrization as
g0 = eTiz
i
T
(x)eBiz
i
B
(x) , (2.38)
where the zi are the Goldstone bosons of the symmetry generators Ti and Bi,
and U = eKiz
i
K
(x), where U is parametrized by the Goldstone bosons of the Gi
symmetry transformations.
As a warming up example let us consider the basic commutators of the AdS
algebra given by (A = 0, 1, . . . , D − 1)
[
MAB,MCD
]
= 2ηA[CMD]B − 2ηB[CMD]A ,
[
MAB, PC
]
= −2ηC[APB] ,
[
PA, PB
]
=
1
R2
MAB .
(2.39)
2.4 Non-Linear Realizations 29
Here PA, and MAB, are the generators of spacetime translations, and rotations
respectively and both of them are anti-hermitian.
To make the non-relativistic and the ultrarelativistic contractions in the fol-
lowing sections, we rescale the generators with parameters ω, α and β as follows
(a = 1, . . . , D − 1):
P0 = αH , Ma0 = ωKa , R = βR̃ , (2.40)
and divide the commutators in the following way
[Mab,Mcd] = 2δa[cMd]b − 2δb[cMd]a ,
[
Jab, (P/K)c
]
= −2 δc[a(P/K)b] ,
[
Ka,Kb
]
=
1
ω2
Mab ,
[
Pa, Pb
]
=
1
β2R2
Mab ,
[
Pa,Kb
]
=
α
ω
δabH ,
[
H,Ka
]
=
1
αω
Pa ,
[
H,Pa
]
= − ω
αβ2R2
Ka .
(2.41)
We consider the coset
G
H
=
AdS
SO(D − 1)
, (2.42)
and the coset element g = g0 U, where g0 = eHtePax
a
is the coset representing the
AdS space and U = eKav
a
is a general boost. Here, the complete set of generators
is given by {Gi} = {H,Pa,Ka,Mab}, where
{Ti} = H is the unbroken time translation,
{Bi} = Pa are the broken translation generators,
{Ki} = Ka are the broken AdS boost generators,
{Hi} = Mab
are the generators of the stability group
in this case the spatial rotations.
The xa (a = 1, ..., D − 1) are the Goldstone bosons of broken translations, t
is the Goldstone boson of the unbroken time translation6 and U is parametrized
by the va Goldstone bosons of the broken AdS boost transformations.
The reason to consider the coset element in terms of g0 and U is because in
this way we have that for a general symmetric spacetime g0 is the coset element
representing the ‘empty’ (super-)spacetime, while U represents the broken sym-
metries that are due to the presence of a dynamical object, in our case a particle,
6The unbroken translation P0 generates via a right action [67, 68] a transformation which is
equivalent to the world-line diffeomorphisms.
30 Preliminaries
in the ‘empty’ (super-)spacetime. For the case of a particle U is given by the
general rotation that mixes the ‘longitudinal’ time direction with the ‘transverse’
space directions. If we would like to consider as a dynamical object a p-brane, we
should consider as U the general rotations that mix the longitudinal and tranverse
directions [68].
First we write out the Maurer-Cartan form Ω0 associated to the AdS space
Ω0 = g−1
0 dg0 = He0 + Pae
a +Kaω
a0 +Mabω
ab , (2.43)
where (e0, ea) and (ωa0, ωab) are the space and time components of the Vielbein
and spin connection 1-forms of the AdS space, respectively. If we parametrize the
AdS space as eHtePax
a
, the Vielbein and spin-connection 1-forms corresponding
to the AdS space are given by
e0 = dt cosh
x
βR
,
ea =
βR
x
dxa sinh
x
βR
+
1
x2
xaxbdxb
(
1 − βR
x
sinh
x
βR
)
,
ωa0 = − ω
αβxR
dtxa sinh
x
R
,
ωab =
1
x2
x[bdxa]
(
cosh
x
βR
− 1
)
.
(2.44)
We now insert a particle in the empty AdS space and consider the Maurer-Cartan
form of the combined system:
Ω = g−1dg = U−1Ω0U + U−1dU = GALA . (2.45)
Using these formulae we find that the Maurer-Cartan form Ω is given by
LH = e0 cosh
v
ω
+
α
v
sinh
v
ω
vae
a ,
LP
a = ea +
α
v
sinh
v
ω
e0va +
βR
v2
(
cosh
v
ω
− 1
)
vavbeb ,
LK
a =
ω
v
sinh
v
ω
dva +
1
v2
(
1 − ω
v
sinh
v
ω
)
vavbdvb + ωa0 cosh
v
ω
+
1
v2
(
1 − cosh
v
ω
)
ωb0vbv
a +
2ω
v
sinh
v
ω
vb ω
ab ,
LM
ab = ωab +
1
ωv
sinh
v
ω
v[bωa]0 +
1
v2
(
cosh
v
ω
− 1
)
v[bdva]
− 2
v2
(
cosh
v
ω
− 1
)
vcv
[bωa]c .
(2.46)
2.4 Non-Linear Realizations 31
Finally, the construction of the most general action with the lowest number
of derivatives is obtained by taking the pull-back of all the L’s invariant under
the subgroup H of the representations that arise in the Cartan form. The result
is unique up to a few possible constants.
2.4.1. Limits vs Non Linear Realizations
There are two ways of obtaining the (super)particle action starting from the
AdS algebra (see Table 2.11).
AdS
Algebra
AdS
Action
AC
Algebra
NH
Algebra
AC
Action
NH
Action
c→0
c→∞
NLR
c→0
c→∞
NLR
Table 2.11
Contruction of the AC and the NH action.
The first method consist in using the non-linear realizations method (NLR)
over the AdS algebra (2.41) written in terms of the contraction parameters and
taking the AC or the NH limits from the Maurer-Cartan one-forms (2.46) as
follows
Newton–Hooke limit: define the contraction parameters as
α = 1
ω , β = ω , ω → ∞ . (2.47)
Carroll limit: in this case, the parametes are given by
α =
ω
2
, β = 1 , ω → ∞ . (2.48)
After taking these limits the results are the same as those given in sections 6 and
7 respectively. Even though this is a general procedure from where we can obtain
the results of both limits, a drawback arises when going to the supersymmetric
case where the calculations become tedious.
32 Preliminaries
The second way, explained in detail in sections 6 and 7 for the non-relativistic
and Carroll limits respectively, consists in taking the NH and AC limits over the
AdS Algebra, leading to the corresponding non-relativistic and ultra-relativistic
algebras, then taking the non-linear realization in each limit to obtain the action
and transformation rules.
2.5. Group averaging method
The group averaging method (see for example [69–71]) is a formalism to derive
the physical Hilbert space for a system with constraints.
Let us assume that there is only an unique constraint Ĝ. For a given arbi-
trary state |ψkin〉 in the kinematical space Hkin (where all the constraints are
well defined) we can obtain an averaged state |ψphys〉 (solution of the constraint
equation, i.e. Ĝ|ψphys〉 = 0) by
|ψphys〉 =
∫ ∞
−∞
dλ e
i
~
λ Ĝ |ψkin〉 , (2.49)
where we introduced an ~ factor for convenience. The Kernel for this transfor-
mation is given by
G(xf , tf ; xi, ti) :=
∫ ∞
−∞
dλ 〈xf , tf |e i~λ Ĝ |xi, ti〉 . (2.50)
If we decompose the evolution in N intervals of length ǫ = 1/N
e
i
~
λ Ĝ =
N
∏
n=1
e
i
~
ǫλ Ĝn , (2.51)
and inserting a set of a complete base of the form 1 =
∫
dtn dxn|xn, tn〉 〈xn, tn|,
we can rewrite the Kernel as
G(xf , tf ; xi, ti) =
∫ ∞
−∞
dλ
N−1
∏
n=1
[∫ ∞
−∞
dtn
∫ ∞
−∞
dxn
] N
∏
n=1
〈xn, tn |e i~ ǫ λ Ĝn |xn−1, tn−1〉 ,
(2.52)
where the final and initial state are 〈xn, tn |≡ 〈xf , tf | and |x0, t0 〉 ≡ |xi, ti 〉
respectively.
On the other hand, we can write the physical state as
ψphys(x, t) = 〈x, t|ψphys〉 =
∫
dx′
∫
dt′G(x, t; x′, t′)ψkin(x′, t′) , (2.53)
2.5 Group averaging method 33
which means that we can use the Kernel G(xf , tf ; xi, ti) to extract a physical
state from a kinematical state.
Now, if |φkin〉 and |ψkin〉 are kinematical states mapped to physical states
(|φphys〉 and |ψphys〉 respectively) according with the definition (2.49) the physical
internal product is defined as
〈φphys|ψphys〉 :=〈φkin|
∫
dλ e
i
~
λ Ĝ |ψkin〉
=
∫
dx dx′ dt dt′ φkin(x, t)G(x, t; x′, t′)ψkin(x′, t′) ,
(2.54)
and all the information of the quantum dynamics is codified in the extraction
amplitude G(xf , tf ; xi, ti), which is the transition amplitude from xi at the time
ti to xf at the time tf .
To exemplify the procedure, in the next section we will show how to use the
averaging method to calculate the Kernel for the relativistic particle to obtain
the usual result. This will be useful for the polymer path integral which we will
discuss in chapter 5.
2.5.1. The relativistic free particle
The action of a relativistic free particle of mass m is
S =
1
2
c
∫ 1
0
dτ
(
1
λ
ẋµẋ
µ −m2λ
)
, (2.55)
and can be written in an equivalent way as
S =
∫ 1
0
dτ
[
pµẋ
µ +
1
2
λ
(
1
c
pµp
µ +m2c
)]
, (2.56)
where xµ(τ) are the coordinates of the particle in a D dimensional Minkowski
space in function of the arbitrary time parameter τ . The parameter λ acts as the
Lagrange multiplier for the constraint
Ĝ =
1
2
(
1
c
p̂µp̂
µ +m2c
)
. (2.57)
Using that
〈xn|e− i
~
ǫλĜ |xn−1〉 =
=
(
1
2π~
)D ∫ ∞
−∞
dDpn exp
{
i
~
[
−ǫλ1
2
(
1
c
pµnp
µ
n
+m2c
)
+ pµn
(
xµ
n
− xµ
n−1
)
]}
.
(2.58)
34 Preliminaries
We can use the group averaging method to calculate the Kernel (2.50) where the
physical internal product is
〈xf |e− i
~
λĜ |xi〉 =
N−1
∏
n=1
[∫ ∞
−∞
dDxn
] N
∏
n=1
〈xn|e− i
~
ǫλĜ |xn−1〉
=
(
1
2π~
)D N
∏
n=1
[∫ ∞
−∞
dDpn
]N−1
∏
n=1


D−1
∏
µ=0
[
δ
(
pµn − pµn+1
)]


exp
{
i
~
[
pµN
(
xµ
N
− xµ
0
)
−
N
∑
n=1
ǫλ
1
2
(
1
c
pµnp
µ
n
+m2c
)
]}
=
(
1
2π~
)D ∫ ∞
−∞
dDpN exp
{
− i
~
[
ǫNλ
1
2
(
1
c
pµNp
µ
N
+m2c
)
− pµN
(
xµ
N
− xµ
0
)
]}
.
(2.59)
The term pµn
(
xµ
n
− xµ
n−1
)
inside the exponential (2.58) combined with the inte-
gral over xn in the product (2.59) would lead to a Dirac delta over the momentum
pµn for each one of the D coordinates. We will only have N − 1 of these Dirac
deltas but N integrals over pn so an integration over the pN momentum would
remain.
Integrating over λ we get the Kernel
G(xf ;xi) =
∫
dλ
(
1
2π~
)D ∫ ∞
−∞
dDpN
exp
{
− i
~
[
ǫNλ
1
2
(
1
c
pµNp
µ
N
+m2c
)
− pµN
(
xµ
N
− xµ
0
)
]}
.
(2.60)
We can simplify this expression by making the integration over the momentum
or over the Lagrange multiplier. In the first case, integrating over pN we obtain
G(xf ;xi) =
(
c
2πi~λ
)
D
2
∫
dλ exp
{
ic
2~
(
XµX
µ
λ
− λm2
)}
, (2.61)
where we used Xµ = xµN − xµ0 and ǫN = 1. This is the result obtained in [72].
2.6 Killing Equations 35
In the second case, integrating over the Lagrange multiplier, we get
G(xf ;xi) =
(
1
2π~
)D ∫ ∞
−∞
dDpN
(
4π~c
τ
)
δ(pµNp
µ
N
+m2c2)e
i
~
pµN (xµ
N
−xµ0 )
=
(
1
2π~
)D−1 (2c
τ
)∫ ∞
−∞
dD−1pN
e
i
~
~pN (~xN−~x0)
2
√
~p 2 +m2c2
[
e
i
~
(x0
N
−x0
0)
√
~p 2+m2c2
+ e− i
~
(x0
N
−x0
0)
√
~p 2+m2c2
]
.
(2.62)
In the last step we integrated over p0 using the Dirac Delta and we used the metric
(−,+,++. . .). This propagator coincides with the usual result (for example given
in [73]).
2.6. Killing Equations
Killing vectors ξµ are vector fields that generate a symmetry of the metric
gµν (the symmetries of a metric are also called isometries of spacetime). Given
a metric, Killing vectors should satisfy the Killing equations. Solutions, if they
exist, represent symmetry transformations of the spacetime. Killing vectors form
a Lie algebra and this resulting algebra of these Killing vectors is the Lie algebra
of the isometry group of the metric.
The Killing equations are applied to two kinds of problems:
1. Determining the metric of a spacetime whose symmetries are assumed.
2. Finding the symmetries of a spacetime.
In this section we focus on the second problem. Given an action we show
the Lagrangian and Hamiltonian procedure to find the Killing equations, whose
solutions will give the symmetries of the system. Both formalisms coincide. As
warm up examples we apply these formalisms to the massive relativistic particle
in the bosonic and in the supersymmetric cases.
2.6.1. Massive Relativistic Particle
The first example that we consider is the bosonic free relativistic particle of
mass M , here µ = 0, 1, . . . D − 1.
36 Preliminaries
Lagrangian Formalism
We consider the action of the free massive relativistic particle
S = M
∫
dτ
√
−ẋµẋµ . (2.63)
Let us consider that the transformation of the coordinates is given by
δxµ = ξµ(x) , (2.64)
the invariance of the variation of the action under the symmetry transformations
δS = 0 lead to the following Killing equations
∂µξν + ∂νξµ = 0 . (2.65)
The solutions of (2.65) are simply ξµ(x) = aµ+λµνxν , where aµ and λµν are con-
stant parameters. These are precisely the infinitesimal translations and Lorentz
transformations of the Poincaré algebra.
Hamiltonian Formalism
The canonical action of the free massive relativistic particle given by
S =
∫
dτ(pµẋµ − e
2
(p2 +M2)) . (2.66)
The basic Poisson brackets of the canonical variables occurring in this action are
{xµ, pν} = δµν , {e, πe} = 1 , (2.67)
Dirac’s Hamiltonian will be
HD =
e
2
(p2 +M2) + λπe . (2.68)
The equations of motion are
ẋµ = epµ , ṗµ = 0 , ė = λ , π̇e = −1
2
(p2 +M2) . (2.69)
We will take as the generator of canonical transformations
G = pµ ξ
µ + γπe , (2.70)
2.6 Killing Equations 37
with parameters ξµ = ξµ(x) and γ = γ(x). Here λ = λ(τ) is an arbitrary
function and πe is constrained π̇e = 0. This leads to the following restrictions on
the parameters:
Ġ = pµẋ
ν∂νξ
µ − 1
2
γ(p2 +M2) = epµp
ν∂νξ
µ − 1
2
γ(p2 +M2) , (2.71)
and the Killing equations that we obtain are
γ = 0 , ∂µξν + ∂νξµ = 0 . (2.72)
As we see, both formalisms lead to the same Killing equations (and thus the same
symmetries) (2.65) and (2.72).
2.6.2. Massive Relativistic Superparticle
It is time to extend the analysis of symmetries to the supersymmetric case.
The superparticle theory has been studied because of its simplicity and its anal-
ogous properties with superstring theory, it has also been considered that the
classical superparticle describes the dynamics of the classical spinning particle or
spin-1/2 particle [74, 75].
Lagrangian Formalism
Now let us consider the supersymmetric case. The action of the free massive
relativistic superparticle is given by
S = M
∫
dτ
√
−(ẋµ + iθ̄γµθ̇)2 . (2.73)
Let us consider that the transformation of the coordinates is given by
δxµ = ξµ(x, θ) , δθ = χ(x, θ) . (2.74)
In order to make the action invariant under the transformation rules, δS = 0, we
find the following conditions
∂µξν + ∂νξµ + iθ̄γµ∂νχ+ iθ̄γν∂µχ = 0 ,
i∂θξµθ̇θ̄γ
µθ̇ − χ̄γµθ̇θ̄γ
µθ̇ − θ̄γµθ̇θ̄γµ∂θχθ̇ = 0 ,
∂θξµ + iχ̄γµ + iθ̄γµ∂θχ+ i∂µξν θ̄γ
ν − θ̄γν∂µχθ̄γ
ν = 0 .
(2.75)
Making the redefinition
hµ = ξµ + iθ̄γµχ , (2.76)
38 Preliminaries
we can rewrite the Killing equations in a more compact form as
∂µhν + ∂νhµ = 0 ,
∂θhµθ̇θ̄γ
µθ̇ = 0 .
(2.77)
The third equation ∂θhµ + i∂µhν θ̄γ
ν = 0 , is a direct consequence of the other
two. The solutions of (2.77) are the transformations of Poincaré symmetry and
supersymmetry:
ξµ(x, θ) = aµ + λµνxν + ǭγµθ , χ =
1
4
λµνγµνθ + iǫ . (2.78)
Hamiltonian Formalism
The canonical action of the free massive relativistic superparticle will be
S =
∫
dτ(pµẋµ + Pθθ̇ − e
2
(p2 +M2) − (P̄θ + ipµθ̄γ
µ)ρ) (2.79)
The basic Poisson brackets of the canonical variables occurring in this action are
{xµ, pν} = δµν , {e, πe} = 1 , {Pαθ , θβ} = −δαβ , {Πα
ρ , ρβ} = −δαβ .
(2.80)
and the corresponding Dirac’s Hamiltonian of this action is given by
HD =
e
2
(p2 +M2) + (P̄θ + ipµθ̄γ
µ)ρ+ λπe + π̄ρΛ . (2.81)
As it occurs in the main text, πe and Πρ are the primary constraints and λ = λ(τ)
and Λ = Λ(τ) are arbitrary functions. The primary hamiltonian equations of
motion are given by
ẋµ = epµ + iθ̄γµρ , ṗµ = 0 , ¯̇Pθ = −ipµρ̄γµ , θ̇ = ρ ,
ė = λ , π̇e = −1
2
(p2 +M2) , ρ̇ = λ , ¯̇Πρ = P̄θ + iPµθ̄γ
µ .
(2.82)
The stability of primary constraints give secondary bosonic and fermionic con-
straints, given by p2 + M2 = 0 and P̄θ + iPµθ̄γ
µ = 0 respectively. By requiring
the stability of these secondary constraints one gets ρ = 0, leading to the same
equations of motion as in the Lagrangian formalism.
We will take as the generator of canonical transformations
G = pµ ξ
µ(x, θ) + P̄θχ(x, θ) + γ(x, θ)πe + Π̄ρ Γ(x, θ) , (2.83)
2.6 Killing Equations 39
with parameters ξµ = ξµ(x, θ), χ = χ(x, θ), γ = γ(x, θ) and Γ = Γ(x, θ) that
have the following restrictions on the parameters:
Ġ = pµ(ẋν∂νξµ + ∂θξ
µθ̇) − 1
2
γ(p2 +M2) − ipµρ̄γ
µχ+ Pθ(∂µχẋµ + ∂θχθ̇)
+ (P̄θ + ipµθ̄γ)Γ ,
= epµp
ν∂νξ
µ + ipµ∂νξ
µθ̄γµρ+ pµ∂θξ
µρ− 1
2
γ(p2 +M2)
− ipµρ̄γ
µχ+ epµPθ∂µχ+ iθ̄γµρP̄θ∂µχ+ P̄θ∂θχρ+ P̄θΓ + ipµθ̄γ
µΓ
(2.84)
and the Killing equations are given by
γ = 0 , ∂µχ = 0 , Γ = ∂θχρ ,
∂µξν + ∂νξµ = 0 ,
∂θξ
µ + +iχ̄γµ + i∂νξ
µθ̄γν + iθ̄γµ∂θχ = 0 .
(2.85)
By making the redefinition (2.76) we obtain again the Killing equations (2.77).
3
Supersymmetric Massive Extension
3.1. Introduction
The first modification from general relativity that we will consider is to search
for a theory with a spin-2 massive particle. Consistent theories for both massless
and massive particles with spin less than two are well known and their quan-
tum versions play an important role in theoretical particle physics. In fact, the
Standard Model is a consistent quantum theory of massive and massless fields
with spin 0, 1/2 and 1, but compared with the lower spin cases, massive spin-2
particles turn out to be more difficult to handle.
One motivation to search for a theory of massive gravity is related with the
cosmological constant problem. It is possible to introduce the cosmological con-
stant as a free parameter in the classical action of a gravitational field as
S =
1
2κ
∫
d4x
√−g(R− 2Λ) + Smatter[gµν , X] (3.1)
where κ ≡ 8πG
c4 ≡ 8π
m2
Pl
≡ 1
M2
Pl
with mPl and MPl being Planck’s mass and the
reduced Planck’s mass respectively; the first term stands for the Einstein-Hilbert
action, the second term is the cosmological constant term (which can be assigned
units of 1 over distance squared) and the third term describes the matter part
where X represents a generic matter field. The variation of equation (3.1) will
lead to the Einstein’s field equations:
Rµν − 1
2
gµνR+ gµνΛ = κTµν (3.2)
here, Tµν is the energy-momentum tensor, this term gathers the ordinary matter
and radiation contributions to the total energy (and momentum) of the universe.
41
42 Supersymmetric Massive Extension
A much less obvious source of energy on the structure of the universe is the
vacuum energy density and is proportional to the cosmological constant.
A nonzero vacuum energy density would have a visible effect on the geometry
of space-time. A large negative(positive) cosmological constant would produce
a space with negative(positive) constant curvature so our Euclidean geometry
would not be valid any longer. If the cosmological constant is nonzero but quite
small we would have to look over large distances to see its effects on space-time
structure. It would also affect the expansion rate of the universe, a negative
cosmological constant would slow the expansion of the galaxies at a constant rate
and a positive cosmological constant would accelerate galaxies away from each
other and increase the expansion rate of the universe.
Now, theoretically, the vacuum energy density is naively expected to be of the
order Λ ∼ km−2 and the acceleration rate of the universe H is computed to be
H ≥ 10−3eV but these values will cause tremendous distortions in the space-time
structure. According to observational data the value of Λ if nonzero has to be of
the order of Λ ∼ 10−52m−2 or similarly the expansion rate has to be less than or
equal to the present-day value of the Hubble parameter H ≤ 10−33eV.
Since the discrepancy between theory and experiment is so big we may be
missing an essential part of the puzzle. Some authors argue that giving mass to
the graviton may solve this unnaturalness problem because long range interac-
tions would be damped exponentially and this would narrow the gap between the
expected and the observed value of the cosmological constant.
There are also purely theoretical motivations of working with a massive theory
of gravity. The main theoretical aim is to better understand our current theory,
to test it and to try to get consistent modifications and to obtain new insights
and points of view.
The idea of adding a mass to a spin-1 and a spin-2 particle is old. Between
1936 and 1941 Proca developed the theory of the massive vector boson fields
governing the weak interaction and the motion of the spin-1 mesons. In 1939 the
free theory for massive spin-2 particles was constructed by Fierz and Pauli [9]. A
massless spin-1 particle enjoys a U(1) symmetry, and in the same way a massless
spin-2 particle has a diffeomorphism symmetry, however a main difference arises
between both cases when coupling with external matter fields. In the spin-2
theory this coupling makes general relativity a fully non-linear theory with non-
linear diffeomorphism invariance, this feature is inherited in the massive theory
(although the symmetry is broken) making it very difficult to obtain.
There are some concerns when dealing with a massive graviton, one of them
is that when taking the massless limit of a massive spin-2 field, it propagates
too many degrees of freedom, and it would seem that we can never get general
3.2 Constructing Massive Actions 43
relativity from it. This phenomena is called the vDVZ discontinuity and it can
be solved by considering non-linear extensions of the theory. A second problem
to overcome is that non-linearities introduce ghosts into the physical spectrum.
Ghosts correspond to fields with negative kinetic energy that lead to instabilities
at the classical level and to non-unitarity at the quantum level.
In the past decade a lot of effort has been put into constructing models that
can cure the vDVZ discontinuity and that are ghost free, for a review see for
example [76]. Here we will concentrate on the three-dimensional theory of massive
gravity called New Massive Gravity. New Massive Gravity (NMG) is a higher-
derivative extension of three-dimensional (3D) Einstein–Hilbert gravity with a
particular set of terms quadratic in the 3D Ricci tensor and Ricci scalar [14]. The
interest in the NMG model lies in the fact that, although the theory contains
higher derivatives, it nevertheless describes, unitarily, two massive degrees of
freedom of helicity +2 and −2. Furthermore, it has been shown that even at
the non-linear level ghosts are absent [19]. The 3D NMG model is an interesting
laboratory to study the validity of the AdS/CFT correspondence in the presence
of higher derivatives. Its extension to 4D remains an open issue and has only
been established so far at the linearized level [77].
A supersymmetric version of NMG was constructed in [20]. Besides the
fourth-order-derivative terms of the metric tensor this model also contains third-
order-derivative terms involving the gravitino. The purpose of this section is
to show how to construct a reformulation of the supersymmetric NMG model
(SNMG) without higher derivatives.
This chapter is organized as follows, first we will show how to construct the
massive actions, then we will give a general review of New Massive Gravity, after
that we will use the spin-1 case as a warm up exercise to study the spin-2 case to
explicitly construct the linearized, massive, off-shell supermultiplet performing a
Kaluza-Klein reduction over a circle and projecting onto the first massive Kaluza
Klein sector. The final form of the 3-dimensional off-shell massive supermultiplet
is then obtained after a truncation and gauge-fixing a few Stückelberg symmetries.
We will look in detail at the massless limit along the construction. Then we will
construct the linearized version of SNMG.
3.2. Constructing Massive Actions
3.2.1. Proca Action
Spin-1 theories share several features with the spin-2 analogues (for example,
unlike the spin-0 or spin-1/2 cases gauge invariance is needed in the massless case
44 Supersymmetric Massive Extension
to obtain the correct number of degrees of freedom), that is why it is instructive
to begin with the simpler spin-1 case.
First we want to construct a relativistic Lagrangian for a massive spin-1 field.
In particle physics, particles with spin-1 are the mediators of interactions and
they are described in field theory as quanta of vector fields. The most impor-
tant examples are the gauge fields of the electromagnetic and strong interactions
(massless photons and gluons respectively) and the ones of weak interactions
(massive W± and Z0 bosons). Thus, we start with a vector field Aµ.
The vector field Aµ has D degrees of freedom which is more than is required
to describe a spin-1 particle (see Table 3.1). Here a distinction has to be made
that distinguishes massless and massive particles. In the massless off-shell case
we can use the gauge transformation to fix one component of Aµ therefore we
will have D − 1 degrees of freedom; in the on-shell case the Lorentz subsidiary
condition (∂µAµ = 0) imposes a constraint on the polarization vectors reducing
again the degrees of freedom by one, so we end with D−2 degrees of freedom. For
example, in electrodynamics, the photon has only two polarization states, both of
which are transverse. For the massive off-shell case the gauge invariance is lost,
because the field Aµ transforms inhomogeneously and thus the mass term in the
Lagrangian is not invariant, so only after imposing the Lorentz condition there
will be left D−1 degrees of freedom. That is why a massive vector field allows an
additional longitudinal polarization state in addition to the two transverse ones.
Aµ Off-Shell 4D 3D On-Shell 4D 3D
Massless D − 1 3 2 D − 2 2 1
Massive D 4 3 D − 1 3 2
Table 3.1
This table gives the counting of degrees of freedom for a vector field Aµ that describes a spin-1
particle.
Here we will consider the massive classical spin-1 field also known as the Proca
field (we will assume that Aµ is a real-valued neutral field, for a charged field Aµ
is complex). That the vector field carries mass m means it satisfies the field
equation
(✷ −m2)Aµ = 0 , (3.3)
that has D − 1 degrees of freedom. The sign of the mass term depends on the
metric choice that we made, in this case we are using the mostly plus signature
3.2 Constructing Massive Actions 45
convention (−,+, . . . ,+). Thus we must impose a constraint to cut down the
number of degrees of freedom by one. The only Lorentz covariant possibility
(linear in Aµ) is
∂µAµ = 0 , (3.4)
Now we make an ansatz for a possible action that contains both (3.3) and (3.4).
The first part of this ansatz will contain the leading Klein–Gordon term, the
second part consist on all possible quadratic terms with the order of derivatives
not higher than the leading term. For this case, the only possibility (for first-order
derivatives) is a (∂µAµ)2 term
SProca =
∫
dDx{1
2
Aµ(✷ −m2)Aµ +
a
2
(∂µAµ)2} , (3.5)
where a is a free parameter that we have to tune in such a way we can obtain
both (3.3) and (3.4).
From the variation δS
δAµ
= 0 we obtain an equation of motion of rank-1
(✷ −m2)Aµ − a∂µ(∂νAν) = 0 , (3.6)
by taking the divergence of the previous equation we obtain an equation of rank-0
(✷ − a✷ −m2)∂µAµ = 0 , (3.7)
setting a = 1 we obtain the divergenceless condition (3.4) and substituting these
into (3.6) the Klein–Gordon equation is also derived.
3.2.2. Fierz-Pauli action
Now let us move on with the graviton case. In this case, particles with spin-2
are described by a second-rank symmetric tensor gµν . Let us count the degrees
of freedom (see Table 3.2), since gµν is a symmetric matrix it has D(D + 1)/2
independent components. In the massless off-shell case, gauge invariance (general
coordinate transformations) fix D components; in the on shell-case the subsidiary
conditions will constrain the polarization tensor reducing the degrees of freedom
by D so we will have at the end D(D − 3)/2 degrees of freedom. These last
constraints are the reason why the polarization states of the massless graviton
are only transverse. For the massive scenario the gauge invariance is broken and
only on-shell we can reduce the degrees of freedom due to the D + 1 constraints
that we obtain from the divergenceless and traceless conditions.
46 Supersymmetric Massive Extension
gµν Off-Shell 4D 3D On-Shell 4D 3D
Massless D(D − 1)/2 6 3 D(D − 3)/2 2 0
Massive D(D + 1)/2 10 6 D(D − 1)/2 − 1 5 2
Table 3.2
This table gives the counting of degrees of freedom for a tensor field gµν that describes a
spin-2 particle.
The next step is to build an action for the massive graviton. We start from
the field equation, the divergenceless equation and the tracelessness equation:
(✷ −m2)hµν = 0 , ∂µhµν = 0 , ηµνhµν = 0 , (3.8)
we need to make an ansatz for a possible action from where we can derive (3.8),
that contains the Klein–Gordon term as the leading term and all quadratic terms
with no more than second-order derivatives
SFP =
∫
dDx{1
2h
µν(✷ −m2)hµν + 1
2h(a✷ − bm2)h+ 1
2ch
µν∂µ∂
ρhρν
+ dhµν∂µ∂νh} ,
(3.9)
where hµν is a symmetric tensor, h ≡ ηµνhµν , and a, b, c and d are parameters
to be tuned.
We obtain an equation of motion of rank-2 from δS
δhµν
= 0
(✷−m2)hµν+ηµν(a✷−bm2)h+c∂(µ∂
ρhν)ρ+d∂µ∂νh+dηµν∂ρ∂σhρσ = 0 , (3.10)
From the divergence of (3.10), i.e. ∂µ( δS
δhµν
) = 0, we get a rank-1 equation
(✷ −m2)∂µhµν + (a✷ − bm2)∂νh+ 1
2c✷∂
ρhνρ + 1
2c∂ν∂
µ∂ρhµρ
+ d✷∂νh+ d∂ν∂
ρ∂σhρσ = 0 ,
(3.11)
from the double divergence ∂ν∂µ( δS
δhµν
) = 0 the rank-0 equation
(✷ −m2)∂µ∂νhµν + (a✷2 − bm2
✷)h+ 1
2c✷∂
µ∂νhµν + 1
2c✷∂
µ∂νhµν
+ d✷2h+ d✷∂µ∂νhµν = 0 ,
(3.12)
3.2 Constructing Massive Actions 47
and from the traceless equation ηµν( δS
δhµν
) = 0, another rank-0 equation
(✷ −m2)h+D(a✷ − bm2)h+ c∂µ∂νhµν + d✷h+ dD∂µ∂νhµν = 0 , (3.13)
It is possible to tune the parameters
a = −1 , b = −1 , c = −2 and d = 1 , (3.14)
where the choice of d should satisfy −2 + dD 6= 0 (here we only consider cases
D ≥ 3). Thus the action becomes
SFP =
∫
dDx{1
2
hµν(✷ −m2)hµν − 1
2
h(✷ −m2)h− hµν∂µ∂
ρhρν + hµν∂µ∂νh} ,
(3.15)
There is another way of understanding the tuning, let us focus on the mass
terms and let us change the coefficients by −1
2m
2(hµνhµν + (1 − α)h2), a proper
Lagrangian should be ghost and tachyon free (free fields should have positive
energy and should not propagate faster than the speed of light) but this change
will give an action describing an additional scalar ghost satisfying the equation
(
✷ − 1 + (α− 1)D
(2 −D)α
m2
)
h = 0 , (3.16)
for α 6= 0. Taking the limit α → 0 the mass of the scalar ghost goes to infinity so
it becomes non-dynamical. As a last step we rearrange the terms in (3.15) using
the linearized Einstein tensor
Glin
µν(h) = ✷hµν − 2∂(µ∂
ρhν)ρ + 2∂µ∂νh− ηµν✷h , (3.17)
so the action reads
SFP =
1
2
∫
dDx{hµνGlin
µν(h) −m2(hµνhµν − h2)} . (3.18)
The kinetic term of the Fierz-Pauli action (3.18) comes from the Einstein term
and is invariant under diffeomorphisms δhµν = ∂µζν + ∂νζµ but the mass term
in the Lagrangian breaks the gauge invariance. Stückelberg’s trick consists in
introducing additional gauge fields to restore the gauge symmetry which had
been broken by the mass term.
Now, it is natural to think that due to the precision of the predictions of
general relativity, massive gravity theories should be deformations of it. How-
ever, when we take the limit of the mass going to zero in the Fierz-Pauli theory
48 Supersymmetric Massive Extension
not all of the experimental outcomes of the theory approach to those of gen-
eral relativity, in particular, the light-bending angle obtained from the massive
theory has a disagreement of 25 percent with respect to the one predicted by
general relativity. Besides, taking the massless limit of the Fierz-Pauli system
coupled to a conserved energy-momentum tensor does not lead to massless grav-
ity, but rather to linearized Einstein gravity plus extra degrees of freedom. This
discontinuity in the physical predictions of general relativity and the massless
limit of Fierz-Pauli theory is known as the van Dam-Veltman-Zakharov (vDVZ)
discontinuity. This vDVZ discontinuity can be verified introducing the Stück-
elberg fields mentioned above and coupling the massive field hµν to matter via
a conserved energy-momentum tensor Tµν , in the massless limit one observes a
non-vanishing coupling of the trace h to the trace of Tµν , therefore, the FP theory
gives linearized general relativity plus a scalar mode that does not decouple.
Consider the FP Lagrangian (3.18) of a massive graviton coupled to the
energy-momentum tensor Tµν
L = LFP + hµνT
µν =
1
2
{hµνGlin
µν(h) −m2(hµνhµν − h2)} + hµνT
µν , (3.19)
now we introduce Stückelberg fields to preserve the gauge symmetry
hµν → hµν + ∂µVν + ∂νVµ , Vµ → Vµ + ∂µφ . (3.20)
Inserting (3.20) into (3.19), together with the requirement that the energy-mo-
mentum tensor is conserved (∂µTµν = 0), considering the scalings Vµ → 1
mVµ,
φ → 1
mφ and taking the limit m → 0 we obtain the action
LFP =
1
2
hµνGlin
µν(h) − 1
2
FµνF
µν − 2(hµν∂µ∂νφ− h∂2φ) + hµνT
µν (3.21)
Here we see that the vector Vµ decouples, but we get off-diagonal terms mixing
the scalar φ and the tensor hµν . This action, however, can be diagonalized by
shifting hµν → hµν + 2
D−2ηµνφ, to obtain
LFP =
1
2
hµνGlin
µν(h) − 1
2
FµνF
µν − 2
D − 1
D − 2
∂µφ∂
µφ+hµνT
µν +
2
D − 2
φT, , (3.22)
where the gauge transformations are simply δhµν = ∂µζν + ∂νζµ and δVµ = ∂µΛ.
Here we see explicitly the coupling between the scalar and the trace of the energy-
momentum tensor. This is the origin of the vDVZ discontinuity. We will illustrate
how a supersymmetric version of this discontinuity arises in section 3.5.2 (see
also [78] for an earlier discussion).
3.3 New Massive Gravity 49
Lets do the counting of degrees of freedom, we have D(D − 3)/2 from the
canonical massless graviton, D − 2 from the canonical massless vector, and one
from the canonical massless scalar, giving in total the D(D− 1)/2 − 1 degrees of
freedom from the massive graviton.
After the discovery of the vDVZ discontinuity, Vainshtein realized that the
discontinuity can be cured in a fully nonlinear Fierz-Pauli theory, but shortly
afterwards Boulware and Deser claimed that any nonlinear completion of Fierz-
Pauli theory leads to a ghost [79].
Here is where it becomes important the construction of a three dimensional
theory of massive gravity. For a massless graviton in three dimensions there are
no propagating degrees of freedom (see Table 3.1), so any potentially ghost-like
feature connected to them would be harmless, since it will not constitute any
physical degree of freedom. On the other hand, for a massive graviton in three
dimensions there are two degrees of freedom, therefore, it is possible to write
down an action with "healthy" massive spin-2 mode and a massless spin-2 ghost
mode which is pure gauge, such a theory is called New Massive Gravity (NMG)
and we will briefly discuss about it in the next section.
3.3. New Massive Gravity
It is possible to solve the differential constraint of Fierz Pauli action (the
divergenceless equation, see (3.8)) by increasing the number of derivatives, to do
that we rewrite hµν in terms of a new symmetric spin-two field h′
µν
hµν = G(lin)
µν (h′) , (3.23)
so the remaining equation of motion and tracelessness equation (3.8) can be
written as
(✷ −m2)G(lin)
µν (h′) = 0 , R(lin)(h′) = 0 , (3.24)
where G(lin)(h′) is the trace of the linearized Einstein tensor which in three di-
mensions is proportional to the linearized Ricci scalar, therefore we can write
together both equations of motion in a single equation of the form
G(lin)
µν − 1
m2
[
✷G(lin)
µν − 1
4
(∂µ∂ν − ηµν✷)R(lin)
]
= 0 . (3.25)
An action that reproduces equation (3.25) at the linear level is
SNMG =
1
κ2
∫
d3x
√−g
(
R+
1
m2
K
)
, K = RµνR
µν − 3
8
R2 , (3.26)
50 Supersymmetric Massive Extension
where Rµν is the Ricci tensor, R its trace and κ has mass dimension [κ] = −1/2
in fundamental units, κ will be the 3D analog of the square root of Newton’s
constant, while m is a relative mass parameter. Note that the Einstein-Hilbert
term has a "wrong" sign but this term together with the curvature-squared in-
variant constructed from K avoid ghosts in the theory. This is the action of New
Massive Gravity (NMG), the spectrum in three dimensions consists of a prop-
agating massive spin-two mode and a non-propagating massless spin-two mode.
The tuning −3
8 of the second term of K suppress a massive spin-0 mode.
In short, at the linearized level the NMG model decomposes into the sum
of a massless spin-2 Einstein–Hilbert theory and a massive graviton with two
propagating degrees of freedom (spin-2 modes of helicity +2 and -2) [14].
For many purposes, it is convenient to work with a formulation of the model
without higher derivatives, see, e.g. [21]. This can be achieved by introducing
an auxiliary symmetric tensor that couples to (the Einstein tensor of) the 3D
metric tensor and has an explicit mass term [14]. Using auxiliary fields, we can
make manifest the connection of NMG with the FP theory, at the level of the
Lagrangian. Using a symmetric auxiliary field fµν with trace f = gµνfµν we can
write the action (3.26) as
Saux-NMG =
1
κ2
∫
d3x
√−g[R+ fµνGµν − m2
4
(
fµνfµν − f2)
]
, (3.27)
expanding about a Minkowski background at the linearized level, the Lagrangian
will take the form
Llin-aux-NMG =
(
f̂µν − 1
2
hµν
)
Glin
µν(h) − m2
4
(
f̂µν f̂µν − f̂2) , (3.28)
where f̂µν denotes the perturbation of fµν , rearranging the terms in last equation
and defining h̃µν = hµν − f̂µν we obtain
LFP-NMG = −1
2
h̃µνGlin
µν(h̃) +
1
2
[
f̂µνGlin
µν(f̂) − m2
2
(
f̂µν f̂µν − f̂2)
]
, (3.29)
From here we can clearly identify the non-propagating Einstein mode h̃µν which
is a ghost, and the decoupled, ghost free, massive spin-2 mode f̂µν given by a FP
Lagrangian.
We can consider now a supersymmetric generalization of New Massive Gravity
(SNMG). There are two ways of doing that. It can be done using the metric and
a higher-derivative action like in [20], or one may consider the auxiliary field
version of NMG, this requires that besides an auxiliary symmetric tensor, we
3.3 New Massive Gravity 51
introduce further auxiliary fermionic fields that effectively lower the number of
derivatives of the gravitino kinetic terms. In the supersymmetric case we need
a 3D massless and a 3D massive spin-2 supermultiplet. The massless multiplet
is already known [14] and here we will show how to construct the linearized,
massive, off-shell spin-2 supermultiplet.
For the SNMG we only consider the case of simple N = 1 supersymme-
try. Having constructed the off-shell massive spin-2 supermultiplet, it is rather
straightforward to construct a linearized version of SNMG without higher deriva-
tives, by appropriately combining a massless and a massive spin-2 multiplet.
We will show the procedure explicitly for the easier spin-1 case in section
3.4. In section 3.5 we will show the results for the Fierz–Pauli case. Along the
construction of the multiplets we will look in detail at the massless limit.
As we saw in chapter 2, three dimensional spin-2 supermultiplets contain
gravitini, so for the sake of completeness we show now the counting of degrees
of freedom for a 3/2 field ψµ: ψµ is a complex spinor with 2[D/2]D components
(where 2[D/2] = 2D/2 for D even and 2[D/2] = 2(D−1)/2 for D odd) for spacetime
dimension D, since we can use supersymmetry to fix 2[D/2] this means that we
have (D−1)2[D/2] independent degrees of freedom. These are the off-shell degrees
of freedom.
Now, the action of a free 3/2 field (also called free Rarita-Schwinger field) is
S = −
∫
dDxψ̄µγ
µνρ∂νψρ (3.30)
and the equation of motion obtained from (3.30) reads
γµνρ∂νψρ = 0 , (3.31)
with the gauge condition
γiψi = 0 , (3.32)
where i stands only for the spatial indices. The spatial components ψi satisfy the
Dirac equation
γ · ∂ψi = 0 , (3.33)
however, there is an additional constraint from contracting this last equation with
∂i which give
∂iψi = 0 , (3.34)
which led us to find 3 × 2[D/2] independent constraints. These constrains imply
that there are only 2[D/2](D−3) classical degrees of freedom. The on-shell degrees
of freedom are half this number (see Table 3.3).
52 Supersymmetric Massive Extension
ψµ Off-Shell 4D 3D On-Shell 4D 3D
Massless (D − 1)2[D/2] 12 4 1
2 (D − 3)2[D/2] 2 0
Table 3.3
This table gives the counting of degrees of freedom for gravitino field ψµ that describes a
spin-3/2 particle.
3.4. Supersymmetric Proca
In this section we show how to obtain the 3D supersymmetric Proca theory
from the KK reduction of an off-shell 4D N = 1 supersymmetric Maxwell theory
and a subsequent truncation to the first massive KK sector. This is a warming-up
exercise for the spin-2 case which will be discussed in the next section.
3.4.1. Kaluza–Klein reduction
Our starting point is the 4D N = 1 supersymmetric Maxwell multiplet which
consists of a vector V̂µ̂, a 4-component Majorana spinor ψ̂ and a real auxiliary
scalar F̂ . We indicate fields depending on the 4D coordinates and 4D indices
with a hat. We do not indicate spinor indices. The supersymmetry rules, with a
constant 4-component Majorana spinor parameter ǫ, and gauge transformation,
with local parameter Λ̂, of these fields are given by
δV̂µ̂ = −ǭΓµ̂ψ̂ + ∂µ̂Λ̂ , δψ̂ =
1
8
Γµ̂ν̂F̂µ̂ν̂ǫ+
1
4
iΓ5F̂ ǫ , δF̂ = iǭΓ5Γµ̂∂µ̂ψ̂ ,
(3.35)
where F̂µ̂ν̂ = ∂µ̂V̂ν̂ − ∂ν̂ V̂µ̂ .
In the following, we will split the 4D coordinates as xµ̂ = (xµ, x3), where x3
denotes the compactified circle coordinate. Since all fields are periodic in x3, we
can write them as a Fourier series. For example:
V̂µ̂(xµ̂) =
∑
n
Vµ̂,n(xµ)einmx
3
, n ∈ Z , (3.36)
where m 6= 0 has mass dimensions and corresponds to the inverse circle radius.
The Fourier coefficients Vµ̂,n(xµ) correspond to three-dimensional (un-hatted)
fields. We first consider the bosonic fields. The reality condition on the 4D
vector and scalar implies that only the 3D (n = 0) zero modes are real. All other
modes are complex but only the positive (n ≥ 1) modes are independent, since
Vµ̂,−n = V ⋆
µ̂,n , F−n = F ⋆n , n 6= 0 . (3.37)
3.4 Supersymmetric Proca 53
In the following we will be mainly interested in the n = 1 modes whose real and
imaginary parts we indicate by
V (1)
µ ≡ 1
2
(
Vµ,1 + V ⋆
µ,1
)
, V (2)
µ ≡ 1
2i
(
Vµ,1 − V ⋆
µ,1
)
,
φ(1) ≡ 1
2
(
V3,1 + V ⋆
3,1
)
, φ(2) ≡ 1
2i
(
V3,1 − V ⋆
3,1
)
,
F (1) ≡ 1
2
(
F1 + F ⋆1
)
, F (2) ≡ 1
2i
(
F1 − F ⋆1
)
.
(3.38)
Similarly, the Majorana condition of the 4D spinor ψ̂ implies that the n = 0
mode is Majorana but that the independent positive (n ≥ 1) modes are Dirac.
This is equivalent to two (4-component, 3D reducible) Majorana spinors which
we indicate by
ψ(1) =
1
2
(
ψ1 +B−1ψ⋆1
)
, ψ(2) =
1
2i
(
ψ1 −B−1ψ⋆1
)
. (3.39)
Here B is the 4 × 4 matrix B = iCΓ0, where C is the 4 × 4 charge conjugation
matrix.
Substituting the harmonic expansion (3.36) of the fields and a similar expan-
sion of the gauge parameter Λ̂ into the transformation rules (3.35), we find the
following transformation rules for the first (n = 1) KK modes:
δφ(1) = −ǭΓ3ψ
(1) −mΛ(2) −mξφ(2) , δφ(2) = −ǭΓ3ψ
(2) +mΛ(1) +mξφ(1) ,
δV (1)
µ = −ǭΓµψ(1) + ∂µΛ(1) −mξV (2)
µ , δV (2)
µ = −ǭΓµψ(2) + ∂µΛ(2) +mξV (1)
µ ,
δF (1) = iǭΓ5Γµ∂µψ(1) − imǭΓ5Γ3ψ
(2) −mξF (2) ,
δF (2) = iǭΓ5Γµ∂µψ(2) + imǭΓ5Γ3ψ
(1) +mξF (1) ,
δψ(1) =
1
8
ΓµνF (1)
µν ǫ+
1
4
ΓµΓ3∂µφ
(1)ǫ+
i
4
Γ5F
(1)ǫ+
m
4
ΓµΓ3V
(2)
µ ǫ−mξψ(2) ,
δψ(2) =
1
8
ΓµνF (2)
µν ǫ+
1
4
ΓµΓ3∂µφ
(2)ǫ+
i
4
Γ5F
(2)ǫ− m
4
ΓµΓ3V
(1)
µ ǫ+mξψ(1) ,
where we have defined
Λ(1) =
1
2
(
Λ1 + Λ⋆1
)
, Λ(2) =
1
2i
(
Λ1 − Λ⋆1
)
. (3.40)
Apart from global supersymmetry transformations with parameter ǫ and gauge
transformations with parameters Λ(1), Λ(2), the transformations (3.40) also con-
tain a global SO(2) transformation with parameter ξ, that rotates the real and
54 Supersymmetric Massive Extension
imaginary parts of the 3D fields. This SO(2) transformation corresponds to a
central charge transformation and is a remnant of the translation in the compact
circle direction.1
In order to write the 3D 4-component Majorana spinors in terms of two ir-
reducible 2-component Majorana spinors it is convenient to choose the following
representation of the Γ-matrices in terms of 2 × 2 block matrices:
Γµ =
(
γµ 0
0 −γµ
)
, Γ3 =
(
0 1
1 0
)
, Γ5 =
(
0 −i
i 0
)
. (3.41)
The 3D 2 × 2 matrices γµ satisfy the standard relations {γµ, γν} = 2ηµν and can
be chosen explicitly in terms of the Pauli matrices by
γµ = (iσ1, σ2, σ3) . (3.42)
In this representation the 4D charge conjugation matrix C is given by
C =
(
ε 0
0 −ε
)
, with ε =
(
0 1
−1 0
)
, (3.43)
where ε is the 3D charge conjugation matrix.
Using the above representation the 4-component Majorana spinors decompose
into two 3D irreducible Majorana spinors as follows:
ψ(1) =
(
χ1
χ2
)
, ψ(2) =
(
ψ1
ψ2
)
, ǫ =
(
ǫ1
ǫ2
)
. (3.44)
In terms of these 2-component spinors the transformation rules (3.40) read
δφ(1) = −ǭ1χ2 + ǭ2χ1 −mΛ(2) −mξφ(2) ,
δφ(2) = −ǭ1ψ2 + ǭ2ψ1 +mΛ(1) +mξφ(1) ,
δV (1)
µ = −ǭ1γµχ1 − ǭ2γµχ2 + ∂µΛ(1) −mξV (2)
µ ,
δV (2)
µ = −ǭ1γµψ1 − ǭ2γµψ2 + ∂µΛ(2) +mξV (1)
µ ,
δF (1) = −ǭ1γµ∂µχ2 + ǭ2γ
µ∂µχ1 −m(ǭ1ψ1 + ǭ2ψ2) −mξF (2) ,
δF (2) = −ǭ1γµ∂µψ2 + ǭ2γ
µ∂µψ1 +m(ǭ1χ1 + ǭ2χ2) +mξF (1) ,
(3.45)
1This is a conventional central charge transformation. Three-dimensional supergravity also
allows for non-central charges from extensions by non-central R-symmetry generators [80], re-
cently discussed in [81].
3.4 Supersymmetric Proca 55
δχ1 =
1
8
γµνF (1)
µν ǫ1 +
1
4
(
γµ∂µφ
(1) + F (1) +mγµV (2)
µ
)
ǫ2 −mξψ1 ,
δχ2 =
1
8
γµνF (1)
µν ǫ2 − 1
4
(
γµ∂µφ
(1) + F (1) +mγµV (2)
µ
)
ǫ1 −mξψ2 ,
δψ1 =
1
8
γµνF (2)
µν ǫ1 +
1
4
(
γµ∂µφ
(2) + F (2) −mγµV (1)
µ
)
ǫ2 +mξχ1 ,
δψ2 =
1
8
γµνF (2)
µν ǫ2 − 1
4
(
γµ∂µφ
(2) + F (2) −mγµV (1)
µ
)
ǫ1 +mξχ2 .
(3.46)
If we take m → 0 in the above multiplet we obtain two decoupled multi-
plets, (φ(1), V (1)
µ , F (1), χ1, χ2) and (φ(2), V
(2)
µ , F (2), ψ1, ψ2). Either one of them
constitutes a massless N = 2 vector multiplet. This massless limit has to be
distinguished from the massless limits discussed in subsections 3.4.3 and 3.5.2,
which refer to limits taken after truncating to N = 1 supersymmetry.
3.4.2. Truncation
In the process of KK reduction, the number of supercharges stays the same.
The 3D multiplet (3.46) we found in the previous subsection thus exhibits four
supercharges and hence corresponds to an N = 2 multiplet, containing two vec-
tors and a central charge transformation. One can, however, truncate it to an
N = 1 multiplet, not subjected to a central charge transformation and containing
only one vector. This truncated multiplet will be the starting point to obtain an
N = 1 supersymmetric version of the Proca theory. The N = 1 truncation is
given by:
φ(2) = V (1)
µ = F (2) = χ2 = ψ1 = 0 , (3.47)
provided that at the same time we truncate the following symmetries:
ǫ1 = Λ(1) = ξ = 0 . (3.48)
Substituting this truncation into the transformation rules (3.46), we find the
following N = 1 massive vector supermultiplet: 2
δφ(1) = ǭ2χ1 −mΛ(2) , δV (2)
µ = −ǭ2γµψ2 + ∂µΛ(2) ,
δψ2 =
1
8
γµνF (2)
µν ǫ2 , δχ1 =
1
4
(
γµ∂µφ
(1) + F (1) +mγµV (2)
µ
)
ǫ2 ,
δF (1) = ǭ2γ
µ∂µχ1 −mǭ2ψ2 .
(3.49)
2Note that the field content given in (3.49) is that of massless N = 2. In the massive case,
however, the scalar field φ will disappear after gauge-fixing the Stückelberg symmetry.
56 Supersymmetric Massive Extension
Redefining ǫ2 → ǫ ,Λ(2) → Λ and
φ(1) → 4φ , V (2)
µ → Vµ , F (1) → −F , ψ2 → ψ , χ1 → χ and m → 4m ,
(3.50)
we obtain
δφ =
1
4
ǭχ−mΛ , δVµ = −ǭγµψ + ∂µΛ , δF = −ǭγµ∂µχ+ 4mǭψ ,
δψ =
1
8
γµνFµνǫ , δχ = γµDµφ ǫ− 1
4
Fǫ ,
(3.51)
where the covariant derivative Dµ is defined as
Dµφ = ∂µφ+mVµ . (3.52)
The transformation rules (3.51) leave the following action invariant:
I1 =
∫
d3x
(
−1
4
Fµν F
µν − 1
2
DµφD
µφ− 2ψ̄ ∂/ψ − 1
8
χ̄ ∂/ χ+ mψ̄χ+
1
32
F 2
)
.
(3.53)
The gauge transformation with parameter Λ is a Stückelberg symmetry, that can
be fixed by imposing the gauge condition
φ = const . (3.54)
Taking the resulting compensating gauge transformation
Λ =
1
4m
ǭχ (3.55)
into account, we obtain the final form of the supersymmetry transformation rules
of the N = 1 supersymmetric Proca theory:
δVµ = −ǭγµψ +
1
4m
ǭ∂µχ , δψ =
1
8
γµνFµνǫ ,
δF = −ǭγµ∂µχ+ 4mǭψ , δχ = mγµǫVµ − 1
4
Fǫ .
(3.56)
The supersymmetric Proca action is then given by
IProca =
∫
d3x
(
−1
4Fµν F
µν − 1
2 m
2Vµ V
µ − 2ψ̄ ∂/ψ − 1
8 χ̄ ∂/ χ+ mψ̄χ+ 1
32F
2
)
.
(3.57)
The supersymmetric Proca theory describes 2+2 on-shell and 4+4 off-shell de-
grees of freedom.
This finishes our description of how to obtain the 3D off-shell massive N = 1
vector multiplet from a KK reduction and subsequent truncation onto the first
massive KK sector of the 4D off-shell massless N = 1 vector multiplet.
3.4 Supersymmetric Proca 57
3.4.3. Massless limit
We end this section with some comments on the massless limit (m → 0).
Taking the massless limit in (3.51), we see that the Proca multiplet splits into a
massless vector multiplet and a massless scalar multiplet. Note that a massless
vector multiplet can be coupled to a current supermultiplet. This is a feature
that we would like to incorporate, in view of the upcoming spin-2 discussion. We
will do so by coupling the above supersymmetric Proca system to a conjugate
multiplet (Jµ,Jψ,Jχ, JF ), where Jµ is a vector, Jψ and Jχ are spinors and JF is
a scalar. Our starting point is then the action
I = IProca + Iint , (3.58)
where the interaction part Iint describes the coupling between the Proca multiplet
and the conjugate multiplet:
Iint = V µJµ + ψ̄Jψ + χ̄Jχ + FJF . (3.59)
Requiring that Iint is separately invariant under supersymmetry, determines the
transformation rules of the conjugate multiplet:
δJµ =
1
4
ǭ γµν∂
νJψ +mǭ γµJχ , δJψ = −γµǫ Jµ − 4mǫJF ,
δJF =
1
4
ǭJχ , δJχ =
1
4m
ǫ∂µJµ + γµǫ ∂µJF .
(3.60)
Taking the massless limit in the action (3.58) and transformation rules (3.56),
(3.60) is non-trivial, due to the factors of 1/m that appear in the transformation
rules. In order to be able to take the limit in a well-defined fashion, we will work
in the formulation where the Stückelberg symmetry is not yet fixed. Note that
this formulation can be easily retrieved from the gauge fixed version, by making
the following redefinition in the action (3.57) and transformation rules (3.56):
Vµ = Ṽµ +
1
m
∂µφ . (3.61)
Applying this redefinition to (3.57) and (3.56) indeed brings one back to the
action (3.53) and to the transformation rules (3.51), whose massless limit is well-
defined. The massless limit of the interaction part Iint (after performing the above
substitution) and of the transformation rules (3.60) of the conjugate multiplet, is
however not well-defined. In order to remedy this, we will impose the constraint
that Jµ corresponds to a conserved current, i.e. that
∂µJµ = 0 . (3.62)
58 Supersymmetric Massive Extension
In order to preserve supersymmetry, we will also take Jχ = 0 and JF = 0.3 The
conjugate multiplet then reduces to a spin-1 current supermultiplet.
The massless limit is now everywhere well-defined. The transformation rules
(3.51) reduce to the transformation rules of a massless vector (Ṽµ, ψ) and scalar
(φ, χ, F ) multiplet, see eqs. (3.128) and (3.123), respectively. Performing the
above outlined procedure and taking the limit m → 0 leads to the action
I =
∫
d3x
[ (
−1
4 F̃µν F̃
µν − 2ψ̄ ∂/ψ + Ṽ µJµ + ψ̄Jψ
)
− 1
2
(
∂µφ∂
µφ+ 1
4 χ̄ ∂/ χ− 1
16F
2
) ]
,
(3.63)
which is the sum of the supersymmetric massless vector and scalar multiplet
actions, see eqs. (3.129) and (3.124), respectively. The vector multiplet action is
coupled to a spin-1 current multiplet. Note that there is no coupling left between
the current multiplet and the scalar multiplet. This will be different in the spin-2
case, as we will see later.
3.5. Supersymmetric Fierz–Pauli
In this section we extend the discussion of the previous section to the spin-2
case, skipping some of the details we explained in the spin-1 case.
3.5.1. Kaluza–Klein reduction and truncation
Our starting point is the off-shell 4D N = 1 massless spin-2 multiplet which
consists of a symmetric tensor ĥµ̂ν̂ , a gravitino ψ̂µ̂ , an auxiliary vector µ̂ and
two auxiliary scalars M̂ and N̂ . This corresponds to the linearized version of the
‘old minimal supergravity’ multiplet. The supersymmetry rules, with constant
spinor parameter ǫ, and gauge transformations of these fields, with local vector
parameter Λ̂µ̂ and local spinor parameter η̂, are given by [82,83]:
δĥµ̂ν̂ = ǭΓ(µ̂ ψ̂ν̂) + ∂(µ̂ Λ̂ν̂) , δM̂ = −ǭΓρ̂λ̂ ∂ρ̂ψ̂λ̂ , δN̂ = −i ǭΓ5 Γρ̂λ̂ ∂ρ̂ψ̂λ̂ ,
δψ̂µ̂= −1
4Γρ̂λ̂ ∂ρ̂ĥλ̂µ̂ǫ− 1
12Γµ̂(M̂ + iΓ5N̂) ǫ+ 1
4 i µ̂Γ5ǫ− 1
12 iΓµ̂Γρ̂Âρ̂Γ5ǫ+ ∂µ̂η̂ ,
δµ̂ =
3
2
i ǭΓ5 Γ ρ̂λ̂
µ̂ ∂ρ̂ ψ̂λ̂ − i ǭΓ5 Γµ̂Γρ̂λ̂ ∂ρ̂ψ̂λ̂ .
(3.64)
3Strictly speaking, preservation of the constraint ∂µJµ = 0 under supersymmetry leads to the
constraint /∂Jχ = 0 and preservation of this new constraint leads to the constraint ✷JF = 0. We
are however interested in the massless limit, in which the conserved currents (Jµ,Jψ) and the
fields (Jχ, JF ) form two separate multiplets, that couple to a massless vector and scalar multiplet
respectively. Since we are mostly interested in the coupling of the supercurrent multiplet (Jµ,Jψ)
to the vector multiplet, we will simply set the fields (Jχ, JF ) equal to zero.
3.5 Supersymmetric Fierz–Pauli 59
Like in the spin-1 case we first perform a harmonic expansion of all fields and local
parameters and substitute these into the transformation rules (3.64). Projecting
onto the lowest KK massive sector we then obtain all the transformation rules of
the real and imaginary parts of the n = 1 modes, like in eq. (3.40) for the spin-1
case. We indicate the real and imaginary parts of the bosonic modes by:
h(1)
µν ≡ 1
2
(
hµν,1 + h⋆µν,1
)
, h(2)
µν ≡ 1
2i
(
hµν,1 − h⋆µν,1
)
,
V (1)
µ ≡ 1
2
(
hµ3,1 + h⋆µ3,1
)
, V (2)
µ ≡ 1
2i
(
hµ3,1 − h⋆µ3,1
)
,
φ(1) ≡ 1
2
(
h33,1 + h⋆33,1
)
, φ(2) ≡ 1
2i
(
h33,1 − h⋆33,1
)
,
P (1) ≡ 1
2
(
A3,1 +A⋆3,1
)
, P (2) ≡ 1
2i
(
A3,1 −A⋆3,1
)
,
M (1) ≡ 1
2
(M1 +M⋆
1 ) , M (2) ≡ 1
2i
(M1 −M⋆
1 ) ,
N (1) ≡ 1
2
(N1 +N⋆
1 ) , N (2) ≡ 1
2i
(N1 −N⋆
1 ) ,
(3.65)
while the fermionic modes decompose into two Majorana modes:
ψ(1)
µ ≡ 1
2
(
ψµ,1 +B−1ψ⋆µ,1
)
, ψ(2)
µ ≡ 1
2i
(
ψµ,1 −B−1ψ⋆µ,1
)
,
ψ
(1)
3 ≡ 1
2
(
ψ3,1 +B−1ψ⋆3,1
)
, ψ
(2)
3 ≡ 1
2i
(
ψ3,1 −B−1ψ⋆3,1
)
.
(3.66)
We next use the representation (3.41) of the Γ-matrices and decompose the
4-component spinors into two 2-component spinors as follows:
ψ(1)
µ =
(
ψµ1
ψµ2
)
, ψ
(1)
3 =
(
χ1
χ2
)
, ψ(2)
µ =
(
χµ1
χµ2
)
, ψ
(2)
3 =
(
ψ1
ψ2
)
,
η(1) =
(
η
(1)
1
η
(1)
2
)
, η(2) =
(
η
(2)
1
η
(2)
2
)
, ǫ =
(
ǫ1
ǫ2
)
.
(3.67)
Furthermore, we perform the following consistent truncation of the fields 4
φ(2) = V (1)
µ = h(2)
µν = M (2) = N (1) = P (2) = A(1)
µ = χ2 = ψ1 = ψµ1 = χµ2 = 0
(3.68)
4If we take the massless limit before the mentioned truncation we find two copies of a N = 2
massless spin-2 multiplet plus two copies of a N = 2 massless spin-1 multiplet, see also text
after (3.46).
60 Supersymmetric Massive Extension
and of the parameters Λ(2)
µ = Λ(1)
3 = ǫ1 = η
(1)
1 = η
(2)
2 = ξ = 0 .
For simplicity, from now on we drop all numerical upper indices, e.g. φ(1) = φ,
and all numerical lower indices, e.g. ψµ1 = ψµ of the remaining non-zero fields
(but not of the parameters). We find that the transformation rules of these
fields under supersymmetry, with constant 2-component spinor parameter ǫ, and
Stückelberg symmetries, with local scalar and vector parameters Λ3 ,Λµ , and
2-component spinor parameters η1 and η2, are given by 5
δhµν = ǭγ(µψν) + ∂(µΛν) ,
δφ = −ǭ χ−mΛ3 , δVµ =
1
2
ǭγµψ − 1
2
ǭχµ +
1
2
∂µΛ3 +
1
2
mΛµ ,
δψµ = −1
4
γρλ∂ρhλµǫ+
1
12
γµMǫ+
1
12
γµPǫ+ ∂µη2 ,
δψ = −1
4
γρλ∂ρVλǫ− 1
12
Nǫ− 1
12
γρAρǫ+mη2 ,
δχµ = −1
4
γρ∂ρVµǫ+
1
4
mγρhρµǫ− 1
12
γµNǫ+
1
4
Aµǫ− 1
12
γµγ
ρAρǫ+ ∂µη1 ,
δχ = −1
4
γρ∂ρφǫ− 1
12
Mǫ+
1
6
Pǫ− 1
4
mγρVρǫ−mη1 ,
δM = −ǭγρ∂ρχ+ ǭγρλ∂ρψλ −mǭγρχρ ,
δN = −ǭγρ∂ρψ − ǭγρλ∂ρχλ +mǭγρψρ ,
δP = ǭγρ∂ρχ+
1
2
ǭγρλ∂ρψλ +mǭγρχρ ,
δAµ =
3
2
ǭγµ
ρλ∂ρχλ − ǭγµγ
ρλ∂ρχλ +
1
2
ǭγµ
ρ∂ρψ − ǭ∂µψ − 1
2
mǭγµ
ρψρ +mǭψµ .
(3.69)
The action invariant under the transformations (3.69) is given by
Im =
∫
d3x
{
hµνGlin
µν(h) −m2
(
hµνhµν − h2
)
+2hµν∂µ∂νφ− 2h∂α∂αφ− FµνFµν + 4mhµν∂(µVν) − 4mh∂µVµ
−4ψ̄µγµνρ∂νψρ − 4χ̄µγµνρ∂νχρ + 8ψ̄γµν∂µχν + 8ψ̄µγµν∂νχ+ 8mψ̄µγµνχν
−2
3M
2 − 2
3N
2 + 2
3P
2 + 2
3AµA
µ
}
,
(3.70)
5The 4D analogue of this multiplet, in superfield language, can be found in [84].
3.5 Supersymmetric Fierz–Pauli 61
where h = ηµνhµν and Glin
µν(h) is the linearized Einstein tensor. We observe that
the action is non-diagonal in the bosonic fields (hµν , Vµ , φ) and the fermionic
fields (ψµ , χ) and (χµ , ψ).
Finally, we fix all Stückelberg symmetries by imposing the gauge conditions
φ = const , Vµ = 0 , ψ = 0 , χ = 0 . (3.71)
Taking into account the compensating gauge transformations
Λ3 = 0 , Λµ =
1
m
ǭχµ ,
η1 = − 1
12m
(M − 2P ) ǫ , η2 =
1
12m
(N + γρAρ) ǫ ,
(3.72)
we obtain the final form of the supersymmetry rules of the 3D N = 1 off-shell
massive spin-2 multiplet:
δhµν = ǭγ(µψν) +
1
m
ǭ∂(µχν) ,
δψµ = −1
4
γρλ∂ρhλµǫ+
1
12
γµ(M + P )ǫ+
1
12m
∂µ(N + γρAρ)ǫ ,
δχµ =
1
4
mγρhρµǫ+
1
4
Aµǫ− 1
12
γµ(N + γρAρ)ǫ− 1
12m
∂µ(M − 2P )ǫ ,
δM = ǭγρλ∂ρψλ −mǭγρχρ ,
δN = −ǭγρλ∂ρχλ +mǭγρψρ ,
δP =
1
2
ǭγρλ∂ρψλ +mǭγρχρ ,
δAµ =
3
2
ǭγµ
ρλ∂ρχλ − ǭγµγ
ρλ∂ρχλ − 1
2
mǭγµ
ρψρ +mǭψµ .
(3.73)
These transformation rules leave the following action invariant:
Im6=0 =
∫
d3x
{
hµνGlin
µν(h) −m2(hµνhµν − h2)
− 4ψ̄µγµνρ∂νψρ − 4χ̄µγµνρ∂νχρ + 8mψ̄µγµνχν
− 2
3
M2 − 2
3
N2 +
2
3
P 2 +
2
3
AµA
µ
}
.
(3.74)
62 Supersymmetric Massive Extension
This action describes 2+2 on-shell and 12+12 off-shell degrees of freedom. The
first line is the standard Fierz–Pauli action. The fermionic off-diagonal mass term
can easily be diagonalized by going to a basis in terms of the sum and difference
of the two vector-spinors.6
The above action shows that the three scalars M, N, P and the vector Aµ
are auxiliary fields which are set to zero by their equations of motion. We thus
obtain the on-shell massive spin-2 multiplet with the following supersymmetry
transformations:
δhµν = ǭγ(µψν) +
1
m
ǭ∂(µχν) , δψµ = −1
4
γρσ∂ρhµσǫ , δχµ =
m
4
γνhµνǫ .
(3.75)
It is instructive to consider the closure of the supersymmetry algebra for
the above supersymmetry rules given the fact that, unlike in the massless case,
the symmetric tensor hµν does not transform under the gauge transformations
δhµν = ∂µΛν + ∂νΛµ and the only symmetries left to close the algebra are the
global translations. We find that the commutator of two supersymmetries on hµν
indeed gives a translation,
[δ1, δ2]hµν = ξρ∂ρhµν , (3.76)
with parameter
ξµ =
1
2
ǭ2γ
µǫ1 . (3.77)
To close the commutator on the two gravitini requires the use of the equa-
tions of motion for these fields. From the action (3.74) we obtain the following
equations:
γµνρ∂νχρ = mγµνψν , (3.78)
and a similar equation for ψµ. These equations of motion imply the standard
spin-3/2 Fierz–Pauli equations
R(1)
µ ≡ ∂/χµ +mψµ = 0 , ∂µχµ = 0 , γµχµ = 0 , (3.79)
and similar equations for ψµ. A useful alternative way of writing the equations
of motion (3.78) is
R(2)
µν ≡ ∂[µχν] +mγ[µψν] = 0 . (3.80)
Using these two ways of writing the equations of motion as well as the FP con-
ditions that follow from them we find that the commutator on the two gravitini
6The +3/2 and −3/2 helicity states are described by the sum and difference of the two
vector-spinors. See also appendix 3.C.
3.5 Supersymmetric Fierz–Pauli 63
gives the same translations (3.77) up to equations of motion. More specifically,
we find the following commutators
[δ1, δ2]ψµ = ξν∂νψµ − 1
4m
ξα∂µR(1)
α − 1
8m
ξαγα∂µ(γρσ∂ρχσ)
+
1
4m
ξα∂µ∂α(γσχσ) − 1
8
ξαγµγα(γρσ∂ρψσ) ,
[δ1, δ2]χµ = ξν∂νχµ +
1
2
ξνR(2)
µν − 1
8
ξργρR(1)
µ
− 1
8
ξργρ∂µ(γνχν) +
m
8
ξργργµ(γνψν) .
(3.81)
Hence, the algebra closes on-shell.
3.5.2. Massless limit
Finally, we discuss the massless limit m → 0 of the supersymmetric FP theory.
This is particularly interesting in view of the fact that the massless limit of the
ordinary spin-2 FP system, coupled to a conserved energy-momentum tensor does
not lead to linearized Einstein gravity. Instead, it leads to linearized Einstein
gravity plus an extra force, mediated by a scalar that couples to the trace of
the energy-momentum tensor with gravitational strength. This phenomenon is
known as the van Dam–Veltman–Zakharov discontinuity. In the following, we
will pay particular attention to this discontinuity in the supersymmetric case.
In order to discuss the massless limit, it turns out to be advantageous to trade
the scalar fields M and P for scalars S and F , defined by
S =
1
6
(
M + P
)
, F =
4
3
(
M − 2P
)
. (3.82)
This field redefinition will make the multiplet structure of the resulting massless
theory more manifest. In order to discuss the vDVZ discontinuity, we will include
a coupling to a conjugate multiplet (Tµν ,J ψ
µ ,J χ
µ , TS , TN , TF , T
A
µ ), as we did in
the Proca case. Here Tµν is a symmetric two-tensor, J ψ
µ , J χ
µ are vector-spinors,
TAµ is a vector and TF , TS , TN are scalars. We will thus start from the action
I = IFP + Iint , (3.83)
where IFP is the supersymmetric FP action (3.74) and the interaction part Iint
is given by
Iint = hµνT
µν + ψ̄µJ µ
ψ + χ̄µJ µ
χ + S TS + F TF +N TN +Aµ T
µ
A . (3.84)
64 Supersymmetric Massive Extension
Requiring that Iint is separately invariant under supersymmetry determines the
transformation rules of the conjugate multiplet:
δTµν =
1
4
ǭ γα(µ∂
αJ ψ
ν) +
m
4
ǭ γ(µJ χ
ν) ,
δJ ψ
µ = γαǫ Tαµ +
1
4
γµαǫ ∂
αTS +mγµǫ TN +
m
2
γµαǫ T
α
A −mǫTAµ ,
δJ χ
µ =
1
m
ǫ∂αTµα − γµαǫ ∂
αTN − 4mγµǫ TF − 3
2
γµαβǫ ∂
αT βA + γµαγβǫ ∂
αT βA ,
δTS =
1
2
ǭ γµJ ψ
µ ,
δTN =
1
12m
ǭ ∂µJ ψ
µ − 1
12
ǭ γµJ χ
µ ,
δTF = − 1
16m
ǭ ∂µJ χ
µ ,
δTAµ = −1
4
ǭJ χ
µ +
1
12
ǭ γµγ
ρJ χ
ρ − 1
12m
ǭ γµ∂
ρJ ψ
ρ .
(3.85)
As in the Proca case, one should go back to a formulation that is still invariant
under the Stückelberg symmetries, in order to take the massless limit in a well-
defined way. This may be achieved by making the following field redefinitions in
the final transformation rules (3.73) and action (3.74) thereby re-introducing the
fields (Vµ , φ′ , χ′ , ψ) that were eliminated by the gauge-fixing conditions (3.71):
hµν = h̃µν − 1
m
(
∂µVν + ∂νVµ
)
+
1
m2
∂µ∂νφ
′ ,
ψµ = ψ̃µ − 1
m
∂µψ , χµ = χ̃µ +
1
4m
∂µχ
′ .
Applying this field redefinition in (3.73) then leads to transformation rules7,
whose massless limit is well-defined. In order to make the massless limit of
the interaction part Iint and of the transformation rules (3.85) well-defined, we
impose that Tµν and J ψ
µ are conserved
∂νTµν = 0 , ∂µJ ψ
µ = 0 , (3.86)
and we put J χ
µ , TF , TN and TAµ to zero in order to preserve supersymmetry and
to obtain an irreducible multiplet in the massless limit. The conjugate multiplet
7These resulting transformation rules are given by the transformation rules (3.69), provided
one makes the following substitution: hµν → h̃µν , ψµ → ψ̃µ, χµ → χ̃µ, φ → −φ′ and χ → χ′/4.
3.5 Supersymmetric Fierz–Pauli 65
(3.85) then reduces to a spin-2 supercurrent multiplet (Tµν ,J ψ
µ , TS) that contains
the energy-momentum tensor Tµν and supersymmetry current J ψ
µ .
As in the Proca case, the massless limit is now well-defined. Performing
the above outlined steps on the action (3.83) and taking the massless limit leads,
however, to an action that is in off-diagonal form. This action can be diagonalized
by making the following field redefinitions:
h̃µν = h′
µν + ηµνφ
′ , ψ̃µ = ψ′
µ +
1
4
γµχ
′ , S = S′ − 1
8
F , χ̃µ = χ′
µ − γµψ .
(3.87)
The resulting action is given by
I =
∫
d3x
{
h′µνGlin
µν(h
′) − 4ψ̄′
µγ
µνρ∂νψ
′
ρ − 8S′2 + h′
µνT
µν + ψ̄′µJ ψ
µ + S′TS
− FµνFµν − 2
3
N2 +
2
3
AµAµ − 4χ̄′
µγ
µνρ∂νχ
′
ρ − 8ψ̄γµ∂µψ
+ 2
[− ∂µφ
′∂µφ′ − 1
4
χ̄′γµ∂µχ
′ +
1
16
F 2]
+ φ′ηµνTµν − 1
4
χ̄′ γµJ ψ
µ − 1
8
FTS
}
.
(3.88)
This is an action for three massless multiplets : a spin two multiplet (h′
µν , ψ
′
µ, S
′),
a mixed gravitino-vector multiplet8 (Vµ, χ′
µ, ψ,N,Aµ) and a scalar multiplet (φ′,
χ′, F ). These multiplets and their transformation rules are collected in appendix
3.B.9 The spin-2 multiplet couples to the supercurrent multiplet in the usual
fashion. Unlike the Proca case however, the supercurrent multiplet does not only
couple to the spin-2 multiplet, but there is also a coupling to the scalar multiplet,
given in the last line of (3.88). Indeed, defining
Tφ = ηµνTµν , J = −1
4
γµJ ψ
µ , TF = −1
8
TS , (3.89)
one finds that the fields (Tφ,J , TF ) form a conjugate scalar multiplet with trans-
formation rules
δTφ = −ǭ γµ∂µJ , δJ = −1
4
ǫ Tφ + γµǫ ∂µTF , δTF =
1
4
ǭJ , (3.90)
8An on-shell version of this multiplet was introduced in [85].
9The transformation rules of the different multiplets can also be found by starting from the
transformation rules of the massive FP multiplet and carefully following all redefinitions as
outlined in the main text, provided one performs compensating gauge transformations.
66 Supersymmetric Massive Extension
such that the last line of (3.88) is invariant under supersymmetry.
We have thus obtained a 3D supersymmetric version of the 4D vDVZ disconti-
nuity. The above discussion shows that the massless limit of the supersymmetric
FP theory coupled to a supercurrent multiplet, leads to linearized N = 1 super-
gravity, plus an extra scalar multiplet that couples to a multiplet that includes the
trace of the energy-momentum tensor and the gamma-trace of the supercurrent.
3.6. Linearized SNMG without Higher Derivatives
Using the results of the previous section we will now construct linearized New
Massive Supergravity without higher derivatives but with auxiliary fields. Fur-
thermore, we will show how, by eliminating the different “non-trivial” bosonic
and fermionic auxiliary fields, one re-obtains the higher-derivative kinetic terms
for both the bosonic and fermionic fields. We remind that by a “non-trivial” aux-
iliary field we mean an auxiliary field whose elimination leads to higher-derivative
terms in the action.
Consider first the bosonic case. The linearized version of lower-derivative
(“lower”) NMG is described by the following action [14]:
I lin
NMG(lower) =
∫
d3x
{
− hµνGlin
µν(h) + 2qµνGlin
µν(h) −m2(qµνqµν − q2)
}
, (3.91)
where hµν and qµν are two symmetric tensors and q = ηµνqµν . The above action
can be diagonalized by making the redefinitions
hµν = Aµν +Bµν , qµν = Bµν , (3.92)
after which we obtain
I lin
NMG [A,B] =
∫
d3x
{
−AµνGlin
µν (A) +BµνGlin
µν (B) −m2
(
BµνBµν −B2
)}
.
(3.93)
Using this diagonal basis it is clear that we can supersymmetrize the action in
terms of a massless multiplet (Aµν , λµ, S) and a massive multiplet (Bµν , ψµ, χµ,
M,N,P,Aµ). Transforming this result back in terms of hµν and qµν and making
the redefinition
λµ = ρµ − ψµ (3.94)
3.6 Linearized SNMG without Higher Derivatives 67
we find the following linearized lower-derivative supersymmetric NMG action
I lin
SNMG(lower) =
∫
d3x
{
− hµνGlin
µν(h) + 2qµνGlin
µν(h) −m2(qµνqµν − q2) + 8S2
− 2
3
M2 − 2
3
N2 +
2
3
P 2 +
2
3
AµA
µ
+ 4ρ̄µγµνρ∂νρρ − 8ψ̄µγµνρ∂νρρ − 4χ̄µγµνρ∂νχρ + 8mψ̄µγµνχν
}
.
(3.95)
This action describes 2+2 on-shell and 16+16 off-shell degrees of freedom. It is
invariant under the following transformation rules
δhµν = ǭγ(µρν) , δS =
1
4
ǭγµνρµν − 1
4
ǭγµνψµν ,
δρµ = −1
4
γρσ
(
∂ρhµσ
)
ǫ+
1
2
Sγµǫ+
1
12
γµ(M + P )ǫ ,
(3.96)
where
ρµν =
1
2
(
∂µρν − ∂νρµ
)
, ψµν =
1
2
(
∂µψν − ∂νψµ
)
, (3.97)
plus the transformation rules for the massive multiplet (qµν , ψµ , χµ , M, N, P,
Aµ) which can be found in eq. (3.73), with hµν replaced by qµν . We have deleted
1/m terms in the transformation of hµν and ρµ since they take the form of a
gauge transformation. Note also that the auxiliary field S transforms to the
gamma trace of the equation of motion for ρµ.
The action (3.95) contains the trivial auxiliary fields (S, M, N, P, Aµ) and the
non-trivial auxiliary fields (qµν , ψµ , χµ). The elimination of the trivial auxiliary
fields does not lead to anything new. These fields can simply be set equal to zero
and disappear from the action. Instead, as we will show now, the elimination of
the non-trivial auxiliary fields leads to higher-derivative terms in the action. To
start with, the equation of motion for qµν can be used to solve for qµν as follows:
qµν =
1
m2
Glin
µν(h) − 1
2m2
ηµνG
lin
tr (h) , (3.98)
where Glin
tr (h) = ηµνGlin
µν(h). One of the vector-spinors, ψµ, occurs as a Lagrange
multiplier. Its equation of motion enables one to solve for χµ:
χµ = − 1
2m
γρσγµ ρρσ . (3.99)
The equation of motion of the other vector-spinor, χµ, can be used to solve for
ψµ in terms of χµ:
ψµ = − 1
2m
γρσγµ χρσ , (3.100)
68 Supersymmetric Massive Extension
and hence, via eq. (3.99), in terms of ρµ. One can show that the solution of ψµ
in terms of (two derivatives of) ρµ is such that it solves the constraint
γµνψµν = 0 . (3.101)
We now substitute the solutions (3.98) for qµν and (3.99) for χµ back into the
action and make use of the identity
− 4χ̄µγµνρ∂νχρ =
8
m2
ρ̄µν∂/ ρµν − 2
m2
ρ̄µνγ
µν∂/ γσρρσρ , (3.102)
where we ignore a total derivative term. One thus obtains the following linearized
higher-derivative (“higher”) supersymmetric action of NMG [20]:
I lin
SNMG(higher) =
∫
d3x
{
− hµνGlin
µν(h) + 4ρ̄µγµνρ∂νρρ + 8S2
+
4
m2
(
RµνRµν − 3
8
R2)lin +
8
m2
ρ̄ab∂/ ρab − 2
m2
ρ̄abγ
ab∂/ γcdρcd
}
.
(3.103)
The action (3.103) is invariant under the supersymmetry rules
δhµν = ǭγ(µρν) , δρµ = −1
4
γρσ∂ρhµσǫ+
1
2
Sγµǫ , δS =
1
4
ǭγµνρµν ,
(3.104)
where we made use of the constraint (3.101) to simplify the transformation rule of
S. Under supersymmetry the auxiliary field S transforms to the gamma-trace of
the equation of motion for ρµ, since the higher-derivative terms in this equation
of motion are gamma-traceless and therefore drop out.
Alternatively, the higher-derivative kinetic terms for ρµ can be obtained by
boosting up the derivatives in the massive spin-3/2 FP equations in the same way
as that has been done for the spin-2 FP equations in the construction of New
Massive Gravity [14], except for one subtlety, see appendix 3.C.
This finishes our construction of linearized SNMG. In the next section we will
discuss to which extent this result can be extended to the non-linear case.
3.7. The non-linear case
Supersymmetric NMG without “non-trivial” auxiliary fields, i.e. with higher
derivatives, has already been constructed some time ago [20]. This action only
contains the auxiliary field S of the massless multiplet. A characteristic feature is
that there is no kinetic term for S and in the bosonic terms S occurs as a torsion
contribution to the spin-connection. However, due to its coupling to the fermions
3.7 The non-linear case 69
it cannot be eliminated from the action. Thus, in the non-linear case we cannot
anymore identify S as a “trivial” auxiliary field.
We recall that, apart from the auxiliary field S, in the linearized analysis of
section 3.5 and 3.6 we distinguish between the trivial auxiliary fields (M , N , P ,
Aµ) and the non-trivial ones (qµν , ψµ , χµ). Only the elimination of the latter
ones leads to higher derivatives in the Lagrangian. In the formulation of [20] only
the auxiliary field S occurs. One could now search either for a formulation in
which all other auxiliary fields occur or for an alternative formulation in which
only the non-trivial auxiliary fields (qµν , ψµ , χµ) are present. In this work we
will not consider the inclusion of all auxiliary fields any further. It is not clear
to us whether such a formulation exists. This is based on the fact that our
construction of the linearized massive multiplet makes use of the existence of a
consistent truncation to the first massive KK level. Such a truncation can only
be made consistently at the linearized level.
Before discussing the inclusion of the non-trivial auxiliary fields (qµν , ψµ , χµ)
it is instructive to first consider the linearized case and see how, starting from
the (linearized) formulation of [20] these three non-trivial auxiliary fields can be
included and a formulation with lower derivatives can be obtained. Our starting
point is the higher-derivative action (3.103) and corresponding transformation
rules (3.104). We first consider the bosonic part of the action (3.103), i.e.
I lin
bos(higher) =
∫
d3x
{
− hµνGlin
µν (h) + 8S2 +
4
m2
(
RµνRµν − 3
8
R2)lin
}
. (3.105)
We already know from the construction of the bosonic theory that the derivatives
can be lowered by introducing a symmetric auxiliary field qµν and writing the
equivalent bosonic action
I lin
bos(lower) =
∫
d3x
{
− hµνGlin
µν (h) + 8S2 + 2qµνGlin
µν(h) −m2
(
qµνqµν − q2
)}
.
(3.106)
The field equation of qµν is given by eq. (3.98) and substituting this solution back
into the lower-derivative bosonic action (3.106) we re-obtain the higher-derivative
bosonic action (3.105).
We next consider the fermionic part of the higher-derivative action (3.103),
I lin
ferm(higher) =
∫
d3x
{
4ρ̄µγµνρ∂νρρ+
8
m2
ρ̄ab∂/ ρab− 2
m2
ρ̄abγ
ab∂/ γcdρcd
}
. (3.107)
To lower the number of derivatives we first replace the terms that are quadratic
in ρµν by the kinetic term of an auxiliary field χµ, while adding another term
70 Supersymmetric Massive Extension
with a Lagrange multiplier ψµ to fix the relation between ρµν and χµ:
I lin
ferm(lower) =
∫
d3x
{
4ρ̄µγµνρ∂νρρ − 4χ̄µγµνρ∂νχρ − 8ψ̄µ (γµνρρνρ −mγµνχν)
}
.
(3.108)
The equation of motion for ψµ enables us to express χµ in terms of ρµν . The
result is given in eq. (3.99). Substituting this solution for χµ back into the
action, the terms linear in the Lagrange multiplier ψµ drop out and we re-obtain
the higher-derivative fermionic action given in eq. (3.107).
Adding up the lower-derivative bosonic action (3.106) and the lower-deriva-
tive fermionic action (3.108) we obtain the lower-derivative supersymmetric ac-
tion (3.95), albeit without the bosonic auxiliary fields (M, N, P, Aµ). We only
consider a formulation in which these auxiliary fields are absent.
Having introduced the new auxiliary fields (qµν , ψµ , χµ) we should derive
their supersymmetry rules. They can be derived by starting from the solutions
(3.98), (3.99) and (3.100) of these auxiliary fields in terms of hµν and ρµ and
applying the supersymmetry rules of hµν and ρµ given in eq. (3.104). This leads
to supersymmetry rules that do not contain the auxiliary fields. These can be
introduced by adding to the supersymmetry rules a number of (field-dependent)
equation of motion symmetries. We thus find the intermediate result:
δhµν = ǭγ(µρν) , δρµ = −1
4
γρσ∂ρhµσǫ+
1
2
Sγµǫ , δψµ = −1
4
γρσ∂ρqµσǫ ,
δqµν = ǭγ(µψν) +
1
m
ǭ∂(µχν) , δS =
1
4
ǭγµνρµν , δχµ =
m
4
γνqµνǫ+
1
2m
ǫ∂µS .
(3.109)
These transformation rules are not yet quite the same as the ones given in
eq. (3.96). In particular, the transformation rules of S and χµ are different.
The difference is yet another “on-shell symmetry” of the action eq. (3.95), with
spinor parameter η, given by
δS = −1
4
η̄γµνψµν , δχµ = − 1
2m
η∂µS . (3.110)
The transformation rules in eqs. (3.96) and (3.109) are therefore equivalent up to
an on-shell symmetry with parameter η = ǫ:
δsusy(eq. (3.96)) = δsusy(eq. (3.109)) + δon-shell(η = ǫ) . (3.111)
We now wish to discuss in which sense the previous analysis can be extended
to the non-linear case. For simplicity, we take the approximation in which one
considers only the terms in the action that are independent of the fermions and
3.7 The non-linear case 71
the terms that are bilinear in the fermions. Furthermore, we ignore in the su-
persymmetry variation of the action terms that depend on the auxiliary scalar
S. Since terms linear in S only occur in terms bilinear in fermions this effec-
tively implies that we may set S = 0 in the action. In this approximation the
higher-derivative action of SNMG is given by [20]
Inonlin
SNMG (higher) =
∫
d3x e
{
− 4R (ω̂) +
1
m2
Rµνab (ω̂)Rµνab (ω̂) − 1
2m2
R2 (ω̂)
+ 4ρ̄µγµνρDν (ω̂) ρρ +
8
m2
ρ̄ab (ω̂) /D (ω̂) ρab (ω̂) − 2
m2
ρ̄µν(ω̂)γµν /D(ω̂)γρσρρσ(ω̂)
− 2
m2
Rµνab (ω̂) ρ̄ργµνγρρab (ω̂) − 2
m2
R (ω̂) ρ̄µγνρµν (ω̂)
+ higher-order fermions and S-dependent terms
}
.
(3.112)
Note that we have replaced the symmetric tensor hµν by a Dreibein field eµ
a.
Keeping the same approximation discussed above the action (3.112) is invariant
under the supersymmetry rules
δeµ
a =
1
2
ǭγaρµ , δρµ = Dµ (ω̂) ǫ . (3.113)
We first consider the lowering of the number of derivatives in the bosonic part of
the action. Since the Ricci tensor now depends on a torsion-full spin connection
we need a non-symmetric auxiliary tensor qµ,ν . The action (3.112) can then be
converted into the following equivalent action:
Inonlin
SNMG (higher) =
∫
d3x e
{
− 4R (ω̂) −m2
(
qµ,νqµ,ν − q2
)
+ 2qµ,νGµ,ν (ω̂)
+ 4ρ̄µγµνρDν (ω̂) ρρ +
8
m2
ρ̄ab (ω̂) /D (ω̂) ρab (ω̂) − 2
m2
ρ̄µν(ω̂)γµν /D(ω̂)γρσρρσ(ω̂)
− 2
m2
Rµνab(ω̂)ρ̄ργµνγρρab(ω̂) − 2
m2
R(ω̂)ρ̄µγνρµν (ω̂)
+ higher-order fermions and S-dependent terms
}
.
The equivalence with the previous action can be seen by solving the equation of
motion for qµ,ν :
qµ,ν =
1
m2
Gµ,ν (ω̂) − 1
2m2
gµνG
tr (ω̂) (3.114)
72 Supersymmetric Massive Extension
and substituting this solution back into the action. Note that the solution for
qµ,ν is not super-covariant.
We next consider the lowering of the number of derivatives in the fermionic
terms in the action. Following the linearized case we define an auxiliary vector-
spinor χµ as
χµ = − 1
2m
γρσγµρρσ (ω̂) , (3.115)
or equivalently
ρµν (ω̂) = −mγ[µχν] . (3.116)
The first equation is the non-linear generalization of eq. (3.99). Using this defi-
nition one can show the following identity
8
m2
eρ̄ab(ω̂) /D(ω̂)ρab(ω̂) − 2
m2
eρ̄µν(ω̂)γµν /D(ω̂) [γρσρρσ(ω̂)] =
= −4eχ̄µγµνρDν(ω̂)χρ − 1
m
eRµνab(ω̂)ρ̄ργµνργabγσχσ
+ higher-order fermions and total derivative terms ,
(3.117)
which is the non-linear generalization of the identity (3.102). This identity can
be used to replace the higher-derivative kinetic terms of the fermions by lower-
derivative ones. At the same time we may use eq. (3.116) to replace ρµν by χµ.
This can be done by introducing a Lagrange multiplier ψµ whose equation of
motion allows us to use eq. (3.115). This leads to the following action:
Inonlin
SNMG (lower) =
∫
d3x e
{
− 4R (ω̂) + 2qµ,νGµ,ν (ω̂) −m2
(
qµ,νqµ,ν − q2
)
+ 4ρ̄µγµνρDν (ω̂) ρρ − 4χ̄µγµνρDν (ω̂)χρ − 8ψ̄µγµνρρνρ (ω̂) + 8mψ̄µγµνχν
− 1
m
Rµνab (ω̂) ρ̄ργµνργabγσχσ +
2
m
Rµνab (ω̂) ρ̄ργµνγργaχb
− 1
m
R (ω̂) ρ̄µγµνχν − 2
m
R (ω̂) ρ̄µχµ
+ higher-order fermions and S-dependent terms
}
.
Our next task is to derive the supersymmetry rules of the auxiliary fields
qµ,ν , ψµ and χµ. Using the solutions of the auxiliary fields in terms of eµa and ρµ
we derived these supersymmetry rules. In this way one obtains supersymmetry
rules that do not contain any of the auxiliary fields and, consequently, do not
3.8 Discussion 73
reduce to the supersymmetry rules (3.96) upon linearization. To achieve this,
we must add to these transformation rules a number of field-dependent equation
of motion symmetries, like we did in the linearized case. Since the results we
obtained are not illuminating we refrain from giving the explicit expressions here.
A disadvantage of the present approach is that, although in principle possible
in the approximation we considered, one cannot maintain the interpretation of
S as a torsion contribution to the spin-connection. This makes the result rather
cumbersome. Without further insight the lower-derivative formulation of SNMG,
if it exists at all at the full non-linear level, does not take the same elegant form
as the higher-derivative formulation presented in [20].
3.8. Discussion
In this work we considered the N = 1 supersymmetrization of New Massive
Gravity in the presence of auxiliary fields. All auxiliary fields are needed to close
the supersymmetry algebra off-shell. At the linearized level, we distinguished
between two types of auxiliary fields: the “non-trivial” ones whose elimination
leads to higher derivatives in the Lagrangian (these are the fields qµν , ψµ and χµ)
and the “trivial” ones whose elimination (if possible at all at the full non-linear
level) does not lead to higher derivatives (these are the fields S,M,N, P and
Aµ). We found that at the linearized level all auxiliary fields could be included
leading to a linearized SNMG theory without higher derivatives. At the non-
linear level we gave a partial answer for the case that only the trivial auxiliary
S and the non-trivial auxiliaries qµν , ψµ and χµ were included. To obtain the
full non-linear answer one should perhaps make use of superspace techniques.
The answer without the non-trivial auxiliaries and with higher derivatives can be
found in [20].
We discussed a 3D supersymmetric analog of the 4D vDVZ discontinuity by
taking the massless limit of the supersymmetric FP model coupled to a super-
current multiplet. We showed that in the massless limit there is a non-trivial
coupling of a scalar multiplet (containing the scalar mode φ of the metric) to a
current multiplet (containing the trace of the energy-momentum tensor). This is
the natural supersymmetric extension of what happens in the bosonic case and
supports the analysis of [78].
As a by-product we found a way to “boost up” the derivatives in the spin-3/2
FP equation, see appendix 3.C. The trick is based upon the observation that,
before boosting up the derivatives like in the construction of the NMG model,
one should first combine the equations of motion describing the helicity +3/2 and
−3/2 states into a single parity-even equation with one additional derivative.
74 Supersymmetric Massive Extension
3.A. General Multiplets and Degrees of Freedom
In any realization of a supersymmetry algebra of the form {Q,Q} ∼ P the
number of bosonic degrees of freedom should match the number of fermionic de-
grees of freedom. On-shell, this always holds, but off-shell bosonic and fermionic
degrees of freedom coincide when the theory has auxiliary fields which “close the
algebra off-shell”.
Field Off-Shell 4D 3D On-Shell 4D 3D
φ
Massless
1 1 1 1 1 1
Massive
ψ Massless 2[D/2] 4 2 1
2 2[D/2] 2 1
Aµ
Massless D − 1 3 2 D − 2 2 1
Massive D 4 3 D − 1 3 2
ψµ Massless (D − 1)2[D/2] 12 4 1
2 (D − 3)2[D/2] 2 0
gµν
Massless D(D − 1)/2 6 3 D(D − 3)/2 2 0
Massive D(D + 1)/2 10 6 D(D − 1)/2 − 1 5 2
Table 3.4
This table contains the results for the degrees of freedom for a scalar field φ, a Majorana
fermion ψ, a gauge field Aµ, a Majorana gravitino ψµ and a graviton field gµν .
To illustrate the off-shell and on-shell degrees of freedom, consider for ex-
ample the 3D massless “mixed gravitino-vector” multiplet with field content
{Vµ, Aµ, N, χµ, ψ}, where Aµ and N are auxiliary fields. The off-shell count-
ing of degrees of freedom is established in the following way: the gauge field Vµ
describes two off-shell degrees of freedom, while the auxiliary field Aµ describes
three, since there is no gauge invariance for this field, and the auxiliary field
N describes one degree of freedom (a propagating and an auxiliary scalar fields
always have one degree of freedom). In addition, the gravitino χµ contains four
degrees of freedom, while the spinor ψ describes two. Therefore, the off-shell
counting of bosonic and fermionic degrees of freedom match 6 + 6 giving a total
of twelve. On the other hand, the on-shell degrees of freedom will be: one for the
gauge field Vµ, zero for the auxiliary fields and for the gravitino, and one for the
spinor ψ. Once more the number of bosonic and fermionic degrees of freedom
match 1 + 1 leading to a total of two degrees of freedom.
3.B Off-shell N = 1 Massless Multiplets 75
3.B. Off-shell N = 1 Massless Multiplets
In this section we collect the off-shell formulations of the different 3D massless
multiplets with N = 1 supersymmetry. A useful reference where more properties
about 3D supersymmetry can be found is [86]. The field content of the different
multiplets can be found in Table 3.5.
multiplet fields off-shell on-shell
s = 2 hµν , ψµ , S 4+4 0+0
s = 1 Vµ , N ,Aµ , χµ , ψ 6+6 1+1
s = 0 φ , χ , F 2+2 1+1
gravitino multiplet χµ , Aµ , D 4+4 0+0
vector multiplet Vµ , ψ 2+2 1+1
Table 3.5
This Table indicates the field content and off-shell/on-shell degrees of freedom of the different
massless multiplets. Only the massless multiplets above the double horizontal line occur in the
massless limit of the FP model.
s=2 The off-shell version of the 3D massless spin-2 multiplet is well-known.
The multiplet is extended with an auxiliary real scalar field S. The off-shell
supersymmetry rules are given by
δhµν = ǭγ(µψν) , δψµ = −1
4
γρσ∂ρhµσǫ+
1
2
Sγµǫ , δS =
1
4
ǭγµνψµν ,
(3.118)
where
ψµν =
1
2
(
∂µψν − ∂νψµ
)
. (3.119)
These transformation rules leave the following action invariant:
Is=2 =
∫
d3x
{
hµνGlin
µν (h) − 4ψ̄µγµνρ∂νψρ − 8S2
}
. (3.120)
s=1 The off-shell “mixed gravitino-vector” multiplet consists of a propagating
vector Vµ, an auxiliary vector Aµ, an auxiliary scalar N , a vector spinor χµ and a
spinor ψ. An on-shell version of this multiplet, called “vector-spinor” multiplet,
76 Supersymmetric Massive Extension
has been considered in [85]. The off-shell supersymmetry rules are given by
δVµ = ǭγµψ − 1
2 ǭχµ ,
δψ = −1
8γ
ρλFρλǫ− 1
12Nǫ− 1
12γ
αAαǫ ,
δχµ = −1
4γ
αFαµǫ− 1
8γµγ
ρλFρλǫ− 1
6γµNǫ+ 1
4Aµǫ− 1
6γµγ
αAαǫ ,
δN = ǭγα∂αψ − ǭγαβ∂αχβ ,
δAµ = 3
2 ǭγ
αβ
µ ∂αχβ − ǭγµγ
αβ∂αχβ + ǭγ α
µ ∂αψ + ǭ∂µψ .
(3.121)
Note that this multiplet is irreducible. It cannot be written as the sum of a
gravitino and vector multiplet. These multiplets are given below. The supersym-
metric action for this multiplet is given by
Is=1 =
∫
d3x
{
− FµνFµν − 2
3N
2 + 2
3AµA
µ − 4χ̄µγµνρ∂νχρ − 8ψ̄γµ∂µψ
}
,
(3.122)
with Fµν = ∂µVν − ∂νVµ .
s=0 The off-shell scalar multiplet consists of a scalar φ, a spinor χ and an
auxiliary scalar F . The off-shell supersymmetry rules are given by
δφ = 1
4 ǭχ , δχ = γµǫ (∂µφ) − 1
4Fǫ , δF = −ǭγµ∂µχ . (3.123)
The supersymmetric action for a scalar multiplet is given by
Is=0 =
∫
d3x
{
− ∂µφ∂µφ− 1
4 χ̄γ
µ∂µχ+ 1
16F
2
}
. (3.124)
Besides the massless multiplets discussed so-far there is a separate gravitino
and vector multiplet. The vector multiplet arises in section 3.4 in the massless
limit of the Proca theory. For completeness we give these two multiplets below.
gravitino multiplet The off-shell gravitino multiplet consists of a gravitino χµ,
an auxiliary vector Aµ and an auxiliary scalar D. The off-shell supersymmetry
rules are given by
δχµ =
1
4
γλγµǫAλ +
1
2
γµǫD , δAµ =
1
2
ǭγρσγµχρσ , δD =
1
4
ǭγρσχρσ ,
(3.125)
where
χµν =
1
2
(
∂µχν − ∂µχµ
)
. (3.126)
3.C Boosting up the Derivatives in Spin-3/2 FP 77
These transformation rules leave the following action invariant:
Is=3/2 =
∫
d3x
{
− 4χ̄µγµνρ∂νχρ − 1
2
AµAµ + 2D2
}
. (3.127)
vector multiplet The off-shell vector multiplet consists of a vector Vµ and a
spinor ψ. The off-shell supersymmetry rules are given by
δVµ = −ǭγµψ , δψ =
1
8
γµνǫ Fµν , (3.128)
with Fµν = ∂µVν − ∂νVµ . The supersymmetric action for a vector multiplet is
given by
Is=1 =
∫
d3x
{
− 1
4
FµνFµν − 2ψ̄γµ∂µψ
}
. (3.129)
This finishes our discussion of the massless multiplets in three dimensions.
3.C. Boosting up the Derivatives in Spin-3/2 FP
In this appendix we show how the higher-derivative kinetic terms for the
gravitino ρµ can be obtained by boosting up the derivatives in the massive spin-
3/2 FP equations in the same way as that has been done for the spin-2 FP
equations in the construction of New Massive Gravity [14] except for one subtlety.
Our starting point is the following fermionic action with two massive gravitini,
ψµ and χµ, each of which carries only one physical degree of freedom in 3D,
I [ψ, χ] =
∫
d3x
{
− 4ψ̄µγµνρ∂νψρ − 4χ̄µγµνρ∂νχρ + 8mψ̄µγµνχν
}
. (3.130)
The equations of motion following from this action are given by
γµνρ∂νψρ −mγµνχν = 0 , γµνρ∂νχρ −mγµνψν = 0 . (3.131)
Note that each one of the equations (3.131) can be used to solve for one grav-
itino in terms of the other one. However, this solution does not solve the other
equation. Therefore, one cannot substitute only one solution back into (3.130)
because one would lose information about the differential constraint encoded in
the other equation. After diagonalization
ζ1
µ = ψµ + χµ , ζ2
µ = ψµ − χµ . (3.132)
78 Supersymmetric Massive Extension
we obtain the massive FP equations for a helicity +3/2 and -3/2 state:
(
/∂ +m
)
ζ1
µ = 0 , γµζ1
µ = 0 , ∂µζ1
µ = 0 , (3.133)
(
/∂ −m
)
ζ2
µ = 0 , γµζ2
µ = 0 , ∂µζ2
µ = 0 . (3.134)
To boost up the derivatives in these equations we may proceed in two ways.
One option is to boost up the derivatives in each equation separately by solv-
ing the corresponding differential constraint. In a second step one should then
combine the two higher-derivative equations by a single equation in terms of ρµ
by a so-called “soldering” technique which has also been applied to construct
New Massive Gravity out of two different Topologically Massive Gravities [87].
Alternatively, it is more convenient to first combine the two equations into the
following equivalent second-order equation which is manifestly parity-invariant:
(
✷ −m2
)
ζµ =
(
/∂ ∓m
) (
/∂ ±m
)
ζµ = 0 , γµζµ = 0 , ∂µζµ = 0 . (3.135)
Note that the action corresponding to these equations of motion cannot be used
in a supersymmetric action since the fermionic kinetic term would have the same
number of derivatives as the standard bosonic kinetic term describing a spin-2
state.
We are now ready to perform the procedure of “boosting up the derivatives”
in the same way as in the bosonic theory where it leads to the higher-derivative
NMG theory. To be specific, we solve the divergenceless condition ∂µζµ = 0 in
terms of a new vector-spinor ρµ as follows:
ζµ = Rµ (ρ) ≡ εµ
νρ∂νρρ . (3.136)
Substituting this solution back into the other two equations in (3.135) leads to
the higher-derivative equations
(
✷ −m2
)
Rµ (ρ) = 0 , γµRµ (ρ) = 0 . (3.137)
These equations of motion are invariant under the gauge symmetry
δρµ = ∂µη . (3.138)
Furthermore, they can be integrated to the following action:
I [ρ] =
∫
d3x
{
ρ̄µRµ (ρ) − 1
2m2
ρ̄µ/∂
[
/∂Rµ (ρ) + εµ
στ∂σRτ (ρ)
]
}
. (3.139)
One can show that the equations of motion following from this action implies the
algebraic constraint given in (3.137). The action (3.139) is precisely the fermionic
part of the action (3.103) of linearized SNMG.
4
Symmetry Analysis for Anisotropic
Field Theories
4.1. Introduction
The second modification from general relativity that we will consider implies
to give up on diffeomorphism invariance. Since at the quantum level, general
relativity is non-renormalizable, it should be viewed as an effective theory that
breaks down at some scale. Lorentz symmetry breaking in Hořava-Lifshitz Grav-
ity theory [22–24] can lead to a modification of the graviton propagator in the
UV, thus rendering the theory power-counting renormalizable.
Abandoning Lorentz invariance by introducing higher order spatial derivatives
and forbidding the introduction of any higher order time derivatives, improves
the UV behaviour of the propagator and at the same time, guarantees the absence
of ghosts [88,89].
It is important to point out that there is nothing wrong with using Lorentz
violations as there is no reason why a theory should necessarily exhibit Lorentz
symmetry in the far UV. Our theory will be consistent as long as it allows Lorentz
invariance to be restored as an emergent symmetry in the IR.
Systems without Lorentz invariance are quite common, for example, multifrac-
tal spacetimes [90], modified gravities described in [91,92], aethereal gravity [93],
they all deal with Lorentz violations. Furthermore, a Lifshitz scalar field theory
has been studied in the context of condensed matter physics as a description of
tricritical phenomena involving spatially modulated phases [94, 95] and low en-
ergy Lorentz breaking can be originated by high energy physics such as string
theory [39,96].
On the other hand, theories that are invariant under scale and conformal
79
80 Symmetry Analysis for Anisotropic Field Theories
transformations have been studied for over a hundred years. Current under-
standing, at least on the classical level, of several nonlinear physical systems and
critical phenomena is based on the invariance under global scaling transforma-
tions xµ → bxµ (e.g. [97,98]).
Hořava’s papers devoted to gravity models are characterized by an specific
anisotropic scaling between space and time:
t → bz t , ~x → b ~x ; b > 0 , z ∈ Re . (4.1)
Note that spatial isotropy is still assumed to be kept; the degree of anisotropy
between space and time is measured by z, which plays the role of a dynamical
critical exponent.
Noether’s theorem [99] states that every differentiable symmetry of the action
of a physical system has a corresponding conservation law, it associates to a
Lagrangian symmetry the conserved current whose total differential vanishes on-
shell. In this chapter, we use Noether’s theorem for general classical fields and for
general symmetry transformations in actions that depend on higher order spatial
derivatives to find the correspondent current densities that lead to a conservation
law.
Then, we analyze several important applications: the scalar field, an electro-
dynamics compatible with the transformations (4.1) for z = 2 as in [100, 101],
and higher derivative Chern-Simons extensions (see [102]). For these cases we ob-
tain the canonical energy-momentum tensor and the conserved current associated
with scale invariance. Conclusions are presented in the last section.
4.2. Noether’s Theorem
Let us consider the action S[Φ], for a collection of arbitrary fields Φ = {φa},
in D dimensions, where a represents space-time or internal indices:
S[Φ] =
∫
dDxL (Φ, ∂µΦ, ∂i1,2Φ, ∂i1,3Φ, . . . , ∂i1,nΦ
)
, (4.2)
where ∂nΦ(x)
∂xi1 ... ∂xin
≡ ∂i1,nΦ . Throughout this paper n = 0, 1, 2, . . . indicates the
derivative order, in = 1, 2, . . . , D − 1 stand for spatial indices, while µ = 0, 1, 2,
. . ., D − 1 denotes space-time indices.
Under arbitrary infinitesimal stationary variations δΦ, the equations of mo-
tion are
∂L
∂Φ
− ∂µ
∂L
∂(∂µΦ)
+ ∂i1,2
∂L
∂(∂i1,2Φ)
− · · · + (−1)n∂i1,n
∂L
∂(∂i1,nΦ)
= 0 , (4.3)
4.2 Noether’s Theorem 81
note that they contain the usual first two terms, plus higher order spatial deriva-
tives.
Now, the addition of a fourth divergence
∂µΛµ = ∂µ Θµ(Φ) + ∂m Ξm (Φ, ∂i1Φ, ∂i1 i2Φ, . . . , ∂i1...in−1Φ) (4.4)
to the Lagrangian, does not modify the Euler-Lagrange equations.
For convenience we introduce the notation
δµ
i1
Di1 = δµ
i1
(
D
i1
2 + · · · +D
i1
n
)
, δµ
i1
W i1 = δµ
i1
(
W
i1
2 + · · · +W
i1
n
)
, (4.5)
where
δµ
i1
D
i1
n ≡ δµ
(i1



(n−1)
∑
k=0
(−1)k
(
n
k
)
∂
n−1−k)
[(
∂
k)
∂L
∂(∂i1,n
Φ)
)
δΦ
]



,
δµ
i1
W
i1
n ≡ δµ
(i1



(n−1)
∑
k=0
(−1)k
(
n
k
)
∂
n−1−k)
[(
∂
k)
∂L
∂(∂i1,n
Φ)
)
∂νΦ δxν
]



,
here the lower index on Di1
n and W i1
n indicates the order of the term, the upper
index i1 should be contracted with the derivative index and on the right side
of the expression all the indices should be contracted. The parenthesis on the
subindexes as usual, denotes that these should be symmetric.
On the other hand, if the action integral (4.2) remains invariant up to a
fourth divergence δS =
∫
dDx ∂µδΩµ [103] when the coordinates are subject to
an infinitesimal transformation of the kind x′
µ = xµ+δxµ , Φ′(x′) = Φ(x)+δΦ(x) ,
then the conserved Noether currents (∂µJ µ = 0) are given by
J µ =
(
δµνL − ∂L
∂(∂µΦ)
∂νΦ
)
δxν +
∂L
∂(∂µΦ)
δΦ − δΩµ − δµi1
(
W i1 − Di1
)
, (4.6)
and the corresponding conserved charge Q is
Q =
∫
Vt
d3xJ 0 =
∫
Vt
d3x
[(
δ0
νL − ∂L
∂(∂0Φ)
∂νΦ
)
δxν +
∂L
∂(∂0Φ)
δΦ − δΩ 0
]
,
(4.7)
Expresion (4.7) is exactly the same conserved charge as in the case of a Lagrangian
depending only on the first space-time derivative
L(x) = L (Φ, ∂µΦ) . (4.8)
82 Symmetry Analysis for Anisotropic Field Theories
If the theory is invariant under translations (δxν = aµ), the canonical energy-
momentum tensor is given by
T µ
ν =δµν L − ∂L
∂(∂µΦ)
∂νΦ − δµ(i
∂L
∂(∂i∂jΦ)
∂
j)
∂νΦ + δµ(i ∂j)
(
∂L
∂(∂i∂jΦ)
)
∂νΦ + . . .
(4.9)
4.3. Examples
4.3.1. Scalar Field
For a massless scalar field φ in D dimensions, we will consider the time-related
term:
St[φ] =
∫
dDx
(
∂0φ∂
0φ
)
. (4.10)
We want to find the conditions in order to make 4.10 invariant under a scale
transformation of the form
t′ = bzt , x′
i = bxi , φ′ = b−∆φ , (4.11)
where b > 0, z,∆ ∈ Re and i = 1, . . . , D − 1. We find that
S′
t[φ] = bz+D−1
∫
dDx b−2(z+∆)
(
∂0φ∂
0φ
)
, (4.12)
is scale invariant if
∆ =
D − 1 − z
2
. (4.13)
We can also add an interaction term φn invariant under the scaling symmetry
4.11 when
n =
D − 1 + z
∆
= 2
(
D − 1 + z
D − 1 − z
)
. (4.14)
If we wish to include a term with second derivatives of the field
(
∂i∂
iφ
)m, invari-
ance under the scale rules leads to
m =
D − 1 + z
2 + ∆
. (4.15)
Let us consider the four dimensional case (see Table 4.1). The usual isotropic
scaling is given when z = 1. In the case when the dynamical critical exponent
z = 2 the only interaction term that will preserve the scaling symmetry will be
4.3 Examples 83
φ10 and a quadratic term if we add second order spatial derivatives. When z = 3
we cannot add an interaction term since in this case n → ∞, but we can add to
the action a cubic term of second order derivatives that remains invariant under
the scaling.
z
∆ n m
3D 4D 3D 4D 3D 4D
1 1
2 1 6 4 6
5
4
3
2 0 1
2 - 10 2 2
3 −1
2 0 -10 - 10
3 3
Table 4.1
Anisotropic scaling terms for a three-dimensional and four-dimensional scalar field with
different dynamical critical exponents z.
Now we will find the conserved Noether currents and charges in the four
dimensional case with z = 2
L = ∂0φ∂
0φ+ ∂
i j
φ∂
i j
φ , i, j = 1, 2, 3 ⇒ −∂0∂
0φ+ ∂
i j
∂
i j
φ = 0 . (4.16)
The scalar field theory obtained when i = j gives an action that is a sum of a
kinetic term involving time derivatives, and a potential term which is of a special
form: It can be derived from a variational principle
1
4
(∆φ(x))2 =
(
1
2
δW
δφ(x)
)
(4.17)
where W = 1
2
∫
dD−1x (∂iφ∂iφ). This resulting action is a prototype of a class of
models introduced and studied in the context of tri-critical phenomena in con-
densed matter physics by Lifshitz [94] in 1941, and is consequently referred to
in the literature as the “Lifshitz scalar” field theory [95, 104]. In the context
originally studied by Lifshitz, t is Wick rotated to become one of the spatial di-
mensions, and the Lifshitz scalar then describes the tricritical point connecting
the phases with a zero, homogeneous or spatially modulated condensate of φ.
In [105] Hořava introduced a new class of nonrelativistic gravity models, charac-
terized by anisotropic scaling between space and time with a nontrivial value of
the dynamical critical exponent z = 2. This anisotropy leads to a change of the
critical dimension of the system to 2 + 1, and makes the theory suitable for the
84 Symmetry Analysis for Anisotropic Field Theories
worldvolume of a membrane where it can be coupled to quantum critical matter
with z = 2, described in the simplest case by Lifshitz scalars with dynamical
critical exponent z = 2.
Using Noether’s theorem as shown before, we obtain the conserved currents
and charges of these kind of field theories. The canonical energy-momentum
tensor (4.9) is (using ∇2 ≡ ∂k∂
k)
T µ
ν =δµνL − 2 δµ0 ∂
0φ∂νφ+ 2 δµ
(i
[
−∂i jφ∂
j)
∂νφ+ (∂
i∇2φ) ∂νφ
]
. (4.18)
This Lagrangian is also invariant under scale transformations
δt = 2b t , δ~x = b ~x , δϕ = −1
2
b ϕ , b > 0 , (4.19)
and its corresponding current can be written as [106]
J µ = 2x0 T µ0 + xs T µs + ∂λ σ
µλ , (4.20)
where
σ0λ = −1
2
g0λϕ2 , σiλ = giλ
(
2ϕ∇ϕ− ∇ϕ2
)
+ gimgλnϕ∂mnϕ . (4.21)
Defining a new tensor
Θµν = T µν +
1
2
[
1
2
(
gµ0gν0 ∂λρ X
λρ
00 + gµigν0 ∂λρ X
λρ
i0
)
+
(
gµigνj ∂λρ X
λρ
ij + gµ0gνj ∂λρ X
λρ
0j
)]
,
Xλρµν = gλρ σµν − gλµ σρν − gλν σµρ + gµν σλρ +
1
3
(
gλµgρν − gλρgµν
)
σα
α ,
(4.22)
which has the following properties
∂µΘµν = 0 , 2 Θ0
0 + Θm
m = 2 T 0
0 + T m
m +
1
2
∂λρX
λρµ
µ , (4.23)
the conserved currents take a simpler form
J µ = 2x0 Θµ0 + xs Θµs . (4.24)
4.3 Examples 85
4.3.2. Electrodynamics
Let us consider the four dimensional electrodynamics Lagrangian with z = 2 [101]:
∂iF
0i = 0L = αF0iF
0i + β (∂kFij) (∂kF ij) , ⇒
α∂0F
0i − 2β∇2∂jF
ji = 0
where Fµν = ∂µAν − ∂νAµ ; α, β ∈ Re ; i, j, k = 1, 2, 3 ; and µ, ν = 0, 1, 2, 3, is
invariant under translations (δxµ = aµ, δAµ = 0). The canonical tensor is:
T 0
ν = δ0
νL − 2αF 0m∂νAm , T k
ν = δkνL − 2αF k0∂νA0 − Wk
ν , (4.25)
where Wk
ν = 4β
(
∂kFmn ∂mνAn − ∂νAn ∂
k
mF
mn
)
.
For the transformation of the fields
A′
i = b−1/2Ai , A′
0 = b−3/2A0 , (4.26)
the Lagrangian remains invariant and the corresponding conserved currents are
J 0 =2x0 T 0
0 + xs T 0
s − αF 0iAi , (4.27)
J p =2x0 T p
0 + xs T p
s − 3αF p0A0 − 2β δp(m
(
3 ∂(mFn)l ∂n)Al − ∂n)∂
(mFn)lAl
)
.
(4.28)
4.3.3. Higher Derivative Chern-Simons Extensions
In this example we will consider three dimensions and an arbitrary dynamical
critical exponent z. The action [102] will take the form
S
ECS
= (2m)−1
∫
d3x ǫαβγ ∇2Aα ∂βAγ , ⇒ ǫαβγ ∇2Fβγ = 0 , (4.29)
where Fβγ = ∂βAγ−∂γAβ. The theory is invariant under scaling transformations
when
A′
0 = b1−zA0 , A′
i = Ai . (4.30)
Moreover, it is also gauge invariant (δxν = 0 , δAα = ∂αΛ) , up to a 4th diver-
gence
J µ =(2m)−1
[
−ǫµαν ∇2 Aα∂νΛ + δµ
iǫ
βαν
(
∂βAα ∂
i∂νΛ − ∂i∂βAα ∂νΛ
)]
− δΩ µ
where δΩµ = (2m)−1 ǫµαν Aα ∇2 ∂νΛ .
86 Symmetry Analysis for Anisotropic Field Theories
4.4. Discussion
The principal goal of this chapter was to study an anisotropic theory which can
be applied to several systems where the Lorentz symmetry is broken. Noether’s
theorem for actions with an anisotropic dependence on time and space was ap-
plied. The general form of the conserved currents and charges for arbitrary sym-
metry transformations was obtained. In particular, for the case of translations,
the canonical energy-momentum tensor was presented.
An interesting result was that the usual Noether’s charge (when the action
depends only on the first space-time derivatives) is not modified even if the La-
grangian depends on additional higher order spatial derivatives.
We made the symmetry analysis for three anisotropic systems, for the free
scalar field and electrodynamics we studied translation and scaling invariance
and for the Chern-Simons theory scaling and gauge transformations. With the
method presented here, it is possible to give the explicit form of the conserved
currents.
5
Path Integrals and Polymer
Quantum Mechanics
5.1. Introduction
Loop quantum gravity is a theory that attempts to merge quantum mechanics
and general relativity. The theory is based on quantum geometry, the essential
discreteness of which permeates all constructions and results. The fundamental
excitations are 1-dimensional and polymer-like.
Polymer quantum mechanics is an alternative formalism of quantum mechan-
ics that supports the idea of the existence of a minimal length scale µ, known as
the polymer length, which enters into the Hamiltonian of the system to deform
its functional form into a new one called the polymer Hamiltonian. Since no
discreteness of space has been detected, this fundamental length must be much
smaller than the natural length parameters associated to a mechanical system.
In contrast to the standard Schrödinger picture, this representation gives an ul-
traviolet cutoff due to the existence of the polymer length as a possible minimum
length scale for the system under consideration. [26,107–110]
Polymer quantization was re-introduced by Ashtekar, et. al. [26] in the
context of Loop quantum cosmology to construct a scheme not equivalent to
Schrödinger’s representation. This formalism has been successfully applied to
the quantization of several cosmologies, avoiding the big bang singularity, but
also for quantum theories of one degree of freedom like the free particle, a parti-
cle in a box [111], the harmonic oscillator [112], as well as some other potentials.
For such quantization the first step is to build a representation of the Weyl al-
gebra on a polymer Hilbert space. The next step is to implement a Hamiltonian
constraint on this space which can not be done directly because of the discreet-
87
88 Path Integrals and Polymer Quantum Mechanics
ness of space, so the procedure introduces a length scale (apart from Planck’s
constant) into the quantum theory.
It is interesting to study the polymer version of different simple toy systems
in order to learn about some of the properties of the loop quantization such as
the quantum constraints, the structure of the physical Hilbert space, the path
integral, the quantum solutions, the effective action and properties of the observ-
ables.
This chapter is organized as follows. In section 5.2 we summarily review the
polymer Quantum Mechanics scheme introducing the Weyl–Heisenberg algebra.
Section 5.3 contains the construction of the path integral for the polymer
quantization. This quantization procedure has the advantage of being quite easy
generalized to the polymer case and furthermore this scheme provides us with an
effective action that we can use to understand more deeply the non-perturbative
aspects of a dynamical system at classical and quantum levels. Besides, it is pos-
sible to analyze carefully the measure of the integral, and we found the interesting
result that the measure for the spatial momenta is modified with a final bounded
integral.
We study the polymer version of several systems: the non-relativistic free
particle, the relativistic particle, the harmonic oscillator and cosmology. We
found some interesting properties in the phase space of this systems such as non
trivial restrictions over the energy and the phase space variables.
5.2. Polymer Quantum Mechanics
In this section we are going to describe the Weyl-Heisenberg algebra which we
are going to use to construct a new representation known as polymer representa-
tion (which is not unitarily equivalent to Schödinger’s representation) [26,111].
In Schrödinger’s representation the Hilbert space HSch is the space of the
quadratically integrable functions (HSch = L2(Re, dx)) and the action of this
operators is given by
Ûλ ψ(x) = eiλ xψ(x) and V̂µ ψ(x) = ψ(x+ µ) , (5.1)
for all ψ ∈ HSch. The unitary groups Ûλ and V̂µ (which depend only in one
parameter) are weakly continuous in λ and µ. This property assures that the
autoadjoint operators x̂ and p̂ exist in HSch such that
Ûλ := eiλ x̂ and V̂µ := ei
µ
~
p̂ . (5.2)
Now, the Stone–von Neumann theorem stablished that any irreducible represen-
tation with commutation relations Ûλ V̂µ = eiλµ V̂µ Ûλ is unitarily equivalent to
5.2 Polymer Quantum Mechanics 89
Schrödinger’s representation (5.1), so all the measures dx that can be obtain from
the action of the operators Û and V̂ are equivalent [113].
The polymer representation of the Weyl–Heisenberg algebra is not equivalent
to Schrödinger’s representation because it violates one of the assumptions of the
Stone–von Neumann theorem, in the new representation, the operator V̂µ is not
weakly continuos in µ due to the discrete structure of the space, therefore the
autoadjoint operator p̂ which satisfies V̂µ := e
i
~
µp̂ does not exist (in other words,
the operator V̂µ is not related with an Hermitian operator as an infinitesimal
generator). This is what we expected from the absence of an spatial continuous,
because if the Riemannian geometry is discrete we do not expect that the operator
p̂ = −i~ d
dx exists in a fundamental level.
The central difference between Schrödinger and polymer quantization is the
choice of a non-separable Hilbert space HPoly. A Hilbert space with an uncount-
able orthonormal basis characterized by kets |xj〉 labelled by real numbers xj .
In this representation {|xj〉} is an orthonormal base, and it’s internal product is
defined as
〈xi|xj〉 =
µ
2π~
∫ π~
µ
−π~
µ
dpi e
i
~
pi (xi−xj) = δxi, xj , (5.3)
note that in the right side we have a Kronecker delta instead of a Dirac delta.
The action of the basic operators Û and V̂ is given by
Ûλ |xj〉 = eiλ xj |xj〉 and V̂µ |xj〉 = |xj − µ〉 , (5.4)
in this case Ûλ is weakly continuous in λ and therefore the autoadjoint operator
x̂ exists in HPoly with Ûλ = eiλ x̂. This action can be written as x̂|xj〉 = xj |xj〉.
However there is an important difference with respect with Schrödinger’s rep-
resentation, now the eigenkets of x̂ are normalizable and also the elements of the
Hilbert space. In this sense the eigenvalues are 'discrete'.
On the other hand, although the family V̂µ has an unitary group HPoly, the
operator is not weakly continuous in µ. This is because no matter how small is
µ, |xj〉 and V̂µ |xj〉 are always orthonormal
lim
µ 7→0
〈xj |V̂µ |xj〉 = 0 . (5.5)
The commutation relation between these operators is
[
x̂, V̂µ
]
= −µV̂µ . (5.6)
90 Path Integrals and Polymer Quantum Mechanics
To define the analogue of the Schrödinger momentum operator we expand
e− i
~
µ p (since µ
~
p is small)
p =
~
µ
e
i
~
µ p − e− i
~
µ p
2i
+ O
(
µ2
~2
p2
)
, (5.7)
so we define the operator momentum in a similar way choosing µ small and as a
fix scale and we define the operator in HPoly as
p̂ ≈ ~
µ
V̂µ − V̂ †
µ
2i
. (5.8)
Therefore, the square of the momentum operator will be simply
p̂2
x ≈ − ~
2
4µ2
(
V̂ 2
µ + V̂ † 2
µ − V̂µ V̂
†
µ − V̂ †
µ V̂µ
)
, (5.9)
and since the action of the operator V̂µ over a spatial state |xn〉 produces a shift
on it as shown in eq. (5.4), the action of p̂2
x will be
p̂2
x |xn〉 =
~
2
4µ2
(2|xn〉 − |xn − 2µ〉 − |xn + 2µ〉) . (5.10)
5.3. Path Integrals for Polymer Quantization
In this section we will imitate Feynman’s formalism using the evolution op-
erator e
i
~
λ Ĝ and integrating over λ in the context of the polymer quantization in
order to find the matrix elements of the evolution operator.
Let us remember that in the polymer construction the solutions of the con-
straint equation are decomposed in sectors in which the wave functions are only
defined in grids of the spatial variable x (in other words x has only discrete
values), so we need to impose that x = nµ where n is an integer over the grid.
Note that in the usual systems the variables x and p are symmetric (because
both of them are quadratic) so one can think alternatively on x or p as the
configuration variable. Here, we will consider x as the momentum variable of the
particle over a circle, while p is in the interval (0, π ~
µ)).
As in section 2.5, we decompose the evolution in N intervals of length ǫ = 1/N
e
i
~
λ Ĝ =
N
∏
n=1
e
i
~
ǫλ Ĝn , (5.11)
5.3 Path Integrals for Polymer Quantization 91
but this time, the complete base contains a discrete part due to the spatial variable
x
1 =
∫ ∞
−∞
dtn
∞
∑
xn=−∞
|xn, tn〉 〈xn, tn| , (5.12)
and the Kernel will have the form
G(xf , tf ; xi, ti) =
∫ ∞
−∞
dλ
N−1
∏
n=1
[∫ ∞
−∞
dtn
]N−1
∏
n=1
[
∞
∑
xn=−∞
]
N
∏
n=1
〈xn, tn |e i
~
ǫ λ Ĝn |xn−1, tn−1〉 .
(5.13)
In the next sections we will study the polymer cases of the non-relativistic and
relativistic particle, the harmonic oscillator and some cosmologies.
5.3.1. Polymer non-relativistic free particle
The action for a non-relativistic free particle of mass m is
S =
∫
dτ
[
pt ṫ+ px ẋ− λ
(
pt +
1
2m
p2
x
)]
, (5.14)
with the constraint
Ĝ = p̂t +
1
2m
p̂2
x . (5.15)
It is possible to treat independently the spatial and the temporal coordinates.
For the momentum p̂t the propagator will be
〈tn|e− i
~
ǫλp̂t |tn−1〉 =
1
2π
∫ ∞
−∞
dptn exp
[
− i
~
ǫλptn +
i
~
ptn(tn − tn−1)
]
. (5.16)
To compute the spatial propagator we need the definition of the delta function
(5.3) and to apply the action of the square of the spatial momentum operator
over a spatial state (5.10). Using these results we obtain
〈xn|e− i
~
ǫλ
2m
p̂2
x |xn−1〉 =
µ
2π~
∫ π~
µ
− π~
µ
dpxn
exp
[
i
~
pxn
(xn − xn−1) − i
~
ǫ λ~2
2mµ2
sin2 µ pxn
~
]
,
(5.17)
therefore the complete propagator will be
〈xf , tf |e− i
~
ǫ λ Ĝ |xi, ti〉 = δ (λ− (tN − t0))
N
∏
n=1
[
µ
2π~
∫ π~
µ
−π~
µ
dpxn
]
N−1
∏
n=1
[
∞
∑
xn=−∞
]
exp
[
i
~
N−1
∑
n=1
xn(pxn − pxn+1) + H
]
.
(5.18)
92 Path Integrals and Polymer Quantum Mechanics
Since the temporal part is the same as in the usual case, we obtained a Dirac
delta directly from it of the form δ (λ− (tN − t0)). Besides, we defined
H =
i
~
[
−x0 px1 + xNpxN − ǫλ~2
2mµ2
N
∑
n=1
sin2 µ pxn
~
]
. (5.19)
Applying Poisson’s summation formula, we can transform the coordinate sum to a
proper path integral adding an infinite sum with guarantees the cyclic invariance
on the momentum (see [114])
∞
∑
xn=−∞
e
i
~
xn(pxn−pxn+1 ) =
∞
∑
ln=−∞
∫ ∞
−∞
dxne
i
~
[xn(pxn−pxn+1 )+2πxnln] , (5.20)
so the propagator becomes
〈xf , tf |e− i
~
ǫ λ Ĝ |xi, ti〉 = δ (λ− (tN − t0))
N
∏
n=1
[
µ
2π~
∫ π~
µ
−π~
µ
dpxn
]
N−1
∏
n=1


∞
∑
ln=−∞
∫ ∞
−∞
dxn

 exp
{
i
~
N−1
∑
n=1
[
xn(pxn − pxn+1) + 2πxnln
]
+ H
}
.
(5.21)
In the last expression the sums over ln can be absorbed into the variables pn by
extending their range of integration from (−π~/µ, π~/µ) to (−∞,∞). Only in
the last integral µ
2π~
∫
π~
µ
−π~
µ
dpxN this is impossible and we arrive at
〈xf , tf |e− i
~
ǫ λ Ĝ |xi, ti〉 = δ (λ− (tN − t0))
µ
2π~
∫ π~
µ
−π~
µ
dpxN
N−1
∏
n=1
[
µ
2π~
∫ ∞
−∞
dpxn
]
N−1
∏
n=1
[∫ ∞
−∞
dxn
]
exp
{
i
~
N−1
∑
n=1
xn(pxn − pxn+1) + H
}
.
(5.22)
We can simplify the last expression by obtaining N − 1 Dirac deltas on the
momentum to obtain
〈xf , tf |e− i
~
ǫ λ Ĝ |xi, ti〉 = δ (λ− (tN − t0))
µ
2π~
∫ π~
µ
−π~
µ
dpxN exp
{
− i
~
[
ǫNλ~2
4mµ2
(
1 − cos
2µ pxn
~
)
− pxN (xN − x0)
]}
,
(5.23)
5.3 Path Integrals for Polymer Quantization 93
and by making the following definitions
z =
ǫNλ~2
4mµ2
, ζ =
2µ pxn
~
, xN − x0 = 2∆µ , (5.24)
we obtain the integral form of a Bessel function J∆(z)
〈xf , tf |e− i
~
ǫ λ Ĝ |xi, ti〉 = δ (λ− (tN − t0))
1
2π
e−iz
∫ 2π
0
dζei(z cos ζ+∆ζ)
= δ (λ− (tN − t0)) i∆J∆(z)e−iz .
(5.25)
To obtain the Kernel of the non-relativistic free particle in the polymer case, we
just have to integrate over λ
G(xf , tf ; xi, ti) = i
1
2µ
(xN−x0)
J 1
2µ
(xN−x0)
(
~
2
4mµ2
(tN − t0)
)
e
− ~
2
4mµ2 (tn−t0)
,
(5.26)
where we used ǫN = 1.
This is the result obtained by the method of images on [115]. It is necessary to
stress that the residual bounded integral in the measure for the spatial momenta
in (5.23) is needed in order to obtain the right result.
Equations of motion
The last step in the construction of the path integral consists in taking the
limit N → ∞. Generally in this limit we have to make the following substitutions
ǫ
∑N
n=0 → ∫
dτ , (tn − tn−1) /ǫ → dt/dτ and (xn − xn−1) /ǫ → dx/dτ to obtain
a continuos action (over τ). Therefore, the Kernel for the free particle in the
polymer case is
G(xf , tf ; xi, ti) :=
∫
dλ 〈xf , tf |e− i
~
ǫ λ Ĝ |xi, ti〉
=
∫
dλ
∫
[Dx(τ)] [Dt(τ)] [Dpt(τ)] [Dpx(τ)] e
i
~
S ,
(5.27)
where S represents the effective polymer action, in this case is given by
S =
∫ 1
0
dτ
[
ptṫ+ px ẋ− λ
(
pt +
~
2
2mµ2
sin2 µ px
~
)]
. (5.28)
94 Path Integrals and Polymer Quantum Mechanics
The equations of motion of the polymer action for the free particle given in (5.28)
are
ṗt = 0, ṗx = 0 , (5.29a)
ṫ = λ, px =
~
2µ
arcsin
(
2mµẋ
λ~
)
. (5.29b)
Now we can use this equations of motion to rewrite the effective action (using
the redefinition x′ = ẋ/λ)
S =
∫ t2
t1
dt


~
2µ
x′ arcsin
(
2mµ
~
x′
)
− ~
2
4mµ2

1 −
√
1 − 4m2µ2
~2
x′2



 , (5.30)
and the Hamiltonian
H =
∂L
∂x′
x′ − L =
~
2
4mµ2

1 −
√
1 − 4m2µ2
~2
x′2

 =
~
2
2mµ2
sin2
(
µpx
~
)
, (5.31)
in the last step we used that p = ∂L
∂x′ = ~
2µ arcsin(2mµ
~
x′). Since the energy H is
constant, we write the velocity x′ in terms of it
x′2 =
2H
m
(
1 − 2mµ2
~2
H
)
. (5.32)
Figure 5.1
Velocity graph. Note that both the velocity and the energy are bounded.
5.3 Path Integrals for Polymer Quantization 95
In order that the velocity stay well defined, lets note that the values of the
energy are restricted by
0 < H <
~
2
2mµ2
, (5.33)
in other words, the velocity of the free particle has a limit
− ~
2mµ
< x′(H) <
~
2mµ
, (5.34)
when the energy takes the values H = 0 or H = ~
2
2mµ2 the velocity of the particle
is zero; on the other hand when the energy is equal to half of it’s maximum value
H = ~
2
4mµ2 the velocity is maximum as we can see in the fig. 5.1.
We can integrate the velocity eq. (5.32) to obtain the solutions for the system
tpoly(x) = ± 1
√
2H
m (1 − 2mµ2
~2 H)
x+ t0poly , (5.35)
where t0poly is a constant.
In the continuous limit, when µ → 0, the energy has to be positive H > 0
and the velocity is unbounded, in other words we recover the case of the classical
non-relativistic free particle. Furthermore, in this limit, the polymer trajectories
tpoly±
would be the classical solutions tclas(x) = ±
√
m
2H x+ t0.
5.3.2. Polymer relativistic particle
The constraint in the case of the relativistic particle is
Ĝ =
1
2
(
1
c
p̂µp̂
µ +m2c
)
, (5.36)
let us assume that only the spatial variables are discrete, while the temporal
variable remains continuous, we are only going to consider a space-time in D =
(1, 1) dimensions with metric (−,+). Then, we can compute the propagator
〈x1
f , x
0
f |e− i
~
ǫλĜ |x1
i, x
0
i〉 (where the first index denote the space-time component
96 Path Integrals and Polymer Quantum Mechanics
and the second index the evolution partition) as follows
〈x1
f ,x
0
f |e− i
~
ǫλĜ |x1
i, x
0
i〉 =
=
N−1
∏
n=1
[∫ ∞
−∞
dx0
n
]N−1
∏
n=1


∞
∑
x1
n=−∞


N
∏
n=1
〈x1
n, x
0
n|e− i
~
ǫλĜn |x1
n−1, x
0
n−1〉
=
N−1
∏
n=1
[∫ ∞
−∞
dx0
n
] N
∏
n=1
[
1
2π~
∫ ∞
−∞
dp0
n
]N−1
∏
n=1


∞
∑
x1
n=−∞


N
∏
n=1
[
µ
2π~
∫ π~
µ
−π~
µ
dp1
n
]
exp
{
i
~
N
∑
n=1
[
−ǫλ1
2
(
1
c
p0,np
0
n +
~
2
cµ2
sin2 µp
1
n
~
+m2c
)
+p0,n
(
x0
n − x0
n−1
)
+ p1,n
(
x1
n − x1
n−1
)
]}
.
(5.37)
We will use here all the tricks that we learned from the previous example. For
the temporal coordinate it is possible to perform all the integrals except for the
last one over the momentum p0
N . For the spatial coordinate first we convert all
the summations into integrals over x1
n using Poisson’s summation formula given
on eq. (5.20) which will add an extra infinite sum. We will use these additional
sums to extend the range of integration on the spatial momentum variables p1
n.
We can do this except for p1
N so this integral will remain bounded. Therefore
the Kernel will have the form
G(x1
f , x
0
f ;x1
i, x
0
i) =
∫ ∞
−∞
dλµ
(
1
2π~
)2 ∫ ∞
−∞
dp0
N
∫ π~
µ
−π~
µ
dp1
N
e
− i
~
[
ǫNλ
2c
(
p0,Np
0
N
+~
2
µ2 sin2 µp1
N
~
+m2c2
)
−p0,N (x0
N
−x0
0)−p1,N (x1
N
−x1
0)
]
.
(5.38)
Now we can integrate over λ to obtain a Dirac delta
G(x1
f , x
0
f ;x1
i, x
0
i) = µ
(
1
2π~
)2 (4π~c
ǫN
)∫ ∞
−∞
dp0
N
∫ π~
µ
−π~
µ
dp1
N
δ
(
p0,Np
0
N +
~
2
µ2
sin2 µp
1
N
~
+m2c2
)
e
i
~
[p0,N (x0
N
−x0
0)+p1,N (x1
N
−x1
0)] .
(5.39)
5.3 Path Integrals for Polymer Quantization 97
Finally, we integrate over p0
N and take ǫN = 1 to obtain
G(x1
f , x
0
f ;x1
i, x
0
i) =
2µc
π~
∫ π~
µ
−π~
µ
dp1
N
e
i
~
p1,N (x1
N
−x1
0)
2
√
~2
µ2 sin2 µp1
N
~
+m2c2



e
i
~
(x0
N
−x0
0)
√
~2
µ2 sin2
µp1
N
~
+m2c2
+ e
− i
~
(x0
N
−x0
0)
√
~2
µ2 sin2
µp1
N
~
+m2c2



.
(5.40)
Equations of motion
Taking the limit N → ∞ and substituting ǫ
∑N
n=0 → ∫
dτ ,
(
x0
n+1 − x0
n
)
/ǫ →
dx0/dτ and
(
x1
n+1 − x1
n
)
/ǫ → dx1/dτ we get
G(x1
f , x
0
f ; x1
i , x
0
i ) :=
∫
dλ 〈xf , tf |e− i
~
ǫ λG |x1
i , x
0
i 〉
=
∫
dλ
∫
[
Dx0(τ)
] [
Dx1(τ)
] [
Dp0(τ)
] [
Dp1(τ)
]
e
i
~
S ,
(5.41)
where the effective action is given by
S =
∫ 1
0
dτ
[
p0ẋ
0 + p1 ẋ
1 − λ
2
(
1
c
p0p
0 +
~
2
cµ2
sin2 µp
1
~
+m2c
)]
. (5.42)
The equations of motion corresponding to the effective action of the relativistic
particle (5.42) would be
ṗ0 = 0, ṗ1 = 0 , (5.43a)
p0 =
c
λ
ẋ0, p1 =
~
2µ
arcsin
(
2cµẋ
λ~
)
, (5.43b)
substituting the equations of motion (5.43a) in the action (5.42), using the metric
(−,+) and choosing the coordinates (x0, x1) = (c t, x), this action has the form
S =
∫
dτ
[
− c3
2λ
ṫ2 +
~
2µ
ẋ arcsin
(
2cµ
λ~
ẋ
)
− λ~2
4cµ2

1 −
√
1 − 4c2µ2
λ2~2
ẋ2

− 1
2
λm2c

 ,
(5.44)
98 Path Integrals and Polymer Quantum Mechanics
now we are going to eliminate the Lagrange multiplier λ with it’s equation of
motion
− cm2
2
+
c3
2λ2
ṫ2 − ~
2
4cµ2

1 −
√
1 − 4c2µ2
λ2~2
ẋ2

 = 0 , (5.45)
this is a fourth order equation in λ so there are four solutions for the Lagrange
multiplier, fixing ṫ = τ we get
λ = ±
√
√
√
√
c2 + 2c4m2µ2
~2 − ẋ2 ±
√
c4 − 2c2ẋ2 + ẋ4 − 4c4m2µ2
~2 ẋ2
2(m2 + c2m4µ2/~2)
, (5.46)
once more the velocity values are bounded, we can find that the velocity must
satisfy that
ẋ2 ≤ c2 +
2c4m2µ2
~2
− 2
√
c6m2µ2
~2
+
c8m4µ4
~4
, (5.47)
now we can get µ in terms of the maximum velocity ẋmax = ν (remember that µ
is the length of the grid and it has to be positive µ > 0)
µ =
~(c2 − ν2)
2c2mν
. (5.48)
For a proton accelerated in the LHC at 7 TeV and substituting the values of its
mass, the speed of light and Planck’s constant
c = 299 792 458 m/s ,
~ = 1.054 571 726 × 10−34 J · s ,
m = 1.672 621 777 × 10−27 kg ,
(5.49)
we obtain for the length of the grid
µ = 1.889 × 10−24 m, (5.50)
this is 1011 lP where lP is Plank’s length (lP = 1.616 199 × 10−35 m).
5.3.3. Polymer Harmonic Oscillator
We will now show how to quantize the polymer harmonic oscillator using the
path integral, the action for the parameterized system is
S =
∫
dτ
[
pt ṫ+ px ẋ− λ
(
pt +
1
2m
p2
x +
k
2
x2
)]
, (5.51)
5.3 Path Integrals for Polymer Quantization 99
with the constraint given by
G = pt +
1
2m
p2
x +
k
2
x2 ≈ 0 . (5.52)
In the polymer bra-ket space we can evaluate the harmonic potential, we thus
get
〈xn
∣
∣
∣
k
2
x2
∣
∣
∣xn−1〉 =
k
2
xnxn−1δxn,xn−1
=
k
2
xn xn−1
µ
2π~
∫ π~
µ
−π~
µ
dpxn e
i
~
pxn (xn−xn−1) ,
(5.53)
in consequence for the spatial part of the constraint we obtain
〈xn|e− i
~
ǫλ( 1
2m
p2
x+ k
2
x2) |xn−1〉 =
= 〈xn|1 − i
~
ǫ λ
(
~
2
2mµ2
sin2 µ px
~
+
k
2
x2
)
|xn−1 〉
=
µ
2π~
∫ π~
µ
−π~
µ
dpxn e
i
~
pxn (xn−xn−1)− i
~
ǫ λ
(
~
2
2mµ2 sin2 µ pxn
~
+ k
2
xn xn−1
)
.
(5.54)
Therefore, using the expressions (5.54) and (5.16) we get for the path integral
〈xf , tf |e− i
~
ǫ λG |xi, ti〉 =
=
N−1
∏
n=1
[∫ ∞
−∞
dtn
]N−1
∏
n=1
[
1
2π
∫ ∞
−∞
dptn
]
e
i
~
ǫ
∑N
n=1
(
−λ ptn+ ptn
tn−tn−1
ǫ
)
N−1
∏
n=1
[
∞
∑
xn=−∞
]
N−1
∏
n=1
[
µ
2π~
∫ π~
µ
−π~
µ
dpxn
]
e
i
~
ǫ
∑N
n=1
(
pxn
xn−xn−1
ǫ
− λ~2
2mµ2 sin2 µ pxn
~
− kλ
2
xnxn−1
)
.
(5.55)
Taking the limit N → ∞ and using ǫ
∑N
n=0 → ∫
dτ , (tn − tn−1) /ǫ → dt/dτ ,
(xn − xn−1) /ǫ → , dx/dτ . Then, we conclude
G(xf , tf ; xi, ti) =
∫
dλ〈xf , tf |e− i
~
ǫ λG |xi, ti〉
=
∫
dλ
∫
[Dx(τ)] [Dt(τ)] [Dpt(τ)] [Dpx(τ)] e
i
~
S ,
(5.56)
where
S =
∫ 1
0
dτ
[
ptṫ+ px ẋ− λ
(
pt +
~
2
2mµ2
sin2 µ px
~
+
k
2
x2
)]
. (5.57)
100 Path Integrals and Polymer Quantum Mechanics
In this form, we get the Kernel of the polymer harmonic oscillator. To deduce the
corresponding Schrödinger equation, we use the relations sin2 θ = 1
2(1 − cos 2θ),
k = mω2 and taking into account that the system is quadratic in the potential
we use the px representation. Furthermore, we assume that from the effective
action (5.57), we can read the effective constraint, then the time independent
Schrödinger equation is
E ψ =
~
2
4mµ2
(
1 − cos
2µ px
~
)
ψ − mω2
~
2
2
∂2 ψ
∂p2
x
. (5.58)
Note that if we would want to define de quantum amplitude we should take into
account in 5.56 that the momenta are fixed, while the coordinates are not, as it
was done in [116]. Next, let us define
u =
µpx
~
+ π/2, g = mωµ, α =
2µE
gω~2
− 1
2g2
. (5.59)
Then, the eq. (5.58) is transformed in Mathieu’s differential equation. This
equation has been studied in several works, see for example [30,117], and reference
therein. In this work, they obtained periodic solutions for special values of α
ψ2n(u) = π−1/2cen(1/4g2, u), α = An(g) ,
ψ2n+1(u) = π−1/2sen+1(1/4g2, u), α = Bn(g) .
(5.60)
We denote by cen and sen (n = 0, 1, . . .) the cosine and sine elliptic functions
respectively. For a pendulum with a single potential well m=1 [117]. Note that
only even order ce and se, functions occur. An and Bn correspond to Mathieu
characteristic value functions. for n even cen and sen are π-periodic, but for n
odd are 2π-antiperiodic. Explicitly, the eigenvalues corresponding to the periodic
eigenfunctions (5.60) are:
E2n
ω
=
2g2An(g) + 1
4g
,
E2n+1
ω
=
2g2Bn+1(g) + 1
4g
. (5.61)
Dynamical analysis
In this subsection we want to study the full dynamics of the system in order
to try to find non- pertubative solutions to the action (5.57). The equations of
motion are
ṗt = 0, ṗx = −λ k x , (5.62a)
ṫ = λ , px =
~
2µ
arcsin
(
2mµẋ
λ~
)
. (5.62b)
5.3 Path Integrals for Polymer Quantization 101
Using the eq. (5.62a) and (5.57) and defining the new time x′ = ẋ/λ to de-
parametrize the action we get
S =
∫
dt


~
2µ
x′ arcsin
(
2mµx′
~
)
− ~
2
4mµ2

1 −
√
1 − 4m2µ2
~2
x′2

− k
2
x2

 .
(5.63)
From this expression we derive the effective Hamiltonian
H =
∂L
∂x′
x′ − L =
~
2
4mµ2

1 −
√
1 − 4m2µ2
~2
x′2

− k
2
x2 . (5.64)
It is clear that the values of the velocity x′are bounded by the same limit as in
the free particle case
− ~
2mµ
< x′ <
~
2mµ
. (5.65)
Now, as H is a constant of motion, we can solve for x′
x′2 =
1
m
[
2H
(
1 − 2mµ2
~2
H
)
+ k
(
4mµ2
~2
H − 1
)
− mµ2k2
~2
x4
]
, (5.66)
in consequence we obtain the complete evolution of the system
t = ±
√
m
k(1 − 2mµ2
~2 H)
EllipticF

arcsin


√
k
2H
x

 ,
2mµ2
~2 H
2mµ2
~2 H − 1

+ t0 . (5.67)
To simplify the dynamical analysis we write the system in the phase space. In
terms of the momentum p = ∂L
∂x′ = ~
2µ arcsin(2mµ
~
x′) the Hamiltonian will be
H =
~
2
2mµ2
sin2 µp
~
+
k
2
x2, (5.68)
solving for p from (5.68) results
p = ±~
µ
arcsin
√
2mµ2
~2
(
H − k
2
x2
)
. (5.69)
Now, using as configuration variable the momentum p, the system evolves under
a potential V (p) = ~
2
2mµ2 sin2 µ p
~
. Solving eq. (5.69) for the “momentum variable"
x we get
x = ±
√
1
km
(
2mH − ~2
µ2
sin2 pµ
~
)
. (5.70)
102 Path Integrals and Polymer Quantum Mechanics
Figure 5.2
Graphic in the phase space. The dashed line corresponds to the separatrix of the movement
given by H = ~
2
2mµ2 .
So we see that the potential is periodic in p (with period π~
µ ). Furthermore, it
is always positive and corresponds to a pendulum. Then, the motion in p, and
the separatrix divide the phase space in two regions with phase space curves of
different characteristics, in one region we get libration and in the other rotation
(Figure 5.2). In the case of libration the system approaches to a standard har-
monic oscillator this limit corresponds to g = 0, i.e. µ = 0, in this limit the
quantum states are equally spaced. The case g ≫ 0 ∼ µ ≫ 0 corresponds to a
rigid rotor limit and here we have quadratically spaced energy levels.
On the other hand, the fixed points are found in p = n ~
2µ arcsin(1), where
n ∈ Z; for n even correspond to minima and n odd correspond the maxima of
the potential. In the region of libration, x is found between x2 < 2H
k , whereas p
is bounded between p2 < ~
2
µ2 arcsin2
√
2mµ2
~2 H.
In the rotation region, if n is even, x = +
√
2H
k or x = −
√
2H
k . For n odd we
get that x = +
√
2H
k − ~2
mkµ2 or x = −
√
2H
k − ~2
mkµ2 .
Using the Hamiltonian equation, with Hamiltonian (5.68) we get
ṗ =
∂H
∂x
= kx = ±
√
k
m
(
2mH − ~2
µ2
sin2 µp
~
)
. (5.71)
In the case of libration, with initial conditions given by
tpoly
L

p0 = ±~
µ
arcsin
√
2mµ2
~2
H

 = 0 , (5.72)
5.3 Path Integrals for Polymer Quantization 103
we obtain
tpoly
L
=
~
µ
√
1
2kH
{
±EllipticF
[
pµ
~
,
~
2
2Hmµ2
]
±(4n∓ 1)EllipticF

arcsin
√
2mµ2
~2
H,
~
2
2mµ2H





.
(5.73)
In the case of rotations, for the initial condition
tpolyR (p = 0) = 0 , (5.74)
results
tpoly
R
=
~
µ
√
1
2kH
{
±EllipticF
[
pµ
~
,
~
2
2Hmµ2
]}
+ t0
R
. (5.75)
Figure 5.3
Solutions for the p variable. The upstairs left plot corresponds to the solutions inside the
libration region, the upstairs right plot corresponds to the rotation region and the downstairs
plot correspond to the separatrix solution
.
From figure 5.3 we observe that in the case of libration the movement in p
is bounded and oscillatory. Whereas, in the case of rotation the movement is
104 Path Integrals and Polymer Quantum Mechanics
not bounded but is still periodic (5.3.3). The case H = ~
2
2mµ2 corresponds to
the separatrix, in the quantum case this solution corresponds to the quantum
tunneling and we observe from the (5.3.2) that this solution has the form of a
kink, but we must remember that this solution corresponds to the p variable
and the must be interpreted carefully It will be quite interesting to see if for the
polymer quantum mechanics this kind of solutions can be interpreted as a map
between a coherent state from the oscillatory side to a squeezed pendular state
in the rotor side. This is the case in the quantum pendulum [117]. Furthermore
it will be interesting to analyze the consequence of this kind of solutions in the
case of quantum cosmology.
5.3.4. Polymer Cosmology
Loop quantum cosmology is the application of loop quantum gravity to cos-
mological models. In loop quantum cosmology the geometric observables display
a fundamental discreteness causing that the big bang is replaced by a quantum
bounce (the quantum Hamiltonian constraint of the theory does not break down
when the scale factor vanishes and the classical singularity occurs replacing the
big bang by a quantum bounce).
Almost all phenomenological work in cosmology is based on the flat, homo-
geneous and isotropic Friedmann Robertson Walker (FRW) space-times and per-
turbations thereof. To construct a Hamiltonian formulation of the FRW models
is necessary to introduce a finite elementary cell V [118] in the space with topol-
ogy Re3 and restrict all integrations to it. It is easiest to fix a fiducial flat metric
q̊abdx
adxb = dx2
1 +dx2
2 +dx2
3 (where xa are Cartesian coordinates), that is related
with the physical 3-metric by qab = a2̊qab and a is the scale factor. The volume of
the cell V with respect to the metric q̊ab will be denoted by V0 and the physical
volume will be just V = a3V0.
The standard canonical pairs are (a, pa) for geometry and (φ, p) for the scalar
field. To compare with LQC results [69] it is convenient to replace the scalar
factor a by a variable ν which is proportional to the volume of the cell V and the
conjugate momentum pa by a new variable b defined as
ν =
a3V0
2πG
, b = −4πG
3V0
pa
a2
. (5.76)
With this replacement ν has the dimension of length and it ranges over (−∞,∞)
and b has dimensions of inverse length. Their Poisson bracket is then given by
{ν, b} = −2 while for the matter phase space {φ, p} = 1.
We will discuss in some detail the simplest cosmological space-time, the spa-
tially flat FLRW model with cosmological constant zero (k = 0, Λ = 0). The
5.3 Path Integrals for Polymer Quantization 105
effective action for this model obtained after applying the path integral formu-
lation that we studied in the previous sections, is given in [119] and it takes the
form
S =
∫ 1
0
dτ
{
p φ̇− 1
2
b ν̇ − λ
[
p2 − 3πG~2
µ2
ν2 sin2 µ b
~
]}
, (5.77)
where γ is the Barbero-Immirzi parameter of LQC, λ is a Lagrange multiplier
and µ is the length of the grid as in the previous examples. By making the
redefinition η = 3πG, the effective action will take the form
S =
∫ 1
0
dτ
{
p φ̇− 1
2
b ν̇ − λ
[
p2 − η~2
µ2
ν2 sin2 µ b
~
]}
. (5.78)
In here, we will study the dynamics of the system by analyzing the phase space,
so first we find that the equations of motion will be
ṗ = 0 , ḃ = −4λ νη~2
µ2
sin2 µ b
~
,
φ̇ = 2λ p , b =
~
2µ
arcsin
(
µ ν̇
2λη~ ν2
)
.
(5.79)
Now we substitute these equations in the action (5.78), we make the identification
λ = φ̇ and use that ν ′ = ν̇/λ to obtain
S =
∫
dφ



1
4
− ~ ν ′
2µ
arcsin
(
µ ν′
2η~ ν2
)
+ ν2 η~
2µ2

1 −
√
1 − µ2 ν ′2
4η2~2 ν4





.
(5.80)
Writing the Hamiltonian in terms of the momentum b = ∂L
∂ν′ = − ~
4µ arcsin
(
µν′
2η~ ν2
)
gives H = −ν2 η~2
µ2 sin2 2µ b
~
. If we consider that the energy is constant from (5.3.4)
we get
ν(b) = ±
√
√
√
√
−H
η~2
µ2 sin2 2µ b
~
. (5.81)
From here we see that there are different restrictions over the energy H and the
values of ν are bounded.
When the universe is flat, there are two types of classical trajectories: one
in which the universe begins with a big bang and expands (eternally expanding
universe) and another one when it contracts into a big crunch (eternally contract-
ing universe). However, for the LQC evolution the quantum solution follows the
106 Path Integrals and Polymer Quantum Mechanics
classical one at low densities and curvatures (LQC has good infrared behavior)
but instead of falling into the singularity undergoes a quantum bounce and joins
on to the classical trajectory that was expanding to the future.
The expansion volume will be given then by
ν = ±
√
√
√
√−µ2H
η~2
1
sin2 2µ b
~
. (5.82)
From here we can see the periodicity of the velocity of expansion b and that the
values of H and ν are restricted by H < 0, and ν /∈
[
−
√
−µ2H
η~2 ,
√
−µ2H
η~2
]
. In the
classical this result will be
νclas = ±
√
− H
η b2
clas
, (5.83)
therefore if we go back in time when the expansion velocity increases, then the
volume of the universe goes to zero into the singularity. However, in the polymer
case, when the expansion velocity increases, the volume bounce intend of going
into the singularity and reaches the minimal value
√
−µ2H
η~2 .
Figure 5.4
Phase space diagram in the case k = 0.
5.4 Discussion 107
5.4. Discussion
In the frame of the polymer representation, we obtained the propagators of the
non-relativistic and relativistic particle, the harmonic oscillator and Friedmann
Robertson Walker cosmologies in the cases of flat and positive curvatures.
One of the most important characteristics of the group averaging procedure
that is not commonly address in the literature and that we stress in this work,
is that it gives us the correct measure of the path integral, as we saw there
is always a remaining integration over the momentum even without taking into
account the polymer representation. This result is not relevant when one analyzes
the effective actions of the systems, but is crucial to take it into account in order
to compute the exact propagator.
Since in quantum field theory the effective action is an important tool for the
investigation of properties of the theories such as non-perturbative effects it was
interesting to take it as our starting point in the analysis of the dynamics of our
systems. The first interesting feature that we found, is that the velocity of the
particle in the systems is bounded. Furthermore, in the relativistic case we found
that this velocity is always smaller than the speed of light.
6
Non Relativistic Limit
6.1. Introduction
Non-relativistic theories have been interesting tools in theoretical physics in
recent years. The main motivation is to widen the applications of the conjectured
anti-de Sitter/conformal field theory (AdS/CFT) correspondence and to test if it
also holds away from its original relativistic setting. AdS/CFT correspondence
can be used to understand strongly interacting field theories by mapping them
to classical gravity. In the study of collective phenomena in condensed matter
physics it is quite common to observe this strong-weak coupling duality since a
strongly coupled system reorganizes itself in such a way that new weakly coupled
degrees of freedom emerge dynamically and the system can be better described
in terms of fields representing the emergent excitations. There are reasons to
expect that non-relativistic holography has applications in effective descriptions
of strongly correlated condensed matter systems [39, 120–123]; for a review, see
e.g. [124]. Usually, one only considers a non-relativistic setting at the boundary
but one might also consider non-relativistic models both in the bulk and at the
boundary [40]. Originally, non-relativistic superstring theories and superbranes
were studied as special points in the parameter space of M-theory with non-
relativistic symmetries [125, 126]. Non-relativistic strings were also thought as a
possible soluble sector within string theory or M-theory [127, 128]. This latter
expectation was based on a similar experience with the pp-wave/BMN limit [129].
Furthermore, a formulation of non-relativistic gravity that is invariant un-
der diffeomorphisms was introduced by Cartan [130], see also [131–135]. This
so-called Newton-Cartan gravity can be reformulated as a gauge theory of the
Bargmann algebra [45, 136]. The interest in Galilean-invariant theories with dif-
109
110 Non Relativistic Limit
feomorphism invariance has increased recently due to their relation with con-
densed matter systems [38, 137, 138], see also [139, 140] and references therein.
Galilean-invariant theories have also appeared recently in studies of Lifshitz
holography [141,142].
In this chapter we make a first step in improving our knowledge on non-
relativistic gravity and particle/string/brane theories that underlie a holography
with a non-relativistic setting at both the boundary and the bulk. In this work
we will consider supersymmetric theories and we will restrict to particles, or more
precisely superparticles [143,144], only.
Non-relativistic N = 2 massive superparticles in ten dimensions in a flat
background have already been studied in [127]. In this chapter we wish to extend
this analysis and consider superparticles in a curved non-relativistic supergrav-
ity background. It should be stressed that, independent of the relation with
non-relativistic holography, there is not much literature on non-relativistic su-
persymmetry; however, see e.g. [145–151]. This in itself provides ample reason to
investigate this topic.
Part of this chapter consists of a review of known results on non-relativistic
superparticles in a flat background, we will use the non linear realizations method
to study the dynamics and the symmetries of these particles. To the best of our
knowledge all non-relativistic superparticle actions with background fields are
new.
In the construction of massive (super-)particle actions an important role is
played by symmetries, both global and local ones. In particular, it is well-known
that relativistic massive superparticles have an infinitely-reducible gauge sym-
metry, called κ-symmetry [152, 153], that eliminates half of the fermions. In the
non-relativistic setting this κ-symmetry corresponds to a fermionic gauge shift
symmetry, i.e. a Stückelberg symmetry [127].
When discussing the symmetries of (super-)particles in a curved background
it is important to distinguish between ‘proper’ and ‘sigma model’ symmetries
[154,155]. In the case of proper symmetries the background (super)gravity fields
only transform through their dependence on the embedding coordinates of the
(super)particle. On the other hand, in the case of sigma model symmetries
the background fields have their own transformation rules. We will clarify how
these sigma model transformations are related to the transformations of the (su-
per)gravity fields when viewed as the components of a (super)gravity multiplet
defined in the target space.
To explain our construction of non-relativistic superparticles in a curved back-
ground, we will first consider the bosonic case and take as our starting point a
single massive non-relativistic particle in a flat background. The action of such
6.1 Introduction 111
a particle is invariant under the (global) Galilei symmetries, hence the name
‘Galilean’ particle. We next partially gauge the spatial target space translations
of the Galilean particle such that the constant parameter of a spatial translation
is promoted to an arbitrary function of time. The resulting extended symmetries
are sometimes called the ‘acceleration-extended’ Galilean symmetries. In order
to achieve this partial gauging the particle must move in a curved Galilean back-
ground that is characterized by the Newton potential Φ [156].1 We will call this
particle a ‘Curved Galilean’ particle. Finally, we perform a full gauging of the
Galilean symmetries such that the parameter of spatial translations becomes an
arbitrary function of the spacetime coordinates and the full symmetries of the
particle action are the general coordinate transformations. Actually, to perform
this gauging it is necessary to extend the Galilei symmetries with an additional
central charge transformation2 [45,136,159]. The background is now promoted to
a Newton–Cartan gravity background that is characterized by a spacelike Viel-
bein eµ
a, a timelike Vielbein τµ and a central charge gauge field mµ. We will
call the corresponding particle a ‘Newton–Cartan’ (NC) particle. Figure 6.1 in-
dicates the different backgrounds that we will consider and outlines the how to
move between backgrounds, either by a gauging of symmetries or by gauge fixing
some of the symmetries.
Galilean
Curved
Galilean
Newton
Cartan
G
F
G
F
Figure 6.1
This figure displays the different backgrounds used in this chapter. For each background,
section 6.4 discusses the bosonic case while section 6.5 treats the supersymmetric case. The
upper arrows marked with an F indicate the direction of gauge-fixing, while the lower arrows
marked with a G indicate the gauging direction.
Next, we will consider the superparticle. This requires a supersymmetric
extension of the gravity backgrounds in the first place. Since non-relativistic su-
1There is also a different way of gauging the spatial translations where the background is
given by a vector field rather than a scalar potential [136, 157].
2In section 6.3, where we discuss supersymmetric particle actions, we will restrict ourselves
to d = 3. In that case, it is known that the Bargmann algebra admits a second central charge,
see e.g. [158]. We will however not consider it in this thesis.
112 Non Relativistic Limit
pergravity multiplets to our knowledge have only been explicitly constructed in
three dimensions, we will only consider superparticles in a three-dimensional (3D)
background. A supersymmetric version of the 3D Galilean and NC backgrounds
was recently constructed by gauging the Galilei, or better Bargmann, superalge-
bra [48]. We will make full use of the construction of [48] which, in particular,
explains how to switch between different backgrounds, with different symmetries,
by partial gauging or partial gauge-fixing. Our aim will be to investigate the
action of a 3D superparticle first in a flat background and, next, in a Galilean
and NC supergravity background with and without a cosmological constant. To
indicate the different cases we will use the same nomenclature as in the bosonic
case but with the word particle replaced by superparticle.
This work is organized as follows. In section 6.2 we go through the known
descriptions of the bosonic Newton-Hooke particle to introduce our notation and
the non linear realizations method to find particle actions. Moreover, we use
the Lagrangian formalism to find the Killing equations of this system to study
the symmetries that is contains. We end this section with comments on the flat
limit. In section 6.4 we discuss the different gaugings and gauge-fixings in a
simple setting. Sections 6.3 and 6.5 are devoted to the supersymmetrization of
the theory discussed in section 6.2.
6.2. The Free Newton-Hooke Particle
This section will serve as a useful warm-up exercise to show the bosonic results
before discussing the superparticle case. We consider a negative cosmological
constant Λ < 0 with the AdS radius given by R2 = −1/Λ 3. In global coordinates
the metric of an AdS spacetime is given by
ds2 = −f(r) dt2 +
1
f(r)
dr2 + r2dϕ2 , f(r) = 1 − Λr2 . (6.1)
By taking the non-relativistic limit of a relativistic particle in such an AdS back-
ground we arrive at the action
S =
∫
dτ
m
2
[
ẋiẋi
ṫ
− ṫ xixi
R2
]
, (6.2)
once a total derivative term is eliminated. Physically, this system is equivalent
to the non-relativistic harmonic oscillator.
3In the following we will express everything in terms of the radius R.
6.2 The Free Newton-Hooke Particle 113
The NH algebra with a central charge extension can be derived as a contrac-
tion of the AdS algebra (2.39) extended to the direct sum of the AdS algebra and a
commutative subalgebra spanned by Z. To make the non-relativistic contraction
we rescale the generators with a parameter ω as follows (a = 1, . . . , D − 1):
P0 = ω Z +
1
2ω
H , Ma0 = ωKa , R = ωR̃ . (6.3)
Taking the limit ω → ∞ and dropping the tilde on the R we get the following
Newton–Hooke algebra:
[
Mab,Mcd
]
= 2ηa[cMd]b − 2ηb[cMd]a ,
[
Jab, (P/K)c
]
= −2 δc[a(P/K)b] ,
[
H,Ka
]
= Pa ,
[
H,Pa
]
= − 1
R2
Ka ,
[
Pa,Kb
]
= δab Z .
(6.4)
The generators
{
H ,Jab , Pa ,Ka
}
generate time translations, spatial rotations,
space translations and boost transformations, respectively. The central extension
Z that leads to the Bargmann version of the NH algebra occurs only in the last
commutator. Physically, the occurrence of the central charge transformations
is related to the fact that at the non-relativistic level the mass of a particle is
conserved.
6.2.1. Non-linear Realizations
In this section we obtain the action and transformation rules for the Newton–
Hooke particles by using the method of non-linear realizations. We will derive the
action and transformation rules of the NH superparticle and afterwards we will
take the flat limit to obtain the results for the Galilean particle, see e.g. [68,160].
The starting point is the AdS algebra with a central extension given in section
2.3. We derive the transformation rules for the coordinates (t, xi, s, vi) using the
coset NH/SO(D − 1) with the coset element g = g0U , where g0 = eHtePix
i
eZs
is the coset representing the ‘empty’ NH space with a central charge extension
and U = eKiv
i
is a general NH boost. This leads to the Maurer–Cartan form Ω0
associated to the NH space
Ω0 = g−1
0 dg0 = H e0 + Pie
i + Zez +Kiω
i0 +Mijω
ij , (6.5)
where (e0, ei, ez) and (ωi0, ωij) are the space, time and central charge components
of the Vielbein and spin-connection 1-forms corresponding to the NH space are
given by
e0 = dt , ei = dxi , ez = ds+
x2
2R2
dt , ωi0 = − dt
R2
xi , (6.6)
114 Non Relativistic Limit
where x2 = xixi. By inserting a particle in the empty NH space we obtain the
Maurer-Cartan form of the combined system
Ω = g−1dg = U−1Ω0U+U−1dU = H LH +PiL
i
P +ZLZ +KiL
i
K +MijL
ij , (6.7)
where the explicit 1-forms are given by
LH = e0 , LiP = ei + vi e0 , LZ = ez + vi ei +
1
2
v2 e0 , LiK = ωi0 + dvi .
(6.8)
Note that we can obtain the Maurer–Cartan forms of the NH space by a matrix
representation of the NH boost.
(
LH , LP
i, LZ
)
=
(
e0, ei, ez
)



1 vi v2
2
0 1 vi
0 0 1



. (6.9)
The action of the NH particle is given by the pull-back of all L’s that are invariant
under rotations, hence
S = a
∫
(LH)∗ + b
∫
(LZ)∗ = a
∫
(
e0 )∗ + b
∫ (
ez + vi ei +
1
2
v2 e0
)∗
= b
∫
dτ
[
x2
2R2
ṫ+ viẋi +
v2
2
ṫ+ ṡ
]
.
(6.10)
Note that neglecting total derivative terms, by choosing b = −m and using the
equation of motion of vi = − ẋ
ṫ
, we recover the action given in eq. (6.2).
Using the Maurer–Cartan form one can derive the transformation rules of the
Goldstone fields that realize the NH algebra (6.4):
δt = −ζ , δxi = λikx
k − ai cos
t
R
+ λiR sin
t
R
,
δvi = λikv
k − ai
R
sin
t
R
− λi cos
t
R
, δs =
aixi
R
sin
t
R
+ λixi cos
t
R
.
(6.11)
6.2.2. The Killing Equations of the Newton-Hooke Particle
In this section we find the Killing symmetries of the NH particle using the
Lagrangian formalism. We start from the action (6.10) and consider that the
coordinates transform as:
δxi = ξi(t, x) , δt = ξ0(t, x) , δvi = ζi(t, x, v) , δs = η(t, x) . (6.12)
6.3 The Newton–Hooke Superalgebra 115
After imposing the invariance of the variation of the action (δS = 0) under the
symmetry transformations, we find that the Killing equations are given by
( x2
2R2
+
v2
2
)
∂tξ
0 + vi(∂tξ
i + ζi) +
1
R2
xiξi + ∂tη = 0 ,
( x2
2R2
+
v2
2
)
∂iξ
0 + vj∂iξ
j + ζi + ∂iη = 0 .
(6.13)
It is easy to prove that the solutions of the Killing equations (6.13) correspond
to the symmetry transformations that we found in (6.11).
6.2.3. The Flat Limit
Taking the flat limit R → ∞ of the Newton–Hooke algebra (6.4) we obtain
the Galilean algebra
[
Mab,Mcd
]
= 2ηa[cMd]b − 2ηb[cMd]a ,
[
Jab, (P/K)c
]
= −2 δc[a(P/K)b] ,
[
H,Ka
]
= Pa ,
[
Pa,Kb
]
= δab Z .
(6.14)
In this case, the time, space and central charge components of the Vielbein sim-
plify to
e0 = dt , ei = dxi , ez = ds , (6.15)
and since we are studying the flat case, all the components of the spin-connection
vanish and the action of the Galilean particle is just
S = b
∫ (
ez + vi ei +
1
2
v2 e0
)∗
=
m
2
∫
dτ
ẋiẋi
ṫ
, (6.16)
where we already substituted the equation of motion for vi. This action is invari-
ant under the transformation rules
δt = −ζ , δxi = λikx
k − ai + vi , δvi = λikv
k − vi , δs = vixi . (6.17)
6.3. The Newton–Hooke Superalgebra
In the same way that the non-relativistic bosonic particle is based upon the
Galilei algebra, or its centrally extended version, the Bargmann algebra, the
action and transformation rules of the non-relativistic 3D N = 2 superparticle
are based upon the supersymmetric extension of the Galilei or Bargmann algebra.
It turns out that we need two supersymmetries since one of the supersymmetries
is, like the time translations in the bosonic case, a Stückelberg symmetry.
116 Non Relativistic Limit
The NH superalgebra can be derived as a contraction of the AdS superal-
gebra. In the case of two supersymmetries there are two independent versions
of the latter one, the so-called N = (1, 1) and N = (2, 0) algebras. As we ex-
plained in Section 2, in the presence of a cosmological constant the relativistic
AdS superalgebra in three dimensions is not unique. Instead, in the case of N
supersymmetries, one always finds N different versions, often referred to as (p, q)
AdS superalgebras [58]. In Section 2.3 we give both the (1, 1) and (2, 0) N = 2
AdS superalgebras. As explained in Section 2.3, the Newton–Hooke superalgebra
that we will use below is obtained by contracting the N = (2, 0) AdS superalge-
bra.
We proceed by discussing the contraction of the 3D N = (2, 0) AdS algebra.
The basic commutators are given by (A = 0, 1, 2)
[
MAB,MCD
]
= 2ηA[CMD]B − 2ηB[CMD]A ,
[
MAB, Q
i] = −1
2
γABQ
i ,
[
MAB, PC
]
= −2ηC[APB] ,
[
PA, Q
i] = xγAQ
i ,
[
PA, PB
]
= 4x2MAB ,
[R, Qi] = 2x εijQj ,
{
Qiα, Q
j
β
}
= 2[γAC−1]αβPA δ
ij + 2x[γABC−1]αβMAB δ
ij + 2C−1
αβ ε
ijR .
(6.18)
Here PA,MAB ,R and Qiα are the generators of spacetime translations, Lorentz
rotations, SO(2) R-symmetry transformations and supersymmetry transforma-
tions, respectively. The bosonic generators PA, MAB and R are anti-hermitian
while the fermionic generators Qiα are hermitian. The parameter x is a contrac-
tion parameter. Note that the generator of the SO(2) R-symmetry becomes the
central element of the Poincaré algebra in the flat limit x → 0.
To make the non-relativistic contraction first we define new supersymmetry
charges by
Q±
α =
1
2
(
Q1
α ± γ0Q
2
α
)
, (6.19)
and rescale the generators with a parameter ω as follows:
P0 = ω Z +
1
2ω
H , R = −ω Z +
1
2ω
H , Ma0 = ωKa ,
Q+ =
1√
ω
Q̃+ , Q− =
√
ω Q̃− .
(6.20)
We also set x = 1/(2ωR). Taking the limit ω → ∞ and dropping the tildes on
6.3 The Newton–Hooke Superalgebra 117
the Q± we get the following 3D N = (2, 0) Newton–Hooke superalgebra:
[
Jab, (P/K)c
]
= −2 δc[a(P/K)b] ,
[
H,Ka
]
= Pa ,
[
H,Pa
]
= − 1
R2
Ka ,
[
H,Q−] =
3
2R
γ0Q
−
[
Jab, Q
±] = −1
2
γabQ
± ,
[
H,Q+] = − 1
2R
γ0Q
+ ,
[
Ka, Q
+] = −1
2
γa0Q
− , [Pa, Q
+] =
1
2R
γaQ
− ,
{
Q+
α , Q
+
β
}
= [γ0C−1]αβ H +
1
2R
[γabC−1]αβJab ,
{
Q+
α , Q
−
β
}
= [γaC−1]αβ Pa +
1
R
[γa0C−1]αβKa ,
[
Pa,Kb
]
= δab Z ,
{
Q−
α , Q
−
β
}
= 2[γ0C−1]αβ Z .
(6.21)
The central extension Z that leads to the Bargmann version of the NH superal-
gebra occurs in the last two (anti-)commutation relations.
6.3.1. Non-linear Realizations
In this section we obtain the action and transformation rules for the flat
Galilean and Newton–Hooke superparticles by using the method of non-linear re-
alizations [61,62].4 We will derive in the first subsection the κ-symmetric action
and transformation rules of the Galilean superparticle, see e.g. [68, 160]. In the
second subsection we will do the same for the NH superparticle. The normaliza-
tions of the (to-be) embedding coordinates that occur in this section differ from
those in the main text.
The starting point is the N = 2 Bargmann superalgebra given in section 6.18.
We derive the transformation rules for the coordinates (t, xi, s, θα−, θ
α
+, v
i) using
the coset G/H = N = 2 super Newton–Hooke/SO(D − 1). The coset element is
given by g = g0U , where g0 = eHt ePix
i
eZs eQ
−
α θ
α
− eQ
+
α θ
α
+ is the coset representing
the ‘empty’ N = 2 NH superspace with a central charge extension and U = eKiv
i
is a general NH boost representing the insertion of the particle.
The Maurer–Cartan form associated to the super NH space is given by
Ω0 = g−1
0 dg0 = H E0 +PiE
i +ZEZ +Kiω
i0 +Mijω
ij − Q̄−E− − Q̄+E+ , (6.22)
4For an early application of this method in a different situation than the one considered
in this work, namely to the construction of worldline actions of conformal and superconformal
particles, see [63, 64].
118 Non Relativistic Limit
where (E0, Ei, EZ , E−α, E+α) and (ωi0, ωij) are the time and space superviel-
bein and spin-connection components of the NH superspace given explicitly by
E0 = dt
(
1 − 1
4R
θ̄+θ+
)− 1
2
θ̄+γ
0dθ+ ,
Ei = dxi
(
1 +
1
4R
θ̄+θ+
)
+
dt
2R
(
3θ̄+γ
0iθ− − xkǫki
2R
θ̄+θ+
)− θ̄+γ
idθ− ,
EZ = ds+
dt
2R
(xixi
R
+ 3 θ̄−θ−
)− θ̄−γ
0dθ− ,
ωi0 =
dt
R
(− xi
R
− xi
4R2
θ̄+θ+ +
3
2R
θ̄+γ
iθ−
)− dxkǫki
4R2
θ̄+θ+ − 1
R
θ̄+γ
i0dθ− ,
ωij = −dt ǫab
8R2
θ̄+θ+ − 1
4R
θ̄+γ
abdθ+ ,
E− = dθ−
(
1 − 1
2R
θ̄+θ+
)− dt
2R
(
3γ0θ− − 3
2R
γ0θ−θ̄+θ+ − xi
R
γi0θ+
)− dxi
2R
γiθ+ ,
E+ = dθ+ +
dt
2R
γ0θ+ .
(6.23)
In terms of the supervielbein and the spin-connection the Maurer–Cartan form
of the N = (2, 0) NH superparticle is given by with the L’s given by
LH = E0 , LiP = Ei + viE0 , LZ = EZ + viEi +
vivi
2
E0 ,
LiK = ωi0 − 2vj ωji + dvi , LijJ = ωij ,
L− = E− − vi
2
γi0E+ L+ = E+ .
(6.24)
Note that we can writhe the space-time super-translations in matrix form in terms
of the vielbein of NH superspace as
(
LH , LP
a, L−α, L+α, LZ
)
=
(
E0, Ea, E−α, E+α, EZ
)







1 vi 0 0 v2
2
0 1 0 0 vi
0 0 1 0 0
0 0 viγi0
2 1 0
0 0 0 0 1







(6.25)
allowing to obtain the Maurer–Cartan forms of the NH superspace by a matrix
representation of the NH boost.
6.3 The Newton–Hooke Superalgebra 119
Using the Maurer–Cartan form one can derive the transformation rules of
the Goldstone fields that realize the NH superalgebra (6.21). We find that the
bosonic transformation rules are given by
δt = −ζ ,
δxi = λikx
k − ai cos
t
R
+
akεki
4R
sin
t
R
θ̄+θ+ + viR sin
t
R
+
vkεki
4
cos
t
R
θ̄+θ+ ,
δki = λikk
k − ai
R
sin
t
R
(
1 − 1
4R
θ̄+θ+
)− vi cos
t
R
(
1 − 1
4R
θ̄+θ+
)
,
δs =
aixi
R
sin
t
R
+
ai
2R
sin
t
R
θ̄−γ
iθ+ + vixi cos
t
R
+
vi
2
cos
t
R
θ̄−γ
iθ+ ,
δθ+ =
1
4
λabγabθ+ ,
δθ− =
1
4
λabγabθ− +
ai
2R
sin
t
R
γi0θ+ +
vi
2
cos
t
R
γi0θ+ .
(6.26)
The transformation rules under the ǫ−-supersymmetry transformations are given
by
δθ− = ǫ−(t) = exp
( 3t
2R
γ0
)
ǫ− , δs = ǭ−(t)γ0θ− , (6.27)
while all others fields are invariant. For the ǫ+-transformations we find the fol-
lowing rules
δt =
1
2
ǭ+(t)γ0θ+ , δxi = ǭ+(t)γiθ−
(
1 − 1
4R
θ̄+θ+
)
,
δki =
kk
2R
θ̄+γkiǫ+(t) +
xi
2R2
ǭ+(t)γ0θ+ − 1
R
θ̄−γ
i0ǫ+(t)
(
1 − 1
4R
θ̄+θ+
)
,
δs = − xi
2R
ǭ+(t)γi0θ−
(
1 − 1
4R
θ̄+θ+
)− xixi
4R2
ǭ+(t)γ0θ+ − 1
2R
ǭ+(t)γ0θ+ θ̄−θ− ,
δθ+ = ǫ+(t)
(
1 − 1
8R
θ̄+θ+
)
,
δθ− =
xi
2R
γiǫ+(t)
(
1 +
1
4R
θ̄+θ+
)− 1
2R
γiθ+ǭ+(t)γiθ− +
3
4R
γ0θ−ǭ+(t)γ0θ+ ,
(6.28)
with
ǫ+(t) = exp
(−t
2R
γ0
)
ǫ+ . (6.29)
120 Non Relativistic Limit
6.3.2. The Killing Equations of the Newton-Hooke Superparticle
The action of the N = (2, 0) NH superparticle is given by the pull-back of all
the L’s that are invariant under rotations:
S = a
∫
(LH)∗ + b
∫
(LZ)∗
= a
∫
dτ
(
ṫ
(
1 − 1
4R
θ̄+θ+
)− 1
2
θ̄+γ
0θ̇+
)
+ b
∫
dτ
[
ds+
dt
2R
(xixi
R
+ 3 θ̄−θ−
)− θ̄−γ
0dθ−
+ vi
(
dxi
(
1 +
1
4R
θ̄+θ+
)
+
dt
2R
(
3θ̄+γ
0iθ− − xkǫki
2R
θ̄+θ+
)− θ̄+γ
idθ−
)
+
v2
2
(
ṫ
(
1 − 1
4R
θ̄+θ+
)− 1
2
θ̄+γ
0θ̇+
)]
.
(6.30)
We will consider that the coordinates transform in the following way:
δxi = ξi(t, x, θ±) , δt = ξ0(t, x, θ±) , δθ± = χ±(t, x, θ±) ,
δvi = ζi(t, x, v, θ±) , δs = η(t, x, θ±) .
(6.31)
Since a and b are independent, the invariance of the variation of the action δS = 0
lead to the following sets of Killing equations
0 = ∂tξ
0(1 − 1
4R
θ̄+θ+
)− 1
2R
θ̄+χ+ − 1
2
θ̄+γ
0∂tχ+ ,
0 = ∂jξ
0(1 − 1
4R
θ̄+θ+
)− 1
2
θ̄+γ
0∂tχ+ ,
0 = ∂θ−
ξ0(1 − 1
4R
θ̄+θ+
)− 1
2
θ̄+γ
0∂θ−
χ+ ,
0 = ∂θ+ξ
0(1 − 1
4R
θ̄+θ+
)
+
1
2
χ̄+γ
0 − 1
2
θ̄+γ
0∂θ+χ+ ,
(6.32)
and for the second part of the action
0 =
1
2R
∂tξ
0
(x2
R
+ 3θ̄−θ− + 3θ̄+γ
0iθ− − xkǫki
2R
θ̄+θ+
)
+ vi∂tξ
i(1 +
1
4R
θ̄+θ+
)
+ viζi
(
1 − 1
4R
θ̄+θ+
)− 1
4R2
viξkǫkiθ̄+θ+ + ∂tη +
1
R2
xiξi
+
1
2R
ζi
(
3θ̄+γ
0iθ− − xkǫki
2R
θ̄+θ+
)
+
3
R
θ̄−χ− − θ̄−γ
0∂tχ−
+
3vi
2R
θ̄+γ
0iχ+γ
0iχ− − viθ̄+γ
i∂tχ− − 3
2R
viθ̄−γ
0iχ+ − 1
2R2
vixkǫkiθ̄+χ+ ,
6.3 The Newton–Hooke Superalgebra 121
0 =
1
2R
∂iξ
0
(x2
R
+ 3θ̄−θ− + 3θ̄+γ
0iθ− − xkǫki
2R
θ̄+θ+
)
+ vj∂iξ
j(1 +
1
4R
θ̄+θ+
)
+ ζi
(
1 +
1
4R
θ̄+θ+
)
+ ∂iη − θ̄−γ
0∂iχ− − vj θ̄+γ
j∂iχ− +
1
2R
viθ̄+χ+ ,
0 = ∂θ−
ξ0
(x2
R
+ 3θ̄−θ− + 3θ̄+γ
0iθ− − xkǫki
2R
θ̄+θ+
)
+ vi∂θ−
ξi
(
1 +
1
4R
θ̄+θ+
)
− ζiθ̄+γ
i + ∂θ−
η − χ̄−γ
0 − θ̄−γ
0∂θ−
χ− + vi∂θ−
χ− − viχ̄+γ
i ,
0 = ∂θ+ξ
0
(x2
R
+ 3θ̄−θ− + 3θ̄+γ
0iθ− − xkǫki
2R
θ̄+θ+
)
+ vi∂θ+ξ
i(1 +
1
4R
θ̄+θ+
)
− 1
2
viζiθ̄+γ
0 + ∂θ+η − θ̄−γ
0∂θ+χ− − viθ̄+γ
i∂θ+χ− .
(6.33)
The transformations rules given in eqs. (6.26)-(6.28) are solutions of these sets
of Killing equations.
6.3.3. The Kappa-symmetric Newton Hooke Superparticle
We are now ready to derive the action and κ-transformation rules. To derive
an action that is invariant under κ-transformations we need to find a fermionic
gauge-transformation that leaves LH and/or LZ invariant. 5
It is convenient to define the line-elements
π0 = ṫ
(
1 +
1
8R
θ̄+θ+
)
+
1
4
θ̄+γ
0θ̇+ , (6.34)
πi =
(
ẋi + 1
4 θ̄+γ
iθ̇− + 1
4 θ̄−γ
iθ̇+
) (
1 − 1
8R θ̄+θ+
)− ṫ
4R
(
3θ̄+γ
0iθ− − xkεki
2R θ̄+θ+
)
.
(6.35)
which are related to the Maurer–Cartan form via the pull-backs
(LH)∗ = π0 , (LP )∗ = πi + kiπ0 . (6.36)
Now, the variation of LH and LZ under gauge-transformations is given by
δLH = d[δzH ] − L̄+γ
0[δz+] ,
δLZ = d[δzZ ] − 2L̄−γ
0[δz−] − δab
(
LaP [δzbG] − LaG [δzbP ]
)
.
(6.37)
5The case of the SU(1, 1|2) superconformal particle is discussed in [161].
122 Non Relativistic Limit
For κ-transformations we find, using the explicit expressions for L+ and L−,
(δLH)∗ = [δz̄+]
(
γ0θ̇+ +
ṫ
2R
θ+
)
,
(δLZ)∗ = 2[δz̄−]γ0
(
θ̇−(1 +
1
2R
θ̄+θ+) − ṫ
2R
(
3γ0θ− − 3
2R
γ0θ−θ̄+θ+ − xi
R
γi0θ+
)
− ẋi
2R
γiθ+ − ki
2
γi0
(
θ̇+ +
ṫ
2R
γ0θ+
)
.
(6.38)
It follows that to obtain a κ-symmetric action we need to take the pull-back of
either LH or LZ , with [δz+] = 0 or [δz−] = 0, respectively. We focus here on the
second case, i.e. we choose a = 0 and we take
S =
∫
(LZ)∗ , [δz+] = κ , [δz−] = 0 , (6.39)
with κ an arbitrary (local) parameter. To compare to the action and transfor-
mations rules given in [162] one needs to make the following redefinitions
t → −t , xi → −xi +
1
2
θ̄+γ
iθ− , π0 → −π0 , πi → −πi , (6.40)
rescale all spinors by 1/
√
2, together with R → −R this leads to the κ-symmetric
Newton–Hooke superparticle action:
S =
∫
dτ m
2
[
πiπi
π0 − θ̄−γ
0θ̇− − ṫ x
ixi
R2 + 3ṫ
2R θ̄−θ− + ṫ xi
2R2 θ̄+γ
iθ− − ṫ
16R2 θ̄+θ+ θ̄−θ−
]
,
(6.41)
The κ-transformations read as follows:
δt =
1
4
κ̄γ0θ+ , δxi =
1
4
κ̄γiθ− − πj
8π0
κ̄γjγi0θ+ ,
δθ+ = κ
(
1 − 1
16R θ̄+θ+
)
, δθ− = − πi
2π0 γi0κ− 3
8R γ0θ−θ̄+γ
0κ+ xi
16R2 γ
iκ θ̄+θ+ .
(6.42)
The κ-symmetry can be gauge-fixed by imposing the gauge condition θ+ = 0.
We find that the action of the NH superparticle takes the form
S =
∫
dτ
m
2
[
ẋiẋi
ṫ
− θ̄−γ
0θ̇− − ṫ xixi
R2
+
3ṫ
2R
θ̄−θ−
]
. (6.43)
A realization of the Newton–Hooke superalgebra, whose explicit form is given
in eq. (6.21), on the embedding coordinates is given by the following bosonic
6.3 The Newton–Hooke Superalgebra 123
transformation rules
δt = −ζ , δxi = λikx
k − ξi(t) , δθ− =
1
4
λabγabθ− , (6.44)
supplemented with the following fermionic transformations:
δt = 0 , δxi = −1
2
ǭ+(t)γiθ− , δθ− = ǫ−(t) − ẋi
2ṫ
γ0i ǫ+(t) +
xi
2R
γi ǫ+(t) .
(6.45)
Here, the time-dependence of the parameters ξi(t) and ǫ±(t) is given by
ξi(t) = viR sin
t
R
+ ai cos
t
R
,
ǫ−(t) = exp
( 3t
2R
γ0
)
ǫ− , ǫ+(t) = exp
(
− t
2R
γ0
)
ǫ+ .
(6.46)
6.3.4. The Flat Limit
Since the NH superalgebra is a deformation of the Galilei algebra, by taking
the flat limit R → ∞, the 3D N = 2 Galilei superalgebra is given by the bosonic
commutation relations
[
Jab, Pc
]
= −2 δc[aPb] ,
[
Ka, H
]
= −Pa ,
[
Jab,Kc
]
= −2 δc[aKb] ,
[
Jab, Jcd
]
= 4δ[a[cJd]b] ,
[
Pa,Kb
]
= δab Z , (6.47)
plus the additional relations [150,151]
[
Jab, Q
±] = −1
2
γabQ
± ,
[
Ka, Q
+] = −1
2
γa0Q
− ,
{
Q+
α , Q
+
β
}
= 2 [γ0C−1]αβ H ,
{
Q+
α , Q
−
β
}
= [γaC−1]αβ Pa ,
{
Q−
α , Q
−
β
}
= 2 [γ0C−1]αβ Z .
(6.48)
In the flat case the spin-connection vanishes and the time and spatial components
of the supervielbein simplify to
E0 = dt− 1
2
θ̄+γ
0dθ+ , Ei = dxi − θ̄+γ
idθ− , EZ = ds− θ̄−γ
0dθ− ,
E− = dθ− , E+ = dθ+ .
(6.49)
124 Non Relativistic Limit
and the Maurer–Cartan 1-forms are given by
LH = E0 , LiP = Ei + viE0 , LZ = EZ + viEi +
vivi
2
E0 ,
LiK = dvi , LijJ = 0 , L− = E− − vi
2
γi0E+ L+ = E+ .
(6.50)
The action of the Galilean superparticle is given by the pull-back of all L’s that
are invariant under rotations, hence
S = a
∫
(LH)∗ +
∫
(LZ)∗ =
∫
dτ
[
−a
2
θ̄+γ
0θ̇+ − θ̄−γ
0θ̇− − πiπi
2π0
]
, (6.51)
where the line-elements are
π0 = ṫ− 1
2
θ̄+γ
0θ̇+ , πi = ẋi − θ̄+γ
iθ̇− , (6.52)
Here we replaced the Goldstone field vi by its equation of motion vi = −πi/π0.
This procedure is know as inverse Higgs mechanism [163], see also [164]. The
bosonic transformations of the embedding coordinates are
δt = −ζ , δxi = λikx
k − ai + vit+
vkεki
4
θ̄+θ+ ,
δθ+ =
1
4
λabγabθ+ , δθ− =
1
4
λabγabθ− +
vi
2
γi0θ+ ,
(6.53)
and the supersymmetry transformations are
δt =
1
2
ǭ+γ
0θ+ , δxi = ǭ+γ
iθ− , δθ± = ǫ± . (6.54)
These transformations leave the action (6.51) and all L’s, in particular the line-
elements (6.52), invariant. The κ-transformations of the coordinates are
δt = −1
2
κ̄γ0θ+ , δxi = − πj
2π0
θ̄+γ
iγj0κ , δθ+ = κ , δθ− = − πi
2π0
γi0κ ,
(6.55)
and the corresponding κ-symmetric action is given by
S =
∫
dτ
m
2
[
−2θ̄−γ
0θ̇− − πiπi
π0
]
. (6.56)
6.4 Gauging Procedure: Bosonic Case 125
6.4. Gauging Procedure: Bosonic Case
6.4.1. The Galilean Particle
In this section we will briefly review the gauging procedure for the particle
case as it was done in [165]. The action describing a particle of mass m moving in
a D-dimensional Minkowski spacetime parametrized by the evolution parameter
τ is
S = −m
∫
dτ
√
−ηµν ẋµẋν , µ = 0, 1, . . . D − 1 , (6.57)
This Lagrangian is invariant under worldline reparametrizations and under Poin-
caré transformations
δxµ = λµνx
ν + ζµ , (6.58)
where λµν and ζµ are the parameters for Lorentz transformations and transla-
tions respectively. Taking the non-relativistic limit by rescaling the longitudinal
coordinate x0 ≡ t and the mass m with a parameter ω
t → ωt , m → ωm , (6.59)
and taking the limit ω → ∞, this rescaling is such that the kinetic term remains
finite. In this limit, action (6.57) can be written as
S =
m
2
∫
ẋiẋi
ṫ
dτ . (6.60)
This non-relativistic action is invariant under worldline reparametrizations and
under the Galilei symmetries
δt = ρ(τ)ṫ , δxi = ρ(τ)ẋi ,
δt = −ζ , δxi = λijx
j − vit− ai ,
(6.61)
where (ζ, ai, λij , v
i) parametrize a constant time and space translations, spatial
rotation and a boost transformation and ρ(τ) is the diffeomorphisms parameter.
However, this Lagrangian is invariant under boosts only up to a total τ -derivative,
leading to a modified Nöether charge giving rise to a centrally extended Galilei
algebra containing an extra central charge generator Z. This centrally extended
Galilei algebra is called the Bargmann algebra.
This results apply to free falling frames without any gravitational interactions.
These frames are connected to each other through the Galilei symmetries. We
now wish to extend these results to include frames that are accelerated with
126 Non Relativistic Limit
respect to the free falling frames. There are two procedures to achieve this: the
first one valid in frames with time-dependent acceleration, consists on gauging
the spatial translations and it is described in section 6.4.2. In the second one,
described in section 6.4.3, one gauges all the symmetries of the Bargmann algebra.
6.4.2. The Curved Galilean Particle
The Curved Galilean Particle is valid in frames with constant acceleration.
The procedure to obtain it goes as follows: gauge the spatial translations by
allowing arbitrary time-dependent boosts, with parameters ξi(t). The complete
bosonic and fermionic transformation rules now read. By doing so we obtain a
gauged action containing the gravitational potential Φ(x)
L =
m
2
(
ẋiẋi
ṫ
− 2ṫΦ(x)
)
. (6.62)
Now the Lagrangian is invariant under worldline reparametrizations and the ac-
celeration extended symmetries
δt = −ζ + ρ(τ)ṫ , δxi = λijx
j − ξi(t) + ρ(τ)ẋi , (6.63)
The acceleration-extended symmetries are not a proper symmetry of the action
(6.62). Instead, the Newton potential should be viewed as a background field
and the acceleration-extended symmetries as sigma model symmetries. In par-
ticular, the transformation rule of the background field, that we will denote by
the symbol δbg, lacks the transport terms that are present in the transformation
rule associated to a proper symmetry, denoted in this chapter by δpr: 6
δbg = δpr + δxµ∂µ . (6.64)
Using this we find that the action (6.62) is invariant under the acceleration-
extended symmetries (6.63) provided the Newton potential Φ transforms as fol-
lows:
δΦ =
1
ṫ
d
dτ
(
ξ̇i
ṫ
)
xi + ρ(τ)Φ̇ + σ(t) . (6.65)
The last term represents gives a boundary term in the action.
6Assuming that xµ → xµ + δxµ a transport term is given by −δxµ∂µ so that the second
term in (6.64) cancels the transport term present in the proper transformation rule represented
by the first term.
6.4 Gauging Procedure: Bosonic Case 127
6.4.3. The Newton-Cartan Particle
We now wish to extend the Curved Galilean particle to a NC particle, i.e. a
particle moving in a NC gravity background and invariant under general coordi-
nate transformations, see [45] for the detailed procedure. The first step to obtain
the corresponding action is to gauge all the symmetries of the Bargmann algebra:
time and space translations, spatial rotations, boosts and central charge trans-
formations. Associate a gauge field to each of the symmetries and promote the
constant parameters describing the transformations to arbitrary functions of the
spacetime coordinates xµ.
Besides these transformations all gauge fields transform under general coor-
dinate transformations with parameters ξµ(xµ). Therefore, for each generator we
have associated gauge fields, gauge parameters and curvatures.
The next step in the gauging procedure is to impose a set of constraints
on the curvatures of the gauge fields. With these constraints the time and space
translations become equivalent to general coordinate transformations modulo the
other symmetries of the algebra. This enables one to solve for the gauge fields
of boosts and spatial rotations ωµa and ωµ
ab in terms of the other ones, so the
independent gauge fields will be (τµ, eµ
a,mµ). The gauge fields τµ and eµ
a of
time and spatial translations are identified as the temporal and spatial vielbeins.
The dynamics of the Newton-Cartan point particle is described by the follow-
ing Lagrangian
L =
m
2
(
hµν ẋ
µẋν
τρẋρ
− 2mµẋ
µ
)
, hµν = eµ
aeν
bδab . (6.66)
The theory is invariant under general coordinate transformations plus boosts,
spatial rotations and central charge transformations, all with parameters that
are arbitrary functions of the spacetime coordinates.
The embedding coordinates transform under these general coordinate trans-
formations, with parameters ξµ(xν), in the standard way:
δxµ = −ξµ(xν) . (6.67)
The transformation rules of the background fields τµ , eµa and mµ follow from the
known proper transformation rules by omitting the transport term, see eq. (6.64):
δbgτµ = ∂µξ
ρ τρ ,
δbgeµ
a = ∂µξ
ρ eρ
a + λab eµ
b + λa τµ ,
δbgmµ = ∂µξ
ρmρ + ∂µσ + λaeµ
a .
(6.68)
128 Non Relativistic Limit
Here, λab , λa and σ are the parameters of a local spatial rotation, boost trans-
formation and central charge transformation, respectively.
The proper transformation rules of the background fields τµ , eµa and mµ can
be obtained by gauging the Bargmann algebra, see e.g. [45]. Since the NC back-
ground is the most general background one must be able to obtain the transforma-
tions of the Curved Galilean and flat backgrounds discussed in the two previous
subsections by gauge-fixing some of the general coordinate transformations. This
is discussed in detail in [48].
For the convenience of the reader, we list in table 6.1 the gauge-fixing condi-
tions that need to be imposed on the NC background fields, and the compensat-
ing gauge transformations that come along with it, that bring us to the Curved
Galilean background in terms of the Newton potential Φ.
gauge condition compensating transformations
τµ(xν) = δµ
∅ ξ∅(xν) = ξ∅
ωµ
ab = 0 λab(xν) = λab
ei
a = δi
a ξi(xν) = ξi(t) − λi
jx
j
τi(x
ν) +mi(x
ν) = ∂im(xν)
m(xν) = 0 σ(xν) = σ(t) + ∂tξ
i(t)xi
τi(x
ν) = 0 λi(xν) = −∂tξ
i(t)
m∅(xν) = Φ(xν) ω∅
a = −∂aΦ(xν)
Table 6.1
This table indicates the gauge-fixing conditions, chronologically ordered from top to bottom,
and the corresponding compensating transformations, that lead from the NC particle, to the
Curved Galilean particle. Note that τi(x
ν) ≡ ei
0(xν). The restriction
τi(x
ν) +mi(x
ν) = ∂im(xν) follows from the gauge conditions made at that point.
6.5. Gauging Procedure: Supersymmetric Case
6.5.1. The Galilean Superparticle
The Galilean superparticle was already discussed in [127]. In terms of the
bosonic and fermionic embedding coordinates {t, xi, θ−} the action is given by
S =
∫
dτ
m
2
[
ẋiẋi
ṫ
− θ̄−γ
0θ̇−
]
. (6.69)
6.5 Gauging Procedure: Supersymmetric Case 129
This action is invariant under the following Galilei symmetries:
δt = −ζ , δxi = λijx
j − vit− ai . δθ− =
1
4
λabγabθ− . (6.70)
The same action is invariant under two supersymmetries with constant parame-
ters ǫ+ and ǫ−:
δt = 0 , δxi = −1
2
ǭ+γ
iθ− , δθ− = ǫ− − ẋi
2ṫ
γ0iǫ+ . (6.71)
One may verify that the above set of transformation rules (6.70) and (6.71) closes
off-shell. Note that the transformation with parameter ǫ+ is realized linearly.
Instead, the one with parameter ǫ− is realized non-linearly, i.e. it is a broken
symmetry. This implies that the superparticle corresponds to a 1/2 BPS state.
6.5.2. The Curved Galilean Superparticle
We will now extend the Galilean superparticle to a Curved Galilean super-
particle thereby replacing the flat background by a Galilean supergravity back-
ground. This corresponds to extending the bosonic particle in a Galilean gravity
background, discussed in section 6.4.2, to the supersymmetric case.
Our starting point is the superparticle action in a flat background, see eq.
(6.69). We will now partially gauge the transformations (6.70) and (6.71) to
allow for arbitrary time-dependent boosts, with parameters ξi(t), and arbitrary
supersymmetry transformations, with parameters ǫ−(t). The complete bosonic
and fermionic transformation rules now read
δt = −ζ , δxi = λijx
j − ξi(t) , δθ− =
1
4
λabγabθ− , (6.72)
and
δt = 0 , δxi = −1
2
ǭ+γ
iθ− , δθ− = ǫ−(t) − ẋi
2ṫ
γ0iǫ+ , (6.73)
respectively.
The Galilean supergravity multiplet, that we will use to perform the partial
gauging of the transformations (6.70) and (6.71), was introduced in [48]. Along-
side the Newton potential Φ(x), it also contains a fermionic background field
Ψ(x). The equations of motion for these two background fields are:
∂i∂iΦ = 0 , γi∂iΨ = 0 . (6.74)
130 Non Relativistic Limit
There is a slight subtlety regarding this Galilean supergravity multiplet, stem-
ming from the fact that Ψ(x) is the superpartner of the Newton force Φi ≡ ∂iΦ(x)
and not of the Newton potential itself. The transformation rules of the Newton
force are, however, compatible with the integrability condition ∂[iΦj] = 0, so that
they can be integrated to transformation rules of the Newton potential Φ(x).
This is done via the introduction of a fermionic prepotential χ(x), that will be
called the ‘Newtino potential’, defined via
∂i χ = γiΨ (i = 1, 2) , γ1∂1χ = γ2∂2χ , (6.75)
where the second equation represents a constraint obeyed by χ(x), as a conse-
quence of its definition. This constraint can be interpreted (upon choosing a
specific basis for the γ-matrices) as the Cauchy–Riemann equations, expressing
holomorphicity of χ1 + iχ2, where χ1,2 are the components of χ. Since the New-
ton potential Φ obeys the Laplace equation in two spatial dimensions, it can also
be seen as the real part of a holomorphic function Φ + i Ξ. The imaginary part
Ξ(x) of this function was called the ‘dual Newton potential’ in [48] and is related
to Φ(x) via the Cauchy–Riemann equations for Φ + i Ξ:
∂iΦ = εij∂
jΞ , ∂iΞ = −εij∂jΦ . (6.76)
The dual Newton potential was introduced in [48] in order to write down the
supersymmetry transformation rule for χ. Since this is a transformation rule for
both real and imaginary parts of χ1 + iχ2, it is natural to expect that it involves
also both real and imaginary parts of Φ + i Ξ and this is indeed the case.
We find that the action of the Curved Galilean superparticle in terms of the
Galilean supergravity background fields Φ and Ψ is given by
S =
∫
dτ
m
2
[
ẋiẋi
ṫ
− θ̄−γ
0θ̇− − 2ṫΦ + 2ṫ θ̄−γ
0Ψ
]
. (6.77)
One may verify that the action (6.77) is invariant under the transformations
(6.72) and (6.73) provided that the background fields transform under the bosonic
symmetries as
δbgΦ =
1
ṫ
d
dτ
(
ξ̇i
ṫ
)
xi + σ(t) , δbgΨ =
1
4
λabγabΨ , (6.78)
and under the fermionic symmetries as
δbgΦ = ǭ−γ
0Ψ +
1
2
ǭ+∂tχ− 1
2
ǭ+γ
iθ− ∂iΦ , (6.79)
δbgΨ =
1
ṫ
ǫ̇− − 1
2
∂iΦγi0ǫ+ − 1
2
ǭ+γ
iθ− ∂iΨ . (6.80)
6.5 Gauging Procedure: Supersymmetric Case 131
The only invariance that is non-trivial to show is the one under the linear
ǫ+-transformations. Varying the action (6.77) under ǫ+-transformations one is
left with the following terms:
δ+S =
∫
dτ
m
2
[
− ǭ+ ṫ ∂tχ− ǭ+ ẋ
i ∂iχ− ṫ
2
ǭ+γ
kθ− θ̄−γ
0i∂i∂kχ
]
. (6.81)
The first two terms combine into a total τ -derivative, since χ is a function of xi
and t and therefore
d
dτ
χ =
(
ṫ ∂t + ẋi ∂i
)
χ . (6.82)
The second term vanishes upon using the equation of motion for the background
field χ.
To calculate the commutator algebra it is important to keep in mind that the
background fields do not transform as fundamental fields but, instead, according
to background fields, see eq. (6.64). This explains the ‘wrong’ sign transport
term in the ǫ+-transformation and the absence of transport terms for all other
symmetries. It also has the consequence that partial derivatives do not commute
with background variations:
[
δbg, ∂µ
]
= −(∂µδx
ν)∂ν . (6.83)
Another subtlety when calculating the commutator algebra is related to the fact
that the parameters ξi and ǫ− are functions of the time t but that t itself is a
scalar function t(τ) of the world-line parameter τ . This implies that when we
calculate commutators we have to vary the t inside the parameters. Keeping the
above subtleties in mind we find that the commutation relations close off-shell
on the embedding coordinates and the background fields.
Imposing the gauge-fixing conditions
Φ = 1 , χ = 0 , (6.84)
we recover the Galilean superparticle with the flat spacetime transformation rules
(6.70) and (6.71). Imposing the additional gauge-fixing condition
t = τ (6.85)
we find agreement with the algebra obtained in [48].
132 Non Relativistic Limit
6.5.3. The Newton–Cartan Superparticle
We wish to extend the result of the previous subsection to arbitrary frames
corresponding to a superparticle in a Newton–Cartan supergravity background.
Due to the complexity of the calculations we only give the result up to quartic
fermions in the action. We find that using this approximation the action is given
by
S =
∫
dτ
m
2
[
ẋµeµ
a ẋνeν a
ẋρτρ
− 2mµẋ
µ − θ̄−γ
0Dτθ− + 2 θ̄−γ
0ψµ−ẋ
µ
− ẋµeµa
ẋρτρ
θ̄−γ
aψν+ẋ
ν
]
,
(6.86)
where the Lorentz-covariant derivative Dτ is defined as
Dτθ− = θ̇− − 1
4
ẋµωµ
abγabθ− . (6.87)
To lowest order in the fermions the action (6.86) is invariant under the following
bosonic and fermionic symmetries of the embedding coordinates:
δxµ = −ξµ(xα) , δθ− =
1
4
λab(xα)γab θ− , (6.88)
and
δxµ = −1
2
ǭ+(xα)γaθ− e
µ
a , δθ− = ǫ−(xα) − ẋµeµ
a
2ẋρτρ
γ0a ǫ+(xα) . (6.89)
In the following we refrain from explicitly denoting the local xµ-dependence of
the parameters.
The transformation rules of the background fields follow from the supergravity
result given in [48] and application of the identity (6.64). We find that the bosonic
transformation rules are given by
δprτµ = 0 , δprmµ = ∂µσ + λaeµ
a ,
δpreµ
a = λab eµ
b + λa τµ , δprψµ+ =
1
4
λabγabψµ+ ,
δprωµ
ab = ∂µλ
ab , δprψµ− =
1
4
λabγabψµ− − 1
2
λaγa0ψµ+ ,
δprωµ
a = ∂µλ
a − λbωµ
ab + λabωµb .
(6.90)
6.6 Discussion 133
To keep the formulas simple we have given here as well as below only the proper
transformation rules. The background transformations are obtained by supple-
menting each of these rules with an additional transformation under general co-
ordinate transformations, see eq. (6.64). For the fermionic transformations we
find the following expressions:
δprτµ =
1
2
ǭ+γ
0ψµ+ , δprmµ = ǭ− γ
0ψµ− ,
δpreµ
a =
1
2
ǭ+γ
aψµ− +
1
2
ǭ−γ
aψµ+ , δprψµ+ = Dµǫ+ ,
δprωµ
ab = 0 , δprψµ− = Dµǫ− +
1
2
ωµ
aγa0ǫ+ .
(6.91)
The variation of ωµab is only zero on-shell, i.e. upon using the equations of motion
of the background fields. The explicit form of these equations of motion are given
in [48]. In the same manner we can write the variation of ωµa as
δprωµ
a =
1
2
ǭ−γ
0ψ̂µ
a
− +
1
2
τµ ǭ−γ
0ψ̂0
a
− +
1
4
eµ
b ǭ+γ
bψ̂a0− +
1
4
ǭ+γ
aψ̂µ0− , (6.92)
where ψ̂µν− is the covariant curvature of ψµ−, see [48]. One may check that, to
lowest order in fermions, the action (6.86) is invariant under the transformations
(6.89), (6.91) and (6.92), upon use of the equations of motion of the background
fields.
As a consistency check we have verified that by imposing the gauge-fixing
conditions of [48] the action and transformation rules of the NC superparticle
reduce to those of the Curved Galilean superparticle.
6.6. Discussion
In this chapter we have constructed with the non-linear realizations method,
the free Newton-Hooke (super)particle actions and we analyzed the dynamics
and the symmetries of a particle moving in such spaces. We also studied the
superparticle actions describing the dynamics of a supersymmetric particle in
a 3D Curved Galilean and Newton–Cartan supergravity background. It is also
possible to construct the actions for a superparticle moving in the cosmological
extension of these backgrounds by including a cosmological constant, see [162].
Due to the computational complexity we gave the action in the Newton–Cartan
case only up to terms quartic in the fermions. The Newton–Cartan background
134 Non Relativistic Limit
is characterized by more fields and corresponds to more symmetries than the
Galilean background. One can switch between the two backgrounds either by a
partial gauging of symmetries (from Galilean to Newton–Cartan) or by gauge-
fixing some of the symmetries (from Newton–Cartan to Galilean). An important
role in the construction is played by symmetries. At several occasions we stressed
that, as far as the background fields are concerned, one should use the background
transformations and not the proper transformations. The latter are used in the
definition of the supergravity multiplet. The relation between the two kind of
transformations is given in eq. (6.64).
A noteworthy feature is that the proof of invariance of the superparticle ac-
tion requires that the background fields satisfy their equations of motion. This
is reminiscent to what happens with the fermionic κ-symmetry in the relativistic
case. We showed that the non-relativistic superparticle also allows a κ-symmetric
formulation but that in the non-relativistic case the κ-symmetry is of a simple
Stückelberg type [127]. Although being rather trivial, we expect that the for-
mulation with κ-symmetry is indispensable for a reformulation of our results in
terms of a non-relativistic superspace and superfields, see e.g. [166]. Such a su-
perspace formulation would be useful to construct the superparticle actions in
the Newton–Cartan background to all orders in the fermions.
“Well, in our country,” said Alice, still panting a little,
“you’d generally get to somewhere else if you run very
fast for a long time, as we’ve been doing.”
“A slow sort of country!” said the Queen. “Now, here,
you see, it takes all the running you can do, to keep
in the same place. If you want to get somewhere else,
you must run at least twice as fast as that!”
Lewis Carroll
7
Carroll Limit
7.1. Introduction
The Carroll limit is defined as the limit when the velocity of light goes to
zero c → 0. This can be interpreted as the shrinking of the light cones till they
collapse into the time axis, in this sense, it is the opposite of the non-relativistic
limit and therefore it is also called ultra-relativistic limit. In this limit, there are
no interactions between spatially separated events and the equations of motion
are trivial, therefore no true motion occurs, except for tachyonic motion.
As we mention in section 2.3, applying the Carroll limit over the Poincaré
group leads to the Carroll group. An interesting feature of this group is that in
the contracted Lie algebra, the generator of time translations H, appears as a
central charge. In other words if Pa generate the spatial translations and Ka the
boosts, the only non-vanishing commutator among these generators is1
[Pa,Kb] = H . (7.1)
Recently, the Carroll group has attracted attention because it appears as the
symmetry group of different systems. For example, some studies have shown that
the Carroll group might have an essential role in the phenomenon of tachyon
1The remaining non-vanishing commutators are those involving spatial rotations, see eq.
(2.12).
135
136 Carroll Limit
condensates in string theory [42]. Sen [167] had a remarkable insight into the
nature of the open bosonic string tachyon, he observed that it should be thought
of as ending on a space-filling D-brane. He pointed out that this D-brane is
unstable in the bosonic theory, as it does not carry any conserved charge, and
he suggested that the open bosonic string tachyon should be interpreted as the
instability mode of the D-brane. This led him to conjecture that the open string
field theory could be used to precisely determine a new vacuum for the open
string, namely one in which the D-brane is annihilated through condensation of
the tachyonic unstable mode [168]. The open string excitations are subject to a
time-like half line collapsed lightcone which corresponds to the Carroll limit [169].
It has also been pointed out, that there is a duality between the Galilean
and the Carroll limits. In [33] it is shown that the Carroll group can be viewed
as a subgroup of the Poincaré group in (D + 1, 1) dimensions and they study
non-Einsteinian electrodynamics in the Carroll limit. They also give an example
where both Galilean and Carrollian symmetries coexist in a Chaplygin gas, a
non-relativistic system in D dimensions that carries a Carroll symmetry.
Additionally, Carroll symmetries allow to build a fully covariant formalism for
warped conformal field theories (WCFTs) [43] (the simplest field theories without
Lorentz invariance that can be described holographically) in curved spaces. In [43]
a procedure to gauge the Carroll algebra is provided, which eventually can lead
to the construction of a Carroll gravity.
Finally, in [170] it is shown that the Bondi-Metzner-Sachs (BMS) group is
the conformal extension of the Carroll group and it is gaining interest with its
applications to conformal field theory.
The aim of this chapter is to study the general structure of the Carroll sym-
metries along the same lines as this has been done for the Galilean symmetries.
This will be done in two stages. As a first step we will study the geometry of
the empty Carroll space considering the coset G/H = AC/Hom AC, where Hom
AC is the homogeneous part of the AC algebra. In a second step we will put a
particle in this Carroll space and construct an action describing its dynamics.
More specifically, in the first part of this chapter we consider the bosonic AC
algebra. In particular, we will construct the action of a particle invariant under
the symmetries corresponding to this algebra using the method of non-linear
realizations [61,62]. This so-called AC particle reduces, in the limit that the AdS
radius goes to infinity, to the Carroll particle that we studied in [41].
A characteristic feature of the free Carroll particle is that it does not move
[33, 41, 171]2. As we will see the AC particle does not move, but unlike the
2If we consider two particles or a particle interacting with Carroll gauge fields the dynamics is
non-trivial. The same phenomenon occurs in tachyon condensation when the tachyon interacts
7.1 Introduction 137
Carroll particle the momenta are not a constant of motion as a consequence of
the AdS-Carroll symmetry. Another difference with the Carroll particle is that
the mass-shell constraint depends on the coordinates of the AC space, therefore
the AC particle ‘sees’ the geometry. This is different from the Carroll case where
the energy of the particle is equal to plus or minus the mass [33, 41]. We find
that only in the massless limit the mass-shell constraint coincides with the flat
Carroll case. Using the AC particle action we will construct the Killing equations
for the AC space. We find that the solution of the Killing equations produces
an infinite-dimensional algebra that contains the symmetries of the AC algebra.
The Lifshitz dilatations are not included in these symmetries. Only in the flat
case the dilatations with z = 0 are part of the infinite dimensional algebra.
In the second part of this chapter we consider the supersymmetric extension
of the Carroll algebras 3. We first construct the N = 1 AC superalgebra in any
dimension (see Tables 2.5 and 7.1, where Q stands for the generator of super-
symmetry). The AC superalgebra in the flat limit contains the supersymmetric
extension of the ‘Lifshitz boost extended Carroll algebra’ introduced in appendix
B of [172]. We construct the AC superparticle action both as the non-relativistic
limit of the relativistic massive superparticle [143, 144] as well as by applying
the non-linear realization technique. As we will see the N = 1 AC superparticle
like in the Relativistic and Galilean case is non-BPS, i.e. the supersymmetries
are non-linearly realized. We will study the super-Killing equations and we find
in general an infinite-dimensional algebra of symmetries thereby extending the
finite N = 1 super AC transformations.
N = 1 [Mab, Q] [Pa, Q] {Qα, Qβ}
AdS-Carroll − 1
2γabQ
1
2RγaQ [γ0C−1]αβH + 2
R [γa0C−1]αβKa
Table 7.1
In this table we give the (anti-)commutators of the N = 1 Newton-Hooke and AdS-Carroll
superalgebras that involve the generators Q of supersymmetry. Note that here is no duality
between the two algebras.
Inspired by the relativistic and Galilei case we will investigate whether the
N = 2 Carroll superparticle is BPS or not. For simplicity we restrict to the
three-dimensional case. We first construct the N = 2 Carroll superalgebra as a
contraction of the N = 2 Poincaré superalgebra. This leads to the result given
with a gauge field [42].
3A first attempt in this direction was done in the unpublished notes [171].
138 Carroll Limit
in Table 7.2. We see that, unlike in the bosonic case, there is no duality in
the supersymmetric case. Next, we construct the action for the N = 2 Carroll
superparticle. This action has two terms, one of them is a Wess-Zumino (WZ)
term. If we properly choose the coefficients of the two terms we find a so-called
kappa gauge symmetry [152, 153] that kills half of the fermions. This gauge
symmetry has the form of a Stückelberg symmetry, similar to what we found in
the Galilean case [127, 162]. We find that after fixing the kappa-symmetry the
super-Carroll action reduces to the action we found in the N = 1 case. The
linearly realized supersymmetry acts trivially on all the fields and therefore the
N = 2 Carroll superparticle reduces to the N = 1 Carroll superparticle and
hence is not BPS. This is rather different from the N = 2 super-Galilei case were
BPS particles do exist. The main difference between the super-Carroll and super-
Galilei cases comes from the kappa symmetry transformations, in the former case
it eliminates the linearized supersymmetry and it the last case it does not.
N = 2 [Ka, Q
+] {Q+
α , Q
+
β } {Q+
α , Q
−
β } {Q−
α , Q
−
β }
Galilei − 1
2γa0Q
− [γ0C−1]αβH [γaC−1]αβPa 2[γ0C−1]αβZ
Carroll 0 1
2 [γ0C−1]αβ(H + 2Z) 0 1
2 [γ0C−1]αβ(H − 2Z)
Table 7.2
In this table we give the (anti-)commutators of the N = 2 Galilei and Carroll supersymmetry
algebras. Both algebras contain also the commutator [Mab, Q
±] = − 1
2
γabQ
±. Note that there
is no duality between these two algebras.
In a separate Appendix we extend our investigations to the N = 2 curved case
and consider the Carroll contraction of the so-called (p, q) AdS superalgebras [58]
for the particular cases of (p, q) = (2, 0) and (p, q) = (1, 1). We find that the
associated particle actions are rather different. While in the (2,0) case we have
kappa-symmetry, we find that this is not the case in the (1,1) case. The two
models have different degrees of freedom.
This chapter is organized as follows. In section 2 we discuss the bosonic
free AC particle thereby extending our previous analysis [41] to the curved case.
In particular, we construct the action and investigate the Killing equations. In
section 3 we consider the N = 1 AC superparticle. At the end of this section we
discuss the flat limit. Finally, in section 4 we investigate the N = 2 Super Carroll
particle. Our conclusions are presented in section 5. Some technical details and
the extension of the N = 2 Super Carroll particle to the curved case, for three
dimensions only, are given in the Appendix.
7.2 The Free AdS Carroll Particle 139
7.2. The Free AdS Carroll Particle
Before discussing the supersymmetric case we will first study in this section
different aspects of the free AdS Carroll (AC) particle.
7.2.1. The AdS Carroll Algebra
In order to write the commutators corresponding to the AC algebra, we will
start with the contraction of the D-dimensional AdS algebra. The basic commu-
tators are given by (A = 0, 1, . . . , D − 1)
[MAB,MCD] = 2ηA[CMD]B − 2ηB[CMD]A , (7.2)
[MAB, PC ] = 2ηC[BPA] , [PA, PB] =
1
R2
MAB , (7.3)
where R is the AdS radius. Here PA and MAB are the (anti-hermitian) generators
of space-time translations and Lorentz rotations, respectively.
To make the Carroll contraction we rescale the generators with a parameter
ω as follows [31,32]:
P0 =
ω
2
H , Ma0 = ωKa . (7.4)
Taking the limit ω → ∞ we find that the commutators corresponding to the
D-dimensional AC algebra are given by (a = 1, . . . , D − 1):
[Mab,Mcd] = 2δa[cMd]b − 2δb[cMd]a , [Mab,Kc] = 2δc[bKa] , (7.5)
[Mab, Pc] = 2δc[bPa] , [Pa,Kb] =
1
2
δabH , (7.6)
[Pa, Pb] =
1
R2
Mab , [Pa, H] =
2
R2
Ka . (7.7)
Notice that the commutation relations of space-time translation coincide with
the same commutation relations of the AdS algebra. The difference between the
AdS and AC algebra is in the different commutation relations that involve the
boost generators. Note that this is not the case for the Newton-Hook algebras.
The AC algebra can be expressed in terms of the left invariant Maurer-Cartan
1-forms La, which satisfy the Maurer-Cartan equations dLC − 1
2 f
C
ABL
BLA = 0.
Explicitly, these equations read
dLH +
1
2
L a
P L
a
K = 0 , dL a
P − 2L b
P L
ab
M = 0 , (7.8)
dL a
K − 2L b
K L
ab
M =
2
R2
LHLP
a , dL ab
M − 2L ca
M L cb
M =
1
2R2
LP
bLP
a . (7.9)
140 Carroll Limit
7.2.2. Non-Linear Realizations
In this subsection we apply the method of non-linear realizations [61,62] and
use the algebra (7.5) to construct the action of the AC particle.
We consider the coset G/H = AC/SO(D-1) and the coset element g = g0 U,
where g0 = eHtePax
a
is the coset representing the AC space and U = eKav
a
is
a general Carroll boost. The xa (a = 1, ...D − 1) are the Goldstone bosons of
broken translations, t is the Goldstone boson of the unbroken time translation4
and U is parametrized by the Goldstone bosons of the broken Carroll boost
transformations.
The reason to consider the coset element in terms of g0 and U is because in
this way we have that for a general symmetric space-time g0 is the coset element
representing the ‘empty’ space-time, while U represents the broken symmetries
that are due to the presence of a dynamical object, in our case a particle, in the
‘empty’ space-time. For the case of a particle U is given by the general rotation
that mixes the ‘longitudinal’ time direction with the ‘transverse’ space directions,
i.e. the Carroll boosts. If we would like to consider as a dynamical object a p-
brane, we should consider as U the general rotations that mix the longitudinal
and transverse directions [68].
Returning to the AC particle, it is interesting to write out the Maurer-Cartan
form Ω0 associated to the AC space
Ω0 = g−1
0 dg0 = He0 + Pae
a +Kaω
a0 +Mabω
ab , (7.10)
where (e0, ea) and (ωa0, ωab) are the space and time components of the Vielbein
and spin connection 1-forms of the AdS space, respectively. If we parametrize the
AdS space as eHtePax
a
, the Vielbein and spin-connection 1-forms corresponding
to the AC space are given by
e0 = dt cosh
x
R
,
ea =
R
x
dxa sinh
x
R
+
1
x2
xaxbdxb
(
1 − R
x
sinh
x
R
)
,
ωa0 = − 2
xR
dtxa sinh
x
R
,
ωab =
1
2x2
(xbdxa − xadxb)
(
cosh
x
R
− 1
)
.
(7.11)
4The unbroken translation P0 generates via a right action [67] [68] a transformation which is
equivalent to the world-line diffeomorphisms.
7.2 The Free AdS Carroll Particle 141
These 1-forms satisfy the structure equations
de0 +
1
2
e a ω a0 = 0 , de a − 2e b ω ab = 0 , (7.12)
de a − 2ω b0 ω ab =
2
R2
e0ea , dω ab − 2ω ca ω cb =
1
2R2
ebea . (7.13)
We see that the Vielbein satisfies the torsionless condition and that the AC space,
like the ancestor AdS space, has constant negative curvature.
We now insert a particle in the empty AC space and consider the Maurer-
Cartan form of the combined system:
Ω = g−1dg = U−1Ω0U + U−1dU . (7.14)
In order to derive an expression for Ω we need to know how the space-time
translation generators and the boost generators transform under a general Carroll
boost:
U−1H U = H , U−1 Pa U = Pa +
1
2
vaH ,
U−1Ka U = Ka , U−1Mab U = Mab + vbKa − vaKb .
(7.15)
We have also U−1dU = dvaKa. Using these formulae we find that the Maurer-
Cartan form Ω is given by
LH = e0 +
1
2
vae
a , LP
a = ea ,
LK
a = ω0a + dva + 2vb ω
ab , LM
ab = ωab .
(7.16)
We note that that the Maurer-Cartan forms of space-time translations can be
written in matrix-form as follows:
(
LH , LP
a ) =
(
e0, ea
)
(
1 0
1
2va 1
)
. (7.17)
The matrix appearing at the right-hand-side is the most general Carroll boost in
the vector representation.
We now proceed with the construction of an action of the AC particle. An
action with the lowest number of derivatives is obtained by taking the pull-back
of all the L’s that are invariant under rotations, see for example [68]. In this way
we obtain the following action:
S = M
∫
(LH)∗ = M
∫
(
e0 +
1
2
vae
a
)∗
= M
∫
dτ
(
ṫ cosh
x
R
+
R
2x
vaẋ
a sinh
x
R
+
1
2x2
xbvbxaẋ
a
(
1 − R
x
sinh
x
R
)
)
.
(7.18)
142 Carroll Limit
This action is invariant under the following transformation rules with constant
parameters (ζ, ai, λi, λij) corresponding to time translations, spatial translations,
boosts and spatial rotations, respectively:
δt = −ζ +
R
2x
λkxk tanh
x
R
+
t
Rx
akxk tanh
x
R
,
δxi = − 1
x2
(
xiakxk − x
R
coth
x
R
(xiakxk − aix2)
)
− 2λik x
k ,
δvi = −λi − 1
x2
λkxkx
isech
x
R
(
1 − cosh
x
R
)
− 2λij v
j − 2t
R2
ai
− 2t
R2x2
xiakxk sech
x
R
(
1 − cosh
x
R
)
+
2
Rx
vba
[ixb]csch
x
R
(
1 − cosh
x
R
)
.
(7.19)
The equations of motion for t, xa and va read
0 =
1
xR
xaẋa sinh
x
R
,
0 = − R
2x
ẋa sinh
x
R
− 1
2x2
xaxbẋ
b
(
1 − R
x
sinh
x
R
)
,
0 =
R
2x
v̇a sinh
x
R
− 1
xR
ṫxa sinh
x
R
+
1
2x2
xax
bv̇b
(
1 − R
x
sinh
x
R
)
+
ẋb
2x2
(vax
b − xavb)
(
cosh
x
R
− 1
)
.
(7.20)
These equations imply that
ẋa = 0 ,
1
xR
ṫxa sinh
x
R
=
R
2x
v̇a sinh
x
R
+
1
2x2
xax
bv̇b
(
1 − R
x
sinh
x
R
)
.
(7.21)
Notice that the evolution of va is non-trivial. If we take the limit R → ∞ we
recover the flat bosonic equations of motion ẋa = v̇a = 0 and therefore a trivial
dynamics for both xa, va [41].
The energy and spatial momenta of the free AC particle are given by
E = −∂L
∂ṫ
= −M cosh
x
R
,
pa =
∂L
∂ẋa
= M
[
R
2x
va sinh
x
R
+
1
2x2
xax
bvb
(
1 − R
x
sinh
x
R
)
]
.
(7.22)
7.2 The Free AdS Carroll Particle 143
They satisfy the constraint
E2 −M2 cosh2 x
R
= 0 . (7.23)
The canonical action of the AC particle is given by 5
S =
∫
dτ
[
−Eṫ+ paẋ
a − e
2
(
E2 −M2 cosh2 x
R
)]
. (7.24)
Note that if we calculate ṗa and impose both equations of motion (7.21) we obtain
ṗa =
M
Rx
ṫxa sinh
x
R
=
eM2
Rx
xa cosh
x
R
sinh
x
R
. (7.25)
In the last step we have used that ṫ = −eE = eM cosh x
R , see eq. (7.27). This is
the same result one finds using the Hamiltonian form given in eq. (7.42).
7.2.3. The Killing Equations of the AdS Carroll Particle
In order to find the Killing symmetries of the AC space, it is convenient
to consider the symmetries of the canonical action (7.24). The basic Poisson
brackets of the canonical variables occurring in the action (7.24) are given by
{E, t} = 1 , {e, πe} = 1 , {xi, pj} = δij . (7.26)
This leads to the following equations of motion:
ṫ = −eE , ẋi = 0 , Ė = 0 , ṗi =
eM2
2Rx
xi sinh
2x
R
,
π̇e = −1
2
(
E2 −M2 cosh2 x
R
)
, ė = λ .
(7.27)
Here λ = λ(τ) is an arbitrary function and πe is constrained by π̇e = 0.
We take as the generator of canonical transformations
G = −Eξ0(t, ~x, e) + pi ξ
i(t, ~x, e) + γ(t, ~x, e)πe , (7.28)
where ξ0 = ξ0(t, ~x, e), ξi = ξi(t, ~x, e) and γ = γ(t, ~x, e). The condition that this
generator generates a Noether symmetry is that it is a constant of motion and it
5Alternatively, we can obtain this action by taking the Carroll limit of the canonical action
of a massive particle in AdS, see appendix A.
144 Carroll Limit
leads to the following restrictions:
Ġ = 0 = −E(ṫ∂tξ
0 + ė∂eξ
0) + ṗiξ
i + pi(ṫ∂tξ
i + ė∂eξ
i) + γπ̇e
= eE2∂tξ
0 − λE∂eξ
0 +
eM2
2Rx
xiξ
i sinh
2x
R
− eEpi∂tξ
i + λpi∂eξ
i − γ
2
(
E2 −M2 cosh2 x
R
)
.
(7.29)
From this equation we deduce the following equations describing the symmetries
of the AC space:
∂eξ
0 = 0 , ∂eξ
i = 0 , ∂tξ
i = 0 ,
γ = 2e∂tξ
0 ,
e
xR
xiξ
i sinh
x
R
+
1
2
γ cosh
x
R
= 0 .
(7.30)
The last two equations can be combined into the single condition
∂tξ
0 = − 1
xR
xiξ
i tanh
x
R
. (7.31)
The generator G is given by
G = −Eξ0(t, ~x) + pi ξ
i(~x) + γ(t, ~x, e)πe . (7.32)
From the variation of the momenta we can obtain the transformation rules for vi
as follows. First, we use that
δpi = {pi, G} = {pi,−Eξ0(t, ~x) + pi ξ
i(~x) + 2e∂tξ
0(t, ~x)πe}
= E∂iξ
0 − pk∂iξ
k − 2e∂t∂iξ
0πe . (7.33)
Next, using eq. (7.22) and πe = 0 we obtain
δpi = −M cosh
x
R
∂iξ
0−M
[
R
2x
vi sinh
x
R
+
1
2x2
xix
bvb
(
1−R
x
sinh
x
R
)
]
∂iξ
k . (7.34)
Finally, using the expression for pi given in eq. (7.22), we obtain the following
7.2 The Free AdS Carroll Particle 145
transformations of the variables vi:
δvi = − 2x
R
∂iξ
0 coth
x
R
− va∂iξa − 1
Rx
vbx
bxk∂iξ
k
(
1 − R
x
sinh
x
R
)
+
2
Rx
coth
x
R
(
1 − R
x
sinh
x
R
)
xixa∂
aξ0 +
1
Rx
(R
x
− coth
x
R
)
vixbξ
b
− 1
Rx
(R
x
− R2
x2
sinh
x
R
)(
xixbξ
b − 1
x2
xixa∂
aξkv
k
)
+
1
Rx3
csch
x
R
(
− R
x
sinh
x
R
− R2
x2
sinh2 x
R
+ 1 + cosh
x
R
)
xixbξ
bxkv
k
+
1
Rx
csch
x
R
(
− 2
R
x
sinh
x
R
− R2
x2
sinh2 x
R
+ 1 + cosh
x
R
)
xixbξ
bxkv
k
+
1
Rx
csch
x
R
(R
x
sinh
x
R
− 1
)
ξixbv
b .
(7.35)
We see that the free Carroll particle in an AdS background has an infinite-
dimensional symmetry. A possible solution to these equations is given by eq.
(7.19) which are the symmetry transformations of the Carroll group. We do not
find any Lifshitz dilatations in this case i.e., a transformation with parameters
ξi = xi, ξ0 = zt.
The Massles Limit
Using the canonical action
S =
∫
dτ
[
−Eṫ+ paẋ
a − e
2
(
E2 −M2 cosh2 x
R
)]
, (7.36)
it is straightforward to take the massless limit M → 0 and obtain the action
S =
∫
dτ
(
−Eṫ+ paẋ
a − e
2
E2
)
. (7.37)
We see that in the massless limit the R-dependence of the AC particle has dis-
appeared. This means that the massive Carroll particles are affected by the
geometry but the massless Carroll particles are not. Consequently, in the mass-
less limit there is no difference between particles in an AdS or flat background.
Furthermore, the isometries should be given by the most general conformal Car-
roll group as it was analyzed in [41]. In this case dilatations are included i.e.,
with parameters ξi = xi, ξ0 = zt.
146 Carroll Limit
7.2.4. The Carroll action as a Limit of the AdS action
In this section we show how to obtain the action of the D-dimensional free
AdS Carroll particle starting from the massive particle moving in a D-dimensional
AdS spacetime and to take the Carroll limit. The canonical form of the action
before taking the limit is given by
S =
∫
dτ [pµẋµ − ẽ
2
(gµνp
µpν +m2)] , (7.38)
where τ is the evolution parameter, gµν is the metric of an AdS space and ẽ is
a Lagrange multiplier. We use that the signature of the metric is (−,+,+, . . .)
and that the AdS line element is given by
ds2 = − cosh2 x
R
(dx0)2 +
R2
x2
sinh2 x
R
(dxa)2 −
(
R2
x2
sinh2 x
R
− 1
)
(dx)2 , (7.39)
where x =
√
xaxa. To take the Carroll limit we first consider a re-scaling of the
variables
x0 =
t
ω
, p0 = ωE , m = ωM , ẽ = − e
ω2
, (7.40)
and next take the limit ω → ∞ to obtain
S =
∫
dτ [−Eṫ+ paẋa − e
2
(E2 −M2 cosh2 x
R
)] . (7.41)
The equations of motion are given by
ṫ = −eE , Ė = 0 ,
ẋa = 0 , ṗa =
eM2
Rx
xa cosh
x
R
sinh
x
R
,
ė = λ , πe = −1
2
(E2 −M2 cosh2 x
R
) .
(7.42)
Note that although the dynamics of x is trivial, i.e. ẋa = 0 (the particle is not
changing its position), the momentum is changing over τ because ṗa 6= 0. In the
flat limit (the limit when R → ∞) the particle is at rest and does not move.
Finally, the mass-shell constraint reads
E2 −M2 cosh2 x
R
= 0 . (7.43)
7.3 The N = 1 AdS Carroll Superalgebra 147
7.3. The N = 1 AdS Carroll Superalgebra
We start by taking the contraction of the D-dimensional N = 1 AdS algebra.
The basic commutators are given by (A = 0, 1, . . . , D − 1)
[MAB,MCD] = 2ηA[CMD]B − 2ηB[CMD]A ,
[MAB, PC ] = 2ηC[BPA] , [PA, PB] = 4x2MAB ,
[MAB, Q] = −1
2
γABQ , [PA, Q] =
1
2R
γAQ ,
{Qα, Qβ} = 2[γAC−1]αβPA +
1
R
[γABC−1]αβMAB ,
(7.44)
where R is the AdS radius and PA,MAB and Qα are the generators of space-
time translations, Lorentz rotations, and supersymmetry transformations, re-
spectively. The bosonic generators PA and MAB are anti-hermitian while the
fermionic generator Qα is hermitian.
To make the Carroll contraction we rescale the generators with a parameter
ω as follows:
P0 =
ω
2
H , Ma0 = ωKa , Q =
√
ωQ̃ ; a = 1, 2, . . . , D − 1
(7.45)
Taking the limit ω → ∞ and dropping the tildes on the Q we get the following
N = 1 AdS Carroll superalgebra:
[Mab, Pc] = 2δc[bPa] , [Mab,Kc] = 2δc[bKa] ,
[Pa, Pb] =
1
R2
Mab , [Pa,Kb] =
1
2
δabH , [Pa, H] =
2
R2
Ka ,
[Pa, Q] =
1
2R
γaQ , [Mab, Q] = −1
2
γabQ ,
{Qα, Qβ} = [γ0C−1]αβH +
2
R
[γa0C−1]αβKa .
(7.46)
The Maurer-Cartan equation dLC − 1
2 f
C
ABL
BLA = 0 in components reads
dLH =−1
2L
a
P L
a
K − 1
2 L̄Qγ
0LQ , dL a
P = 2L b
P L
ab
M ,
dL a
K =2L b
K L
ab
M + 2
R2LHLP
a − 1
R L̄Qγ
a0LQ , dL ab
M = 2L ca
M L cb
M + 1
2R2LP
bLP
a ,
dLQ =1
2γabLQ L
ab
M − 1
2RγaLQLP
a .
(7.47)
148 Carroll Limit
7.3.1. Superparticle Action
We now use the algebra (7.46) to construct the action of the AC superparticle
with the coset
G
H
=
N = 1 AdS Carroll
SO(D-1)
(7.48)
that is locally parametrized as g = g0 U, where g0 = eHtePax
a
eQαθ
α
is the coset
representing the ‘empty’ curved AC Carroll superspace and U = eKav
a
is a general
Carroll boost representing the particle inserted in the empty space. The Maurer-
Cartan form Ω0 associated to the empty AC superspace is given by
Ω0 = g−1
0 dg0 = HE0 + PaE
a +Kaω
a0 +Mabω
ab − Q̄E , (7.49)
where (E0, Ea, Eα) and (ωa0, ωab) are the time and space components of the
supervielbein and the spin connection of super-AdS if we parametrize the AdS
superspace as eHtePax
a
eQαθ
α
. The explicit expressions for these components are
given by
E0 = dt cosh
x
R
− 1
2
θ̄γ0dθ − 1
2
ωabθ̄γabγ
0θ ,
Ea =
R
x
dxa sinh
x
R
+
1
x2
xaxbdxb
(
1 − R
x
sinh
x
R
)
,
ωa0 = − 2
xR
dtxa sinh
x
R
− 1
R
θ̄γa0dθ − 1
2R2
θ̄γabγ
0θEb ,
ωab =
1
2x2
(xbdxa − xadxb)
(
cosh
x
R
− 1
)
,
Eα = dθα − 1
2R
[γaθ]αE
a +
1
2
ωab[γabθ]α .
(7.50)
In this case we have torsion given by T0 = −1
2Ē
αγ0Eα and a non-vanishing spin
connection. The Maurer-Cartan form for the N = 1 AC superparticle inserted
in the AC superspace is given by
Ω = g−1dg = U−1Ω0U + U−1dU , (7.51)
where
LH = E0 +
1
2
vaE
a , LP
a = Ea ,
LK
a = ωa0 + dva + 2vb ω
ab , LM
ab = ωab ,
LQα = Eα .
(7.52)
Note that the Maurer-Cartan forms of the spacetime supertranslations can be
written in matrix form in terms of the Supervielbein components of the AC
7.3 The N = 1 AdS Carroll Superalgebra 149
superspace as follows:
(
LH , LP
a, LQα
)
=
(
E0, Ea, Eα
)


1 0 0
1
2va 1 0
0 0 1

 . (7.53)
Like in the bosonic case the Maurer-Cartan forms of the supertranslations of
the AC superparticle can be obtained from the Maurer-Cartan forms of the AC
superspace by a matrix representation of the Carroll boost.
The action of the N = 1 AC superparticle is given by the pull-back of all the
L’s that are invariant under rotations:
S = M
∫
(LH)∗ = M
∫
(
E0 +
1
2
vaE
a
)∗
=
= M
∫
dτ
(
ṫ cosh
x
R
+
R
2x
vaẋ
a sinh
x
R
+
1
2x2
xbvbxaẋ
a
(
1 − R
x
sinh
x
R
)
− 1
2
θ̄γ0θ̇ − 1
4x2
xbẋaθ̄γabγ
0θ
(
cosh
x
R
− 1
)
)
.
(7.54)
The equations of motion corresponding to this action can be written as follows
ẋi = 0 , θ̇ = 0 ,
1
xR
ṫxa sinh
x
R
=
R
2x
v̇a sinh
x
R
+
1
2x2
xax
bv̇b
(
1 − R
x
sinh
x
R
)
.
(7.55)
We can write a Hamiltonian version of this action with the momenta given by
pt = M cosh
x
R
,
pa = M
[
R
2xva sinh x
R + 1
2x2xax
bvb
(
1 − R
x sinh x
R
)
− 1
4x2x
bθ̄γabγ
0θ
(
cosh x
R − 1
)]
,
P̄θ =
M
2
θ̄γ0 .
(7.56)
Then, the canonical form of (7.54) is
S =
∫
dτ
[
−ṫE + ẋap
a + ¯̇θPθ − e
2
(
E2 −M2 cosh2 x
R
)
−
(
P̄θ cosh x
R + 1
2Eθ̄γ
0
)
ρ
]
.
(7.57)
The bosonic transformation rules for the coordinates with constant parameters
(ζ, ai, λi, λij) corresponding to time translations, spatial translations, boosts and
150 Carroll Limit
rotations, respectively, are given by
δt = −ζ +
R
2x
λkxk tanh
x
R
+
t
Rx
akxk tanh
x
R
,
δxi = − 1
x2
(
xiakxk − x
R
coth
x
R
(xiakxk − aix2)
)
− 2λik x
k ,
δvi = −λi − 1
x2
λkxkx
isech
x
R
(
1 − cosh
x
R
)
− 2λij v
j − 2t
R2
ai
− 2t
R2x2
xiakxk sech
x
R
(
1 − cosh
x
R
)
+
2
Rx
vba
[ixb]csch
x
R
(
1 − cosh
x
R
)
,
δθ = −1
2
λabγabθ +
1
2Rx
akxbγkbθcsch
x
R
(
1 − cosh
x
R
)
.
The fermionic transformation rules with constant parameter ǫ corresponding to
the supersymmetry transformation are given by
δt =
1
2
ǭγ0θsech
x
R
cosh
x
2R
− 1
2x
xk ǭγk0θsech
x
R
sinh
x
2R
,
δxi = 0 ,
δvi =
1
Rx
xi tanh
x
R
(
ǭγ0θ cosh
x
2R
− 1
x
xk ǭγk0θ sinh
x
2R
)
− 1
Rx
xiǭγ0θ sinh
x
2R
+
1
R
ǭγi0θ cosh
x
2R
+
1
Rx
xk ǭγik0θ sinh
x
2R
,
δθ = ǫ cosh
x
2R
+
1
x
xkγkǫ sinh
x
2R
.
(7.58)
7.3.2. The Super Killing Equations
The basic Poisson brackets of the canonical variables are given by
{E, t} = 1 , {e, πe} = 1 , {xi, pj} = δij ,
{Pαθ , θβ} = −δαβ , {Πα
ρ , ρβ} = −δαβ ,
(7.59)
and the corresponding Dirac Hamiltonian of the action (7.57) is given by
HD =
e
2
(
E2 −M2 cosh2 x
R
)
+ λπe +
(
P̄θ cosh
x
R
+
1
2
Eθ̄γ0
)
ρ+ π̄ρΛ , (7.60)
πe = 0 and Πρ = 0 are the primary constraints, λ = λ(τ) and Λ = Λ(τ) are
arbitrary functions. The corresponding primary Hamiltonian equations of motion
7.3 The N = 1 AdS Carroll Superalgebra 151
are given by
ṫ = −eE − 1
2
θ̄γ0ρ , Ė = 0 ,
ẋi = 0 , ṗi =
eM2
xR
xi cosh
x
R
sinh
x
R
− 1
xR
xi sinh
x
R
P̄θρ ,
(7.61)
π̇e = −1
2
(
E2 −M2 cosh2 x
R
)
, ė = λ ,
θ̇ = −ρ cosh
x
R
, ¯̇Pθ = −1
2
Eρ̄γ0 , ρ̇ = −Λ , ¯̇Πρ = P̄θ cosh
x
R
+
1
2
Eθ̄γ0 .
(7.62)
The stability of primary constraints give as secondary constraint the mass-shell
condition E2 −M2 cosh2 x
R = 0 and the fermionic constraint P̄θ cosh x
R + 1
2Eθ̄γ
0 =
0. If we require the stability of the secondary constraints we get ρ = 0. Substitut-
ing this into (7.62) and using the canonical momenta (7.56) we obtain equations
(7.55).
The generator of canonical transformations has a bosonic and a fermionic part
given by
G = −Eξ0(t, ~x, θ)+pi ξ
i(t, ~x, θ)+γ(t, ~x, θ)πe− P̄θχ(t, ~x, θ)+Π̄ρΓ(t, ~x, θ) , (7.63)
the parameters ξ0 = ξ0(t, ~x, θ), ξi = ξi(t, ~x, θ), χ = χ(t, ~x, θ), γ = γ(t, ~x, θ) have
the following restrictions
0 = Ġ
= −E(ṫ∂tξ0 + ∂θξ
0θ̇) + ṗiξ
i + pi(ṫ∂tξi + ∂θξ
iθ̇) + γπ̇e − ¯̇Pθχ− P̄θ(∂tχṫ+ ∂θχθ̇) + ¯̇ΠρΓ
= eE2∂tξ
0 +
1
2
E∂tξ
0θ̄γ0ρ+ E∂θξ
0ρ cosh
x
R
+
eM2
xR
xiξi cosh
x
R
sinh
x
R
− 1
xR
xiξi sinh
x
R
P̄θρ− eEpi∂tξ
i − 1
2
pi∂tξ
iθ̄γ0ρ− pi∂θξ
iρ cosh
x
R
− 1
2
γ
(
E2 −M2 cosh2 x
R
)
+
E
2
ρ̄γ0χ+ eEP̄θ∂tχ+
1
2
P̄θ∂tχθ̄γ
0ρ
+ P̄θ∂θχρ cosh
x
R
+ P̄θΓ cosh
x
R
+
E
2
θ̄γ0Γ .
(7.64)
152 Carroll Limit
From this equation we derive the super-Killing equations
γ = 2e∂tξ
0 , Γ = −∂θχρ+
1
xR
xiξi tanh
x
R
ρ ,
∂tξ
0 = − 1
xR
xiξi tanh
x
R
, ∂θξ
0 =
1
2
χ̄γ0sech
x
R
+
1
2
θ̄γ0∂θχsech
x
R
,
∂tξ
i = 0 , ∂θξ
i = 0 , ∂tχ = 0 .
(7.65)
The solution to this equations is given by eqs. (7.58) and (7.58) with the symmetry
generator G given by
G = −Eξ0(~x, θ) + pi ξ
i(t, ~x) + 2e∂tξ
0(~x, θ)πe − P̄θχ(~x, θ)
+ Π̄ρ
(
− ∂θχ(~x, θ)ρ+
1
xR
xiξi(~x, θ) tanh
x
R
ρ
)
.
(7.66)
Then, the N = 1 AC superparticle has an infinite dimensional algebra with the
transformation rules given by (7.58) and (7.58).
7.3.3. The Flat Limit
We end this section with some comments on the flat limit (R → ∞) which
can be taken directly from the AC curved case in order to obtain the dynamics
and symmetries of the N = 1 flat Carroll superparticle. In this case, the time
and space components of the supervielbein simplify to
E0 = dt− 1
2
θ̄γ0dθ, Ea = dxa , Eα = dθα . (7.67)
In the R → ∞ limit, the torsion becomes T0 = −1
2dθ̄γ
0dθ and since we are
studying the flat case, the spin connection vanishes. The supertranslations can
be again written in terms of the supervielbein in matrix form as in (7.53) and
the action is given by
S = M
∫
(
E0 +
1
2
vaE
a
)∗
= M
∫
dτ(ṫ− 1
2
θ̄γ0θ̇ +
1
2
vaẋ
a) . (7.68)
The equations of motion that follow from this action are:
~̇x = ~̇v = θ̇ = 0 . (7.69)
Therefore, the superparticle does not move. The transformation rules of the
different variables are given by
δt = −ζ +
1
2
λixi +
1
2
ǭγ0θ , δxi = −ai − 2λij x
j ,
δvi = −λi − 2λij v
j , δθ = −1
2
λijγ
ijθ + ǫ .
(7.70)
7.3 The N = 1 AdS Carroll Superalgebra 153
As we can see from the transformation of θ the N = 1 Carroll superparticle is
not BPS like in the relativistic and Galilean case.
If we rewrite the action (7.68) in Hamiltonian form
S =
∫
dτ
[
−ṫE + ẋap
a + ¯̇θPθ − e
2
(E2 −M2) −
(
P̄θ +
1
2
Eθ̄γ0
)
ρ
]
, (7.71)
it turns out that the super-Killing equations can be obtained as the flat limit of
the equations (7.65)
γ = 0 , Γ = −∂θχρ , ∂tξ
0 = 0 , ∂tξ
i = 0 , ∂θξ
i = 0 , ∂tχ = 0 ,
∂θξ
0 =
1
2
χ̄γ0 +
1
2
θ̄γ0∂θχ ,
(7.72)
where the symmetry generator G is
G = −Eξ0(~x, θ) + pi ξ
i(~x) − P̄θχ(~x, θ) − Π̄ρ∂θχ(~x, θ)ρ . (7.73)
From the variation of the momenta
δpi = {pi, G} = E∂iξ
0 − pk∂iξ
k + P̄θ∂iχ (7.74)
and using that the energy, the spatial momenta and the fermionic momenta are
given by
E = −M , pi =
M
2
vi , P̄θ =
M
2
θ̄γ0 , (7.75)
we find that the transformation rule of vi
δvi = −2∂iξ0 − vk∂iξ
k + θ̄γ0∂iχ . (7.76)
Note that the above symmetries include the dilatations given by
δt = 0, δxa = xa, δθ = 0, δva = −va . (7.77)
These dilatations, together with the super-Carroll transformations, form a super-
symmetric extension of the Lifshitz Carroll algebra [172] with dynamical exponent
z=0. The Lifshitz Carroll algebra with z=0 has appeared in a recent study of
warped conformal field theories [43].
154 Carroll Limit
7.3.4. The super-AdS Carroll action as a limit of the super-AdS
action
In the supersymmetric case, we obtain the action of the free AdS Carroll
superparticle starting from the massive superparticle moving in an AdS spacetime
whose action is given by
S =
∫
dτ [ẋµp
µ + ¯̇φPφ − ẽ
2
(gµνp
µpν +m2) + (P̄φ + gµνp
µφ̄γν)λ] , (7.78)
where gµν is the AdS metric with line element given by eq. (A.2). Rescaling the
variables as
x0 =
t
ω
, p0 = ωE , m = ωM , ẽ = − e
ω2
,
φ =
1√
ω
θ , Pφ =
√
ω Pθ , λ =
1√
ω
ρ ,
(7.79)
allows us to take the Carroll limit with ω → ∞ to obtain
S =
∫
dτ [−ṫE + ẋap
a + ¯̇θPθ − e
2
(
E2 −M2 cosh2 x
R
)
+ (P̄θ cosh
x
R
+ E θ̄γ0)ρ] .
(7.80)
The primary equations of motion are
ṫ = −eE − θ̄γ0ρ , Ė = 0 ,
ẋa = 0 , ṗa =
eM2
Rx
xa cosh
x
R
sinh
x
R
− 1
xR
xa sinh
x
R
P̄θρ ,
ė = λ , πe = −1
2
(E2 −M2 cosh2 x
R
) ,
θ̇ = − cosh
x
R
ρ , ¯̇Pθ = −E ρ̄γ0 ,
ρ̇ = −Λ , ¯̇Πρ = P̄θ cosh
x
R
+ E θ̄γ0 .
(7.81)
After requiring the stability of all the constraints we obtain the equations of
motion (7.55). Like in the bosonic case we find that the dynamics of x is trivial,
ẋa = 0 (the particle is not changing its position), but that the momentum is
changing over τ because ṗa 6= 0.
7.4. The N = 2 Flat Carroll Superparticle
In this Section we extend our investigations to the N = 2 supersymmetric
case. The flat case is discussed in this Section while the curved case will be dealt
with in Appendix C.
7.4 The N = 2 Flat Carroll Superparticle 155
7.4.1. The N = 2 Carroll Superalgebra
Our starting point is the N = 2 super-Poincaré algebra. For simplicity, we
consider 3D only. The basic commutators are (A = 0, 1, 2; i = 1, 2)
[MAB,MCD] = 2ηA[CMD]B − 2ηB[CMD]A ,
[MAB, PC ] = 2ηC[BPA] ,
[MAB, Q
i] = −1
2
γABQ
i ,
{Qiα, Qjβ} = 2[γAC−1]αβPAδij + 2[C−1]αβǫijZ .
(7.82)
To make the Carroll contraction we define new supersymmetry charges by
Q±
α =
1
2
(Q1
α ± γ0Q
2
α) (7.83)
and rescale the different symmetry generators with a parameter ω as follows:
P0 =
ω
2
H , Ma0 = ωKa , Z = ωZ̃ ,
Q+ =
√
ωQ̃+ , Q− =
√
ωQ̃− .
(7.84)
Taking the limit ω → ∞ we obtain the following 3D Carroll algebra
[Mab,Kc] = 2δc[bKa] , [Mab, Pc] = 2δc[bPa] ,
[Ka, Pb] = −1
2
δabH , [Mab, Q̃
±] = −1
2
γabQ̃
±
{Q̃+
α , Q̃
+
β } = [γ0C−1]αβ
(1
2
H + Z̃
)
, {Q̃−
α , Q̃
−
β } = [γ0C−1]αβ
(1
2
H − Z̃
)
.
(7.85)
The Maurer-Cartan equation dLC − 1
2f
C
ABL
B ∧LA = 0 in components reads:
dLH = −1
2
L a
P L
a
K − 1
4
L̄−γ
0L− − 1
4
L̄+γ
0L+ , dL a
P = 2L b
P L
ab
M ,
dLZ = −1
2
L̄+γ
0L+ +
1
2
L̄−γ
0L− , dL a
K = 2L b
K L
ab
M ,
dL− =
1
2
γabL− L
ab
M , dL+ =
1
2
γabL+ L
ab
M ,
dL ab
M = 2L ca
M L cb
M .
(7.86)
156 Carroll Limit
7.4.2. Superparticle Action and Kappa Symmetry
To construct the action of the N = 2 Carrollian superparticle we consider the
following coset:
G
H
=
N = 2 super Carroll
SO(D-1)
. (7.87)
The coset element is given by g = g0 U, where g0 = eHtePax
a
eQ
−
α θ
α
−eQ
+
α θ
α
+eZs is the
coset representing the ‘empty’ N = 2 Carroll superspace with a central charge
extension and U = eKav
a
is a general Carroll boost representing the insertion of
the particle.
The Maurer-Cartan form associated to the super-Carroll space is given by
Ω0 = (g0)−1dg0 = HE0 + PaE
a − Q̄−E− − Q̄+E+ + ZEZ , (7.88)
where (E0, Ea, E−α, E+α, EZ) are the supervielbein components of the Carroll
superspace given explicitly by
E0 = dt− 1
4
θ̄−γ
0dθ− − 1
4
θ̄+γ
0dθ+ , Ea = dxa ,
E−α = dθ−α , E+α = dθ+α ,
EZ = ds+
1
2
θ̄−γ
0dθ− − 1
2
θ̄+γ
0dθ+ .
(7.89)
In terms of the supervielbein the Maurer-Cartan form of the N = 2 Carroll
superparticle is given by
LH = E0 +
1
2
vaE
a , LaP = Ea ,
LZ = EZ , LaK = dva ,
L−α = E−α , L+α = E+α .
(7.90)
As before, we can write the space-time super-translations in matrix form in terms
of the Vielbein of Carroll superspace as follows:
(
LH , LP
a, L−α, L+α, LZ
)
=
(
E0, Ea, E−α, E+α, EZ
)







1 0 0 0 0
1
2va 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1







.
(7.91)
7.4 The N = 2 Flat Carroll Superparticle 157
The action of the N = 2 Carrollian superparticle is given by the pull-back of
all the L’s that are invariant under rotations:
S = a
∫
(LH)∗ + b
∫
(LZ)∗
= a
∫
dτ
(
ṫ− 1
4 θ̄−γ
0θ̇− − 1
4 θ̄+γ
0θ̇+ + 1
2vaẋ
a
)
+ b
∫
dτ
(
ṡ+ 1
2 θ̄−γ
0θ̇− − 1
2 θ̄+γ
0θ̇+
)
.
(7.92)
The equations of motion corresponding to this action are given by
ẋa = 0 , v̇a = 0 , θ̇− = 0 , θ̇+ = 0 . (7.93)
The transformation rules for the coordinates with constant parameters (ζ, η, ai,
λi, λij , ǫ+, ǫ−) corresponding to time translations, Z transformations, spatial
translations, boosts, rotations and supersymmetry transformations, respectively,
are given by
δt = −ζ +
1
2
λixi +
1
4
ǭ−γ
0θ− +
1
4
ǭ+γ
0θ+ , δxi = −ai − 2λij x
j ,
δs = −η − 1
2
ǭ−γ
0θ− +
1
2
ǭ+γ
0θ+ , δvi = −λi − 2λij v
j ,
δθ+ = −1
2
λabγabθ+ + ǫ+ , δθ− = −1
2
λabγabθ− + ǫ− .
(7.94)
To derive an action that is invariant under additional κ-transformations we
need to find a fermionic gauge-transformation that leaves LH and/or LZ invari-
ant. The variation of LH and LZ under gauge-transformations is given by
δLH = d([δzH ]) +
1
2
LaP [δzaK ] +
1
2
LaK [δzaP ] +
1
2
L̄−γ
0[δz−] +
1
2
L̄+γ
0[δz+] ,
δLZ = d([δzZ ]) − L̄−γ
0[δz−] + L̄+γ
0[δz+] .
(7.95)
where [δzaK ] is obtained from LH by changing the 1-forms dt, dθ+, dθ− with the
transformations δt, δθ+, δθ−. In analogous way we can construct the other terms
appearing in (7.95).
For κ-transformations, [δzH ] = 0, [δzaK ] = 0, [δzaP ] = 0,
0 = δLH =
1
2
δθ̄−γ
0[δz−] +
1
2
δθ̄+γ
0[δz+] ,
0 = δLZ = −δθ̄−γ
0[δz−] + δθ̄+γ
0[δz+] .
(7.96)
It follows that to obtain a κ-symmetric action we need to take b = ±1
2a. We
focus here on the case b = −1
2a. With this choice the action and κ-symmetry
158 Carroll Limit
rules are given by
S = a
∫
(LH − 1
2
LZ)∗ , [δz+] = κ , [δz−] = 0 , (7.97)
where κ = κ(τ) is an arbitrary local parameter. Using this we find the following
κ-transformations of the coordinates
δt =
1
4
θ̄+γ
0κ , δxa = 0 , δθ+ = κ ,
δs =
1
2
θ̄+γ
0κ , δva = 0 , δθ− = 0 .
(7.98)
After fixing the κ-symmetry, by imposing the gauge condition θ+ = 0, the action
reduces to
S = a
∫
dτ
(
ṫ− 1
2
ṡ− 1
2
θ̄−γ
0θ̇− +
1
2
vaẋ
a
)
. (7.99)
The residual transformations that leave this action invariant are given by
δt = −ζ +
1
2
λixi +
1
4
ǭ−γ
0θ− , δxi = −ai − 2λij x
j ,
δs = −η − 1
2
ǭ−γ
0θ− , δvi = −λi − 2λij v
j ,
δθ− = −1
2
λabγabθ− + ǫ− .
(7.100)
The linearly realized supersymmetry acts trivially on all the fields and therefore
the N = 2 Super Carroll particle reduces to the N = 1 Super Carroll particle and
hence is not BPS since the kappa-symmetry eliminates the linearized supersym-
metry. This is different from the N = 2 Super Galilei case were BPS particles do
exist.
7.5. Discussion
In this chapter we have investigated the geometry of the flat and curved
(AdS) Carroll space both in the bosonic as well as in the supersymmetric case.
We furthermore have analyzed the symmetries of a particle moving in such a
space. In the bosonic case we constructed the Vielbein and spin connection of the
AdS Carroll (AC) space which shows that this space is torsionless with constant
(negative) curvature. We constructed the action of a massive particle moving in
this space thereby extending the flat case analysis of [41]. Like in the flat case,
we found that the AC particle does not move. However, in the curved case the
7.A The 3D N = 2 AdS Carroll Superparticle 159
momenta are not conserved. Particles moving in a Carroll space, whether flat or
curved, do not have a relation among their velocities and momenta.
Using the symmetries of the AC particle we have computed the Killing equa-
tions of the AC space. We found that these Killing equations allow an infinite-
dimensional algebra of symmetries that, unlike in the flat case, does not include
dilatations. Another difference with the flat case is that there is no duality be-
tween the Newton-Hooke and AdS Carroll algebras. Furthermore, in the curved
case the mass-shell constraint depends on the coordinates of the AC space.
In the second part of this chapter we have extended our investigations to
the supersymmetric case. Unlike the bosonic case, the N = 1 AC superspace
has torsion with constant curvature due to the presence of fermions. Like in the
bosonic case, we found that the N = 1 AC superparticle does not move and the
momenta are conserved. We have constructed the super-Killing equations and
showed that the symmetries form an infinite dimensional superalgebra. After
taking the flat limit we found that among the symmetries of the N = 1 Carroll
superparticle we have a supersymmetric extension of the Lifshitz Carroll algebra
[172] with dynamical exponent z = 0. The bosonic part of this algebra has
appeared as a symmetry of warped conformal field theories [43].
We also showed that the N = 2 Carroll superparticle has a fermionic kappa-
symmetry such that, when this gauge symmetry is fixed, the N = 2 Carroll
superparticle reduces to the N = 1 Carroll superparticle. Apparently, in flat
Carroll superspace the number of supersymmetries is not physically relevant.
This is due to the fact that the kappa gauge symmetry neutralizes the extra
linear supersymmetries beyond N = 1. Unlike the bosonic case, there is no
duality between the N = 2 Super Galilei and Super Carroll algebras.
In a separate appendix we investigated the N = 2 AC superparticle 6. We
studied the so-called (2,0) and (1,1) super-Carroll spaces and the corresponding
superparticles. Physically, the (2,0) and (1,1) cases are different, they have un-
equal degrees of freedom. For instance, only the (2,0) superparticle has a kappa-
symmetry. Apparently, for the AC superparticle the type of supersymmetry one
considers does make a difference.
7.A. The 3D N = 2 AdS Carroll Superparticle
There are two independent versions of the 3D N = 2 AdS algebra, the so-
called N = (1, 1) and N = (2, 0) algebras. Correspondingly, there are two
possible N = 2 AdS Carroll superalgebras which we consider below.
6 For simplicity we did only consider the 3D case.
160 Carroll Limit
7.A.1. The N = (2, 0) AdS Carroll Superalgebra
We will start with the contraction of the 3D N = (2, 0) AdS algebra. The
basic commutators are given by (A = 0, 1, 2; i = 1, 2)
[MAB,MCD] = 2ηA[CMD]B − 2ηB[CMD]A , [MAB, Q
i] = −1
2
γABQ
i ,
[MAB, PC ] = 2ηC[BPA] , [PA, Q
i] = xγAQ
i ,
[PA, PB] = 4x2MAB , [R, Qi] = 2xǫijQj ,
{Qiα, Qjβ} = 2[γAC−1]αβPAδ
ij + 2x[γABC−1]αβMABδ
ij + 2[C−1]αβǫ
ijR .
(7.101)
Here PA,MAB,R and Qiα are the generators of space-time translations, Lorentz
rotations, SO(2) R-symmetry transformations and supersymmetry transforma-
tions, respectively. The bosonic generators PA,MAB and R are anti-hermitian
while the fermionic generators Qiα are hermitian. The parameter x = 1/(2R),
with R being the AdS radius. Note that the generator of the SO(2) R-symmetry
becomes the central element of the Poincaré algebra in the flat limit x → 0.
To take the Carroll contraction we define new supersymmetry charges by
Q±
α =
1
2
(Q1
α ± γ0Q
2
α) (7.102)
and rescale the generators with a parameter ω as follows:
P0 =
ω
2
H , R = ωZ , Ma0 = ωKa , Q± =
√
ωQ̃± . (7.103)
Taking the limit ω → ∞ and dropping the tildes on the Q± we get the following
3D N = (2, 0) Carroll superalgebra:
[Mab, Pc] = 2δc[bPa] , [Mab,Kc] = 2δc[bKa] ,
[Pa, Pb] =
1
R2
Mab , [Pa,Kb] =
1
2
δabH , [Pa, H] =
2
R2
Ka
[Pa, Q
±] =
1
2R
γaQ
∓ , [Mab, Q
±] = −1
2
γabQ
± ,
{Q+
α , Q
+
β } =
1
2
[γ0C−1]αβ (H + 2Z) , {Q−
α , Q
−
β } =
1
2
[γ0C−1]αβ (H − 2Z) ,
{Q+
α , Q
−
β } =
1
R
[γa0C−1]αβKa .
(7.104)
7.A The 3D N = 2 AdS Carroll Superparticle 161
In components the Maurer-Cartan equation dLC − 1
2 f
C
ABL
BLA = 0 reads as
follows:
dLH = − 1
2L
a
P L
a
K − 1
4 L̄−γ
0L− − 1
4 L̄+γ
0L+ , dL a
P = 2L b
P L
ab
M ,
dL a
K =2L b
K L
ab
M + 2
R2LHLP
a − 1
R L̄−γ
a0L+ , dLZ = −1
2 L̄+γ
0L+ + 1
2 L̄−γ
0L− ,
dL− =1
2γabL− L
ab
M − 1
2RγaL+LP
a , dL+ = 1
2γabL+ L
ab
M − 1
2RγaL−LP
a ,
dL ab
M =2L ca
M L cb
M + 1
2R2LP
bLP
a .
(7.105)
7.A.2. Superparticle action
We use the algebra (7.104) to construct the action of the N = 2 Carrollian
superparticle. The coset that we will consider is
G
H
=
N = (2, 0) AdS Carroll
SO(D-1)
, (7.106)
with the coset element g = g0 U, where g0 = eHtePax
a
eQ
−
α θ
α
−eQ
+
α θ
α
+eZs is the coset
representing the N = (2, 0) Carroll superspace with a central charge extension
and U = eKav
a
is a general Carroll boost that represents the superparticle.
The Maurer-Cartan form associated to the super-Carroll space is given by
Ω0 = (g0)−1dg0 = HE0 + PaE
a +Kaω
a0 +Mabω
ab − Q̄−E− − Q̄+E+ + ZEZ ,
(7.107)
where (E0, Ea, E−α, E+α, EZ) and (ωa0, ωab) are the supervielbein and the spin
connection of the Carroll superspace which are given explicitly by
E0 = dt cosh
x
R
− 1
4
(θ̄−γ
0dθ− + θ̄+γ
0dθ+) − 1
4
ωab(θ̄+γabγ
0θ+ + θ̄−γabγ
0θ−)
+
1
4R
θ̄−γ
a0θ+E
a ,
Ea =
R
x
dxa sinh
x
R
+
1
x2
xaxbdxb
(
1 − R
x
sinh
x
R
)
,
EZ = ds+
1
2
θ̄−γ
0dθ− − 1
2
θ̄+γ
0dθ+
− 1
2
ωab(θ̄+γabγ
0θ+ + θ̄−γabγ
0θ−) +
1
2R
θ̄−γ
a0θ+E
a .
ωab =
1
2x2
(xbdxa − xadxb)
(
cosh
x
R
− 1
)
,
162 Carroll Limit
ωa0 = − 2
xR
dtxa sinh
x
R
− 1
R
θ̄+γ
a0dθ− − 1
4R2
(θ̄+γabγ
0θ+ + θ̄−γabγ
0θ−)Eb
− 1
R
ωbcθ̄−γbcγ
a0θ+ ,
E−α = [dθ−]α − 1
2R
[γaθ+]αE
a +
1
2
ωab[γabθ−]α ,
E+α = [dθ+]α − 1
2R
[γaθ−]αE
a +
1
2
ωab[γabθ+]α ,
(7.108)
We can use the supervielbein to write the Maurer-Cartan form of the N = (2, 0)
Carroll superparticle as follows:
LH = E0 +
1
2
vaE
a , LaP = Ea ,
LaK = ωa0 + dva + 2vb ω
ab , LZ = EZ ,
L−α = E−α , L+α = E+α .
(7.109)
7.A.3. Global Symmetries and Kappa symmetry
The action of the Carrollian superparticle is given by the pull-back of all L’s
that are invariant under rotations:
S = a
∫
(LH)∗ + b
∫
(LZ)∗
= a
∫
dτ
(
ṫ cosh
x
R
+
R
2x
vaẋ
a sinh
x
R
+
1
2x2
xbvbxaẋ
a
(
1 − R
x
sinh
x
R
)
− 1
4
θ̄−γ
0θ̇− − 1
4
θ̄+γ
0θ̇+
− 1
8x2
xbẋa(θ̄+γabγ
0θ+ + θ̄−γabγ
0θ−)
(
cosh
x
R
− 1
)
+
1
4x
θ̄−γ
a0θ+
[
ẋa sinh
x
R
+
1
Rx
xaxbẋ
b
(
1 − R
x
sinh
x
R
)]
)
+ b
∫
dτ
(
ṡ+ 1
2 θ̄−γ
0θ̇− − 1
2 θ̄+γ
0θ̇+
− 1
4x2x
bẋa(θ̄+γabγ
0θ+ − θ̄−γabγ
0θ−)
(
cosh x
R − 1
)
+
1
2x
θ̄−γ
a0θ+
[
ẋa sinh
x
R
+
1
Rx
xaxbẋ
b
(
1 − R
x
sinh
x
R
)]
)
,
(7.110)
7.A The 3D N = 2 AdS Carroll Superparticle 163
which is invariant under the following bosonic transformation rules for the coor-
dinates with constant parameters (ζ, η, ai, λi, λij) corresponding to time trans-
lations, Z transformations, spatial translations, boosts, rotations, respectively
δt = −ζ +
R
2x
λkxk tanh
x
R
+
t
Rx
akxk tanh
x
R
,
δxi = − 1
x2
(
xiakxk − x
R
coth
x
R
(xiakxk − aix2)
)
− 2λik x
k ,
δs = −η ,
δvi = −λi − 1
x2
λkxkx
isech
x
R
(
1 − cosh x
R
)− 2λij v
j − 2t
R2
ai
− 2t
R2x2x
iakxk sech x
R
(
1 − cosh
x
R
)
+ 2
Rxvba
[ixb]csch x
R
(
1 − cosh x
R
)
,
δθ+ = −1
2
λabγabθ+ +
1
2Rx
akxbγkbθ−csch
x
R
(
1 − cosh
x
R
)
,
δθ− = −1
2
λabγabθ− +
1
2Rx
akxbγkbθ+csch
x
R
(
1 − cosh
x
R
)
.
(7.111)
The same action is invariant under fermionic transformation rules with constant
parameters (ǫ+, ǫ−) corresponding to the supersymmetry transformations
δt =
1
4
ǭ+γ
0θ+sech
x
R
cosh
x
2R
− 1
4x
xk ǭ+γ
k0θ−sech
x
R
sinh
x
2R
+
1
4
ǭ−γ
0θ−sech
x
R
cosh
x
2R
− 1
4x
xk ǭ−γ
k0θ+sech
x
R
sinh
x
2R
,
δxi = 0 ,
δvi =
1
Rx
xiǭ+γ
0θ+
(1
2
tanh
x
R
cosh
x
2R
− 2 sinh
x
2R
)
+
1
R
ǭ+γ
i0θ− cosh
x
2R
− 1
2Rx2
xixk ǭ+γ
k0θ− tanh
x
R
sinh
x
2R
+
1
2Rx
xiǭ−γ
0θ− tanh
x
R
cosh
x
2R
− 1
Rx
xbǭ−γbγ
i0θ− sinh
x
2R
− 1
2Rx2
xixk ǭ−γ
k0θ+ tanh
x
R
sinh
x
2R
,
δs =
1
2
ǭ+γ
0θ+ cosh
x
2R
+
1
2x
xk ǭ+γ
k0θ− sinh
x
2R
− 1
2
ǭ−γ
0θ− cosh
x
2R
− 1
2x
xk ǭ−γ
k0θ+ sinh
x
2R
,
164 Carroll Limit
δθ+ = ǫ+ cosh
x
2R
+
1
x
xkγkǫ− sinh
x
2R
,
δθ− = ǫ− cosh
x
2R
+
1
x
xkγkǫ+ sinh
x
2R
.
(7.112)
To derive an action that is invariant under κ-transformations we need to
find a fermionic gauge-transformation that leaves LH and/or LZ invariant. The
variation of LH and LZ under gauge-transformations are given by
δLH = d([δzH ]) +
1
2
LaP [δzaK ] +
1
2
LaK [δzaP ] +
1
2
L̄−γ
0[δz−] +
1
2
L̄+γ
0[δz+] ,
δLZ = d([δzZ ]) − L̄−γ
0[δz−] + L̄+γ
0[δz+] ,
(7.113)
where, for example, [δzaK ] is obtained from LH by changing the 1-forms dt, dθ+,
dθ− with the transformations δt, δθ+, δθ−. For κ-transformations we have [δzH ] =
0, [δzaK ] = 0, [δzaP ] = 0 and hence we find
δLH =
1
2
δθ̄−γ
0[δz−] +
1
2
δθ̄+γ
0[δz+] ,
δLZ = −δθ̄−γ
0[δz−] + δθ̄+γ
0[δz+] .
(7.114)
It follows that to obtain a κ-symmetric action we need to take the pull-back of
either LH or LZ , with b = ±1
2a. We focus here on the case b = −1
2a. For this
choice the action and κ-symmetry rules are given by
S = a
∫
(LH − 1
2
LZ)∗ , [δz+] = κ , [δz−] = 0 , (7.115)
where κ = κ(τ) is an arbitrary local parameter. Using this we find the following
κ-transformations of the coordinates
δt =
1
4
sech
x
R
θ̄+γ
0κ , δxa = 0 , δθ+ = κ ,
δs =
1
2
θ̄+γ
0κ , δva =
1
2Rx
xaθ̄+γ
0κ tanh
x
R
, δθ− = 0 .
(7.116)
After κ-gauge fixing (setting θ+ = 0) the action reads
S = a
∫
dτ
(
ṫ cosh
x
R
− 1
2
ṡ+
R
2x
vaẋ
a sinh
x
R
+
1
2x2
xbvbxaẋ
a
(
1 − R
x
sinh
x
R
)
−1
2
θ̄−γ
0θ̇− − 1
4x2
xbẋaθ̄−γabγ
0θ−
(
cosh
x
R
− 1
)
)
.
(7.117)
7.A The 3D N = 2 AdS Carroll Superparticle 165
This action is invariant under the following transformation rules
δt = − 1
4x
xk ǭ+γ
k0θ−sech
x
R
sinh
x
2R
+
1
4
ǭ−γ
0θ−sech
x
R
cosh
x
2R
,
δxi = 0 ,
δvi =
1
R
ǭ+γ
i0θ− cosh
x
2R
− 1
2Rx2
xixk ǭ+γ
k0θ− tanh
x
R
sinh
x
2R
+
1
2Rx
xiǭ−γ
0θ− tanh
x
R
cosh
x
2R
− 1
Rx
xbǭ−γbγ
i0θ− sinh
x
2R
δs = − 1
2x
xk ǭ+γ
k0θ− sinh
x
2R
− 1
2
ǭ−γ
0θ− cosh
x
2R
,
δθ− = ǫ− cosh
x
2R
+
1
x
xkγkǫ+ sinh
x
2R
.
(7.118)
7.A.4. The N = (1, 1) AdS Carroll Superalgebra
We now consider the 3D N = (1, 1) anti-de Sitter algebra which is given by
[MAB,MCD] = 2ηA[CMD]B − 2ηB[CMD]A , [MAB, Q
±] = −1
2
γABQ
± ,
[MAB, PC ] = 2ηC[BPA] , [PA, Q±] = ±xγAQ± ,
{Q±
α , Q
±
β } = 4[γAC−1]αβPA ± 4x[γABC−1]αβMAB , [PA, PB] = 4x2MAB .
(7.119)
Here PA,MAB and Q±
α are the generators of space-time translations, Lorentz
rotations and supersymmetry transformations, respectively. The bosonic gen-
erators PA and MAB are anti-hermitian while the fermionic generators Q±
α are
hermitian. Like in the previous case, the parameter x = 1/(2R) is a contraction
parameter.
To make the Carroll contraction we rescale the generators with a parameter
ω as follows:
P0 =
ω
2
H , Ma0 = ωKa , Q± =
√
ωQ̃± . (7.120)
Taking the limit ω → ∞ and dropping the tildes on the Q± we get the following
3D N = (1, 1) Carroll superalgebra:
166 Carroll Limit
[Mab, Pc] = 2δc[bPa] , [Mab,Kc] = 2δc[bKa] ,
[Pa, Pb] =
1
R2
Mab , [Pa,Kb] =
1
2
δabH , [Pa, H] =
2
R2
Ka
[Pa, Q
±] = ± 1
2R
γaQ
± , [Mab, Q
±] = −1
2
γabQ
± ,
{Q±
α , Q
±
β } = 2[γ0C−1]H ± 4
R
[γa0C−1]αβKa .
(7.121)
The corresponding components of the Maurer-Cartan equation are given by
dLH = −1
2
L a
P L
a
K − L̄+γ
0L+ − L̄−γ
0L− ,
dL a
P = 2L b
P L
ab
M ,
dL a
K = 2L b
K L
ab
M +
2
R2
LHLP
a − 2
R
L̄+γ
a0L+ +
2
R
L̄−γ
a0L− ,
dL ab
M = 2L ca
M L cb
M +
1
2R2
LP
bLP
a ,
dL+ =
1
2
γabL+ L
ab
M − 1
2R
γaL+LP
a ,
dL− =
1
2
γabL− L
ab
M +
1
2R
γaL−LP
a .
(7.122)
7.A.5. Superparticle Action
Taking the algebra (7.121) we consider the following coset
G
H
=
N = (1, 1) AdS Carroll
SO(D-1)
. (7.123)
The coset element is g = g0 U, where g0 = eHtePax
a
eQ
−
α θ
α
−eQ
+
α θ
α
+ is the coset
representing the N = (1, 1) Carroll superspace and U = eKav
a
is a general Carroll
boost representing the insertion of the superparticle..
The Maurer-Cartan form associated to the super-Carroll space is given by
Ω0 = (g0)−1dg0 = HE0 + PaE
a +Kaω
a0 +Mabω
ab − Q̄−E− − Q̄+E+ , (7.124)
7.A The 3D N = 2 AdS Carroll Superparticle 167
where (E0, Ea, E−α, E+α) and (ωa0, ωab) are the supervielbein and the spin con-
nection of the Carroll superspace:
E0 = dt cosh
x
R
− θ̄−γ
0dθ− − θ̄+γ
0dθ+ − ωab(θ̄+γabγ
0θ+ + θ̄−γabγ
0θ−) ,
Ea =
R
x
dxa sinh
x
R
+
1
x2
xaxbdxb
(
1 − R
x
sinh
x
R
)
,
ωa0 = − 2
xR
dtxa sinh
x
R
− 1
R2
(θ̄+γabγ
0θ+ + θ̄−γabγ
0θ−)Eb
− 2
R
(θ̄+γ
a0dθ+ − θ̄−γ
a0dθ−) ,
ωab =
1
2x2
(xbdxa − xadxb)
(
cosh
x
R
− 1
)
,
E−α = [dθ−]α +
1
2R
[γaθ−]αEa +
1
2
ωab[γabθ−]α ,
E+α = [dθ+]α − 1
2R
[γaθ+]αEa +
1
2
ωab[γabθ+]α .
(7.125)
We can use the supervielbein to write the Maurer-Cartan form of the N = (1, 1)
Carroll superparticle as follows:
LH = E0 +
1
2
vaE
a , LaP = Ea ,
LaK = ωa0 + dva + 2vb ωab ,
L−α = E−α , L+α = E+α .
(7.126)
7.A.6. Global Symmetries
The action of the Carrollian superparticle is given by the pull-back of all L’s
that are invariant under rotations:
S = M
∫
(LH)∗
= M
∫
dτ
(
ṫ cosh
x
R
+
R
2x
vaẋ
a sinh
x
R
+
1
2x2
xbvbxaẋ
a
(
1 − R
x
sinh
x
R
)
−θ̄−γ
0θ̇− − θ̄+γ
0θ̇+ − 1
2x2
xbẋa(θ̄+γabγ
0θ+ + θ̄−γabγ
0θ−)
(
cosh
x
R
− 1
)
)
.
(7.127)
This action is invariant under the following bosonic transformation rules for the
coordinates with constant parameters (ζ, ai, λi, λij) corresponding to time trans-
168 Carroll Limit
lations, spatial translations, boosts and rotations, respectively
δt = −ζ +
R
2x
λkxk tanh
x
R
+
t
Rx
akxk tanh
x
R
,
δxi = − 1
x2
(
xiakxk − x
R
coth
x
R
(xiakxk − aix2)
)
− 2λik x
k ,
δvi = −λi − 1
x2
λkxkx
isech
x
R
(
1 − cosh
x
R
)
− 2λij v
j
− 2t
R2
ai − 2t
R2x2
xiakxk sech
x
R
(
1 − cosh
x
R
)
+
2
Rx
vba
[ixb]csch
x
R
(
1 − cosh
x
R
)
,
δθ+ = −1
2
λabγabθ+ +
1
2Rx
akxbγkbθ+csch
x
R
(
1 − cosh
x
R
)
,
δθ− = −1
2
λabγabθ− +
1
2Rx
akxbγkbθ−csch
x
R
(
1 − cosh
x
R
)
.
(7.128)
The same action is invariant under the following fermionic transformation rules
with constant parameters (ǫ+, ǫ−) corresponding to supersymmetry transforma-
tions:
δt = ǭ+γ
0θ+sech
x
R
cosh
x
2R
− 1
x
xk ǭ+γ
k0θ+sech
x
R
sinh
x
2R
+ ǭ−γ
0θ−sech
x
R
cosh
x
2R
+
1
x
xk ǭ−γ
k0θ−sech
x
R
sinh
x
2R
,
δxi = 0 ,
δvi =
2
R
ǭ+γ
i0θ+ cosh
x
2R
− 2
Rx
xk ǭ+γkγ
i0θ+ sinh
x
2R
+
2
xR
xi tanh
x
R
(
ǭ+γ
0θ+ cosh
x
2R
− 1
x
xk ǭ+γ
k0θ+ sinh
x
2R
)
− 2
R
ǭ−γ
i0θ− cosh
x
2R
− 2
Rx
xk ǭ−γkγ
i0θ− sinh
x
2R
+
2
xR
xi tanh
x
R
(
ǭ−γ
0θ− cosh
x
2R
+
1
x
xk ǭ−γ
k0θ− sinh
x
2R
)
δθ+ = ǫ+ cosh
x
2R
+
1
x
xkγkǫ+ sinh
x
2R
,
δθ− = ǫ− cosh
x
2R
− 1
x
xkγkǫ− sinh
x
2R
.
(7.129)
One may not assume the validity of field equations at
very high density of field and matter and one may not
conclude that the beginning of the expansion should
be a singularity in the mathematical sense.
Albert Einstein
8
Conclusions and Outlook
The discovery of general relativity 100 years ago is considered to be one of
the greatest scientific and intellectual achievements of all time. Its importance
and relevance have since then only increased as a result of the number and range
of observations that can be made, and the applications that have since appeared
[173,174].
For example, the prediction of light-bending (which grew into gravitational
lensing), the gravitational redshift and the anomaly in the perihelion advance
of Mercury have made it possible to use satellite navigation systems (GPS), to
study binary pulsars, to use microlensing to infer the distribution of mass within
galaxies and the distribution of dark matter, to detect the presence of new exo-
planets, and to develop radio astronomy, which has since led to the discovery of
quasars and pulsars.
The corresponding mathematical and conceptual progress has also been out-
standing: for example, Einstein’s equations are one of the most interesting and
important systems in the theory of partial differential equations and geometrical
analysis. Moreover, the geometric concepts of connection and curvature have
become fundamental to modern gauge theories.
In spite of its powerful scope, the theory has always had drawbacks in tests
outside the solar system, including the understanding of compact stars such as
white dwarfs, supernovae and neutron stars, in which the enormous nuclear den-
sity counterbalances the gravitational force, which in return requires the simul-
taneous use of quantum theory and GR. So far, there has not been a satisfactory
physical meaning given to a singularity occurring in a given spacetime, and the
169
170 Conclusions and Outlook
modelling of properties of inflation, dark matter and dark energy remains incom-
plete.
Serious efforts to modify GR theory have taken a number of forms. Each
modification adopts a different starting point and perspective, treating certain
aspects of GR as more fundamental, either hoping that once the main difficul-
ties are resolved, the remaining aspects can be handled successfully, or studying
specific features of the theory to better understand it.
Many modifications of GR have been attempted over the last 50 years, such
as supersymmetry, higher dimensions, extended objects, higher spin theories, in-
finite towers of particles and fields, massive gravitons, higher derivative theories,
holography, etc. Many new theories have emerged, such as string theory, M -
theory, twistor theory, Hořava-Lifshitz gravity, loop quantum gravity, etc. Some
of these have attempted to modify Einstein gravity at short distances and thus
focus on the quantum description, while others have concentrated their efforts on
choosing adequate matter that would improve the ultra-violet behavior, or focus
on infrared modifications to try to improve the dark matter and dark energy
problems of cosmology. Others have studied inflationary scenarios or the unifi-
cation of the gauge couplings. Each of these approaches has its own virtues and
drawbacks but together they have instigated the study of a variety of challeng-
ing problems and led to unforeseen applications in diverse areas of mathematics,
cosmology and physics.
In the context of this intricate enterprise, the aim of this thesis was to study
a supersymmetric massive extension of GR, the symmetry properties of Hořava-
Lifshitz anisotropic scaling and the features of some systems using the polymer
representation. We also studied two different (and opposite) supersymmetric
limits of the theory, the non-relativistic limit and the Carroll limit. Throughout
this thesis, we have concentrated our attention on the simplest three-dimensional
cases, not only because these are technically simpler, but also because three-
dimensional gravity is interesting in its own right.
Supersymmetry has been widely studied in the literature because it can be
used in many modern developments in mathematical and theoretical physics: it
automatically appears as a symmetry of string theory, as well as in brane theories,
the AdS/CFT correspondence, the Seiberg-Witten duality, etc.
Massive gravity is a different extension; it considers gravity to be propagated
by a massive spin-2 particle. One of the main motivations for massive gravity is
that adding a mass to the graviton leads to long distance modifications of GR
that affect dark energy and reduce the cosmological constant problem.
Instead of adding new symmetries it is also possible to take a different ap-
proach by modifying the existing GR symmetries. The anomalous scaling in
171
Hořava-Lifshitz theory breaks the spacetime diffeomorphism invariance in such
a way, that the gravitational action is no longer a topological invariant and the
theory becomes dynamical. Nonetheless, this Lorentz symmetry violation does
not represent a problem if the theory recovers this symmetry at energy scales
where it has already been tested.
Another modification that we can study is to give up on the Schrödinger
quantization scheme to try to handle the GR quantization. The motivation for
studying polymer representations is that they provide the simplest analogues of
the representation featuring in loop quantum gravity. It is interesting to study
this approach because in some cosmology models [175–177] it leads to the avoid-
ance of the classical singularity which in loop quantum cosmology is replaced
by a quantum bounce, and is a microscopic basis for calculating the entropy of
some black holes that agrees with the Bekenstein–Hawking’s semiclassical for-
mula [178].
In addition, studying the various limits of GR helps to understand it better
because certain problems are easy to study in limiting cases in which these prob-
lems often become simpler and easier to analyze. Furthermore it is interesting to
study these limiting cases because of the applications that may arise from them.
There are two ways to consider the limits: The first one, known in the lit-
erature as limiting procedure, see [44], obtains the limiting gravity/supergravity
theories directly from the GR theory/supertheory. The second procedure consists
of obtaining the gravity theory from the gauging of the contracted algebra. In
this work, we have chosen the latter approach and investigated the non-relativistic
and ultra-relativistic limits by choosing appropriate contractions of the Poincaré
superalgebra and studying the particle actions that emerge from gauging the
algebras.
The first GR modification that we studied is the supersymmetric-massive
extension. We motivated and discussed the Kaluza Klein formulation to obtain
the 3D, off-shell massive spin-2 supermultiplet. This multiplet, together with
the previously known off-shell massless spin-2 multiplet, can be used to write a
supersymmetric version of linearized NMG in the auxiliary field form.
The massless limit involves a non-trivial coupling of a scalar multiplet to
a current multiplet. In the usual Fierz–Pauli case, it is possible to cure the
vDVZ discontinuity by taking into account its non-linear version, but a non-
linear version of the SNMG is not obvious since the construction of the massive
spin-2 supermultiplet is based on the truncation of the Kaluza Klein expansion
to the first massive level, which can only be performed at the linearized level. To
obtain a non-linear SNMG theory one may have to use superspace techniques.
The second GR modification that we considered deals with the anisotropic
172 Conclusions and Outlook
spacetime scaling that breaks Lorentz symmetry. There are many open issues
regarding Hořava-Lifshitz gravity. For example, renormalizability beyond power
counting has not been explicitly shown, it has not been proved if the theory
approaches GR in the infrared limit or not, the role of matter and its coupling
to gravity have not been fully clarified yet and couplings between the matter and
the scalar graviton could lead to violations of the equivalence principle [91].
Despite these drawbacks, it is interesting to apply the anisotropic scaling to
several systems in order to try to learn more about their features and later on
apply this lessons in the gravity framework. Our study takes a first step on
this direction by the symmetry analysis of some higher derivative toy systems.
After applying Noether’s theorem for actions with an anisotropic dependence
between time and space we obtained the explicit form of the conserved currents
for a scalar field and for the electrodynamics Lagrangian, both systems with
dynamical critical exponent z = 2 and additionally for a higher derivative Chern-
Simons extension. We found that Noether’s charge is unaffected because of the
additional higher order spatial derivatives.
The third GR modification that we took under consideration is the polymer
quantization scheme. In a similar way as in the previous case we studied some
simpler systems in pursuance of gaining a deeper understanding about some fea-
tures of the loop quantum gravity framework. We obtained the path integral
for the polymer non-relativistic free particle, the relativistic particle and the har-
monic oscillator. We calculated the propagators of the systems and obtained that
the velocity of the particle in all the systems is bounded.
The fourth modification that we considered was the non-relativistic superpar-
ticle in a curved background. We first investigated the symmetries of a superpar-
ticle moving on a flat and curved (AdS) non-relativistic space. We then explained
the gauging procedure for constructing the corresponding non-relativistic super-
gravity theory.
The fifth and last modification that we investigated was the supersymmetric-
Carroll limit. Carroll symmetries have recently appeared in a number of contexts.
For instance, an identification of a duality between the Galilean and the Carroll
limits has led to the study of some non-Einsteinian systems, and these symmetries
have also appeared in studies of tachyon condensation and warped conformal field
theories.
We constructed the vielbein and spin connection of the AdS Carroll space, as
well as the action of a massive particle moving in this space and its symmetries.
We then extended the study of the supersymmetric case by focusing on the N =
1, 2 AdS Carroll particle.
Our results can be extended in a number of ways. In the massive case, for in-
173
stance, it would natural to extend the results of this work to the case of extended,
i.e. N > 1, supersymmetry, or to ’cosmological’ massive gravity theories. Higher-
derivative, linearized versions of NMG with extended supersymmetry, or anti-de
Sitter vacua, are given in [56,179]. Of special interest is the case of maximal su-
persymmetry since this corresponds to the KK reduction of the N = 8 massless
maximal supergravity multiplet which only exists in a formulation without (triv-
ial) auxiliary fields. We expect that a formulation of this maximal SNMG theory
without higher-derivatives will be useful in finding out whether this massive 3D
supergravity model has the same miraculous ultraviolet properties as in the 4D
massless case.
As for the polymer quantization it is possible to continue in different direc-
tions. It would be interesting to study some features of the solutions of the
systems, for example, it was recently proved that the polymer harmonic oscilla-
tor possesses instanton solutions [116]. It would be also interesting to apply the
formalism to models with curvature and a cosmological constant different to zero.
Regarding the non-relativistic particle, it would be useful to construct a four-
dimensional analog of our results. In order to do so, one would first have to
be able to construct the Galilean and Newton–Cartan supergravity multiplets in
four spacetime dimensions. This has not yet been done.
Another generalization of our results might involve going from superparticles
to superstrings or even super p-branes. This would require using a ‘stringy’ gen-
eralization of the non-relativistic limits we have considered here, see for example
[165]. The case of a non-relativistic superstring in a flat background was already
considered in [128]. In the case of a non-relativistic curved background one could
apply holography and study the corresponding non-relativistic supersymmetric
boundary theory.
In the case of a particle/superparticle propagating in three spacetime dimen-
sions, the Galilei algebra contains a second central charge that might also be
included. This leads to the notion of a non-commutative non-relativistic parti-
cle/superparticle in which the embedding coordinates are non-commutative with
respect to the Dirac brackets [180, 181]. Finally, we have found that some of
our superparticles are 1/2 BPS, thus corroborating recent results for relativistic
superparticles [182]. It would be interesting to verify whether the statement that
“all superparticles are BPS” applies to non-relativistic superparticles as well.
As a possible continuation of our ideas regarding the Carroll superparticle,
it would be interesting to find the coupling of the AdS Carroll particle, and
the corresponding superparticle, to the (super) AdS gauge fields. As in the flat
Carroll case [41] we would expect the particle/superparticle to have a non-trivial
dynamics.
174 Conclusions and Outlook
It would be also interesting to study whether it is possible to construct a
corresponding Carroll gravity/supergravity theory. This can be approached in
one of two ways. One approach would be to gauge the (super)Carroll algebra
and/or the Lifshitz Carroll algebra with z = 0 by extending the results in [183] in
order to gauge the AdS-Carroll algebra and to extend this in the supersymmetric
case. A second alternative approach would be to try to define an ultra-relativistic
limit of relativistic gravity/supergravity similar to the non-relativistic limit [44].
Finally, in the supersymmetric-massive extension and in the limiting cases, it
would be desirable to construct a four-dimensional analogue of our results, but
so far, this has not yet been achieved.
List of Publications
Lorena Parra, J. David Vergara, Symmetry analysis for anisotropic field
theories, AIP Conf. Proc. 1473 (2011) 243-247.
Eric A. Bergshoeff, Marija Kovacevic, Lorena Parra, Jan Rosseel, Yihao
Yin and Thomas Zojer, New Massive Supergravity and Auxiliary Fields,
Class. Quant. Grav. 30 (2013) 195004, [arXiv:1304.5445].
Eric A. Bergshoeff, Marija Kovacevic, Lorena Parra and Thomas Zojer, A
new road to massive gravity?, PoS, Corfu2012, (2013) 053.
Lorena Parra, Kaluza Klein reduction of supersymmetric Fierz-Pauli, Phys.
Part. Nucl. Lett. 11 (2014) 7, 984-986.
Lorena Parra, J. David Vergara, Polymer quantum mechanics some exam-
ples using path integrals, AIP Conf. Proc. 1577 (2014), 269-280.
Eric A. Bergshoeff, Joaquim Gomis, Marija Kovacevic, Lorena Parra, Jan
Rosseel and Thomas Zojer, The Non-Relativistic Superparticle in a Curved
Background, Phys. Rev. D 90 (2014) 6, 065006, [arXiv:1406.7286]
Eric A. Bergshoeff, Joaquim Gomis and Lorena Parra, The Symmetries of
the Carroll Superparticle, J. Phys. A 49 (2016) 18, 185402
175
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Resumen
La Relatividad General (RG), ha resistido la prueba del tiempo incluso des-
pués de cien años después de su concepción. Desde el punto de vista experimental,
ha sido puesta a prueba con increíble exactitud por experimentos en el sistema
solar y mediciones de pulsar timing, las cuales predicen con precisión, por ejem-
plo, la deflección de la luz, los adelantos en el perihelio de Mercurio y el cambio
de frecuencia gravitacional de la luz. La RG también predice que a lo largo del
tiempo la energía orbital de un sistema binario se convertirá en radiación gravita-
cional y la tasa de decaimiento del periodo orbital medido es casi precisamente el
valor predicho por la teoría. Por otro lado desde el punto de vista teórico, la RG
provee una descripción elegante, consistente y comprensiva del espacio-tiempo,
la gravedad y la materia a nivel macroscópico.
Sin embargo, la RG pierde precisión y predictibilidad no sólo en regímenes
gravitacionales fuertes sino que también existen fenómenos que aún no se han
explicado en los regímenes débiles como el problema que surge de la rotación de
las galaxias y el problema del déficit de masa en cúmulos de galaxias. Además,
no existe un modelo satisfactorio que explique la energía oscura y la expansión
acelerada del universo. También existen deficiencias de la teoría, por ejemplo,
no ha sido posible unificar la gravedad con las fuerzas electrodébil y fuerte, la
predicción de singularidades en el espacio-tiempo y su incompatibilidad con la
mecánica cuántica.
Ha habido muchos esfuerzos que estudian posibles deformaciones de la RG
debido a su importancia experimental y teórica. Es técnicamente muy desafiante
tener una extensión de la RG, debido a que la nueva teoría debe hacer frente
a los problemas a gran escala y al mismo tiempo tomar en cuenta el comporta-
miento que surgirá en la teoría cuántica. En otras palabras, estamos buscando
una teoría matemáticamente consistente (libre de inestabilidades y fantasmas),
capaz de reproducir la RG a escalas cosmológicas, que provea una explicación a
la aceleración actual del universo, pero que al mismo tiempo, sea capaz de ofrecer
una descripción de la gravedad siguiendo los principios de la mecánica cuántica.
Debido a que esta tarea es muy complicada, el objetivo de este trabajo es el de
dar un primer paso en la disección de la RG y de estudiar diferentes extensiones
y límites de la teoría para entenderla mejor. Modificar una teoría conocida es una
de las mejores formas de descubrir nuevas estructuras que podrían tener aplica-
ciones no previstas. En esta tesis abordamos la siguiente pregunta: ¿Cuáles son
las posibles formas de modificar y estudiar los diferentes límites de la gravedad?
En la Tabla 1.1 hemos indicado las extensiones y los límites que hemos estudiado
en esta tesis:
a) Añadir nuevas simetrías: supersimetría.
191
192 Resumen
b) Añadir nuevos parámetros: parámetro de masa.
c) Modificar las simetrías existentes: abandonar la simetría de Lorentz.
d) Modificar el esquema de cuantización: cuantización polimérica.
e) Estudiar el límite no-relativista.
f) Estudiar el límite ultra-relativista.
Con el desarrollo de la mecánica cuántica, los principios de simetría y las
leyes de conservación obtuvieron un papel fundamental en Física. Las simetrías
conocidas en física de partículas son la invariancia de Poincaré (translaciones y
rotaciones), las simetrías internas que pueden dividirse en globales (relacionadas
con los números cuánticos conservados como la carga eléctrica) y las simetrías
locales (que forman la base de las teorías de norma por lo que están relacionadas
con las fuerzas de interacción) y las simetrías discretas (como la conjugación de
carga, la transformación de paridad y la invariancia bajo inversión temporal).
En 1967 Coleman y Mandula demostraron que dadas ciertas suposiciones,
estas son las únicas simetrías posibles para un sistema físico. El teorema de
Coleman-Mandula establece que para teorías cuánticas de campos, las simetrías
espacio-temporales y las simetrías internas sólo pueden combinarse en un pro-
ducto directo del grupo de simetría (en otras palabras, las simetrías internas y
de Poincaré no se mezclan). Hay algunas formas para sobrepasar este teorema
relajando algunas de las suposiciones que requiere. En particular, el teorema su-
pone que los generadores de simetría del álgebra involucran sólo conmutadores.
Relajando esta suposición permitiendo tanto relaciones de conmutación así co-
mo de anticonmutación entre los generadores, uno obtiene la posibilidad de la
supersimetría, en otras palabras se incluyen generadores de simetría bosónicos y
fermiónicos.
La supersimetría ha sido estudiada ampliamente en la literatura debido a que
se ha utilizado en muchos desarrollos modernos en física teórica y matemática,
por ejemplo, emerge automáticamente como una simetría de teoría de cuerdas,
de teorías de branas, en la correspondencia AdS/CFT, en la dualidad de Seiberg-
Witten, etc.
Además de añadir supersimetría, también podemos añadir nuevos parámetros
a la teoría de gravedad. Por ejemplo, un parámetro de masa. La gravedad masi-
va es una extensión diferente que parece natural debido a que sabemos que las
partículas de interacción de las fuerzas electrodébiles deben de adquirir una masa
a través del mecanismo de Higgs. Algunas de las principales motivaciones de la
gravedad masiva provienen del hecho de que agregando una masa al gravitón se
Resumen 193
generarán modificaciones a gran distancia a la RG, afectando la energía oscura
y disminuyendo el problema de la constante cosmológica. Existen algunas preo-
cupaciones cuando se trata con un gravitón masivo, una de ellas surge cuando
se toma el límite no-masivo de un campo de espín-2, debido a que propaga de-
masiados grados de libertad y parecería que nunca se podría obtener a la RG de
este límite. Este fenómeno se conoce como la discontinuidad vDVZ y puede resol-
verse considerando extensiones no lineales de la teoría. Un segundo problema a
superar es que las no-linearidades introducen fantasmas en el espectro físico. Los
fantasmas corresponden a campos con energía cinética negativa que conducen a
inestabilidades a nivel clásico y a la no-unitariedad a nivel cuántico.
En vista de que no ha sido posible aplicar las reglas de la mecánica cuán-
tica a la teoría de la RG, recientemente han surgido diversos caminos hacia la
cuantización. En este trabajo se han estudiado las características de diversos sis-
temas inspirados en dos modelos de cuantización: la gravedad de Hořava–Lifshitz
y la gravedad cuántica de lazos. Cabe mencionar, que todos los posibles modelos
que han surgido necesitan superar grandes problemas formales y conceptuales,
además, con el desarrollo actual en física de partículas y en observaciones cosmo-
lógicas, no es posible validar sus predicciones mediante pruebas experimentales.
A pesar de lo anterior, es posible probar los modelos dentro de escenarios físicos
teóricos plausibles y estudiar sus propiedades y resultados.
La teoría que Hořava propuso está basada en un escalamiento anisotrópico
entre las coordenadas espaciales y temporales que conduce al rompimiento de la
invariancia bajo defeomorfismos en distancias cortas, dónde la RG se recupera en
el límite de baja energía.
La segunda modificación a la RG que consideramos en esta tesis está inspirada
en la violación de Lorentz de la gravedad de Hořava. Utilizamos el teorema de
Noether para estudiar las simetrías de acciones anisotrópicas y encontramos las
cargas y densidades de corriente respectivas.
Por otro lado, nos enfocamos en uno de los resultados de la teoría cuántica
de lazos: la discretización cuántica del espacio. La mecánica cuántica polimérica
es un esquema de cuantización (que no es equivalente al esquema de cuantización
de Schrödinger) que replica alguna de las características de la teoría cuántica de
lazos.
La siguiente modificación de la RG que estudiamos es precisamente este cam-
bio de esquema de cuantización. Analizamos la cuantización polimérica a través
del procedimiento de la integral de trayectoria, para analizar la dinámica de diver-
sos sistemas. Con este esquema podemos obtener una acción efectiva que podemos
utilizar para comprender mejor aspectos no-perturbativos de la dinámica de los
sistemas a nivel clásico y cuántico.
194 Resumen
El siguiente enfoque para estudiar a la RG además de modificar las simetrías
y el esquema de cuantización es el de estudiar diferentes límites de RG para lograr
un mejor entendimiento de la teoría, ya que usualmente podemos obtener claridad
de algunos problemas estudiándolos en sus casos límite donde estos problemas se
vuelven más simples y fáciles de analizar. Es interesante estudiar estos casos
límite por las aplicaciones que pueden surgir de ellos.
En esta tesis consideramos los límites supersimétricos no-relativistas y ultra-
relativistas de la RG: los límites cuando la velocidad de la luz tiende a infinito y a
cero respectivamente. Geométricamente, la transición no-relativista puede verse
como la apertura de los conos de luz (los conos se convierten en hipersuperficies
tipo-espacio) mientras que de forma contraria, la transición ultra-relativista pue-
de entenderse como la contracción de los conos de luz (los conos se convierten en
hipersuperficies tipo tiempo). Mientras que el límite no-relativista está gobernado
por el álgebra de Galileo, el álgebra que le corresponde al límite ultra-relativista
es el álgebra de Carroll, ambos grupos de invariancia pueden obtenerse de con-
tracciones adecuadas del grupo de Poincaré. Cuando se comienza del grupo AdS,
diferentes contracciones conducirán a al grupo no-relativista de Newton-Hooke y
al grupo ultra-relativista de AdS-Carroll.
Existen muchas teorías conformes de campos no-relativistas que describen
sistemas físicos. Estos ejemplos surgen de la física de materia condensada, de la
física atómica y de la física nuclear. Las versiones no relativistas de la corres-
pondencia AdS-CFT se han investigado recientemente porque abren la puerta a
posibles aplicaciones en la dualidad norma-gravedad a una variedad de sistemas
con interacción fuerte. Los grupos de simetría no-relativistas, como los grupos
de simetría conforme de Schrödinger y de Galileo son relevantes en el estudio de
átomos fríos que tienen un dual que posee estas simetrías. Además, en el caso de
las cuerdas, los límites no-relativistas pueden tener aplicaciones en el contexto de
versiones no-relativistas de AdS-CFT.
Por otro lado, las llamadas simetrías de Carroll que surgen en el límite ultra-
relativista han jugado un papel importante en investigaciones recientes. Por ejem-
plo, teorías con simetrías de Carroll surgen es estudios de condensación de ta-
quiones. Recientemente también han aparecido en el estudio de teorías de campo
conformes deformadas.
En esta tesis, consideramos la extensión masiva supersimétrica, el estudio de
las simetrías de sistemas donde la simetría de Lorentz no se conserva, la cuanti-
zación polimérica y los límites supersimétricos no-relativista y ultra-relativista.
La primera modificación de la RG que estudiamos es la extensión masiva
supersimétrica. Motivamos y discutimos la formulación de Kaluza Klein para ob-
tener el supermultiplete espín-2 masivo off-shell tridimensional. Este multiplete
Resumen 195
junto con el multiplete espín-2 no massivo off-shell que ya era conocido, puede
utilizarse para escribir una versión supersimétrica de la nueva gravedad masi-
va linearizada (linearized New Massive Gravity NMG) en la forma de campos
auxiliares.
En el límite no-masivo hay un acoplamiento no-trivial del multiplete escalar
con un multiplete de corriente. En el caso usual de Fierz–Pauli, es posible curar
la discontinuidad de vDVZ tomando en cuenta la versión no-lineal de la teoría,
pero una versión no lineal de la teoría SNMG no es obvia debido a que la cons-
trucción de un supermultiplete masivo de espín-2 se basa en truncar la expansión
de Kaluza–Klein en el primer nivel masivo, lo cual sólo puede realizarse a nivel
lineal. Para obtener la teoría no-lineal de la teoría SNMG, probablemente sea
necesario utilizar técnicas de super-espacio.
La cuarta extensión que consideramos fue la de una superpartícula no-relati-
vista en un fondo curvo. Primero, construimos con el método de realizaciones
no-lineales, las acciones para la (super)partícula libre de Newton-Hooke, analiza-
mos su dinámica y las simetrías de una partícula moviéndose en dichos espacios.
También estudiamos las acciones de una superpartícula que describen la dinámi-
ca de una partícula supersimétrica en un fondo de supergravedad tridimensional
de Galileo con curvatura y de Newton–Cartan. Debido a la complejidad compu-
tacional mostramos la acción en el caso de Newton–Cartan sólo hasta términos
cuadráticos en los fermiones. El fondo de Newton–Cartan está caracterizado por
más campos y corresponde a más simetrías que en el fondo de Galileo. Es posible
cambiar de un fondo a otro ya sea normalizando parcialmente las simetrías (del
fondo de Galileo al de Newton–Cartan) o fijando la norma de algunas simetrías
(de Newton–Cartan a Galileo).
La quinta y última modificación que consideramos fue el límite de Carroll
supersimétrico. Estudiamos partículas cuya dinámica sea invariante bajo la su-
perálgebra de Carroll. En el caso bosónico encontramos que la partícula de Carroll
posee una simetría infinita-dimensional que sólo en el caso plano incluye dilatacio-
nes. La dualidad entre el álgebra de Carroll y el álgebra de Bargmann, importante
en el caso plano, no se extiende en el caso curvo. Sólo en el límite plano encontra-
mos que la acción es invariante bajo una simetría infinito-dimensional que incluye
a la extensión supersimétrica del álgebra de Lifshitz–Carroll.
A lo largo de esta tesis, nos centramos en los casos tridimensionales más
simples, no sólo porque eran técnicamente más sencillos sino también porque la
gravedad tridimensional es interesante por sí misma. En todos los casos sería
deseable construir los resultados análogos en cuatro dimensiones, pero hasta el
momento, no ha sido posible lograrlo.
Agradecimientos
Era muy difícil imaginarse que un doctorado se convertiría en un viaje lleno
de conocimiento, retos, recuerdos y satisfacciones. Me gustaría aprovechar esta
oportunidad para agradecer a las personas que enriquecieron mi camino durante
estos cinco años.
First of all, I owe to my two supervisors Eric Bergshoeff and David Vergara a
debt of gratitude for all the support they gave me during my PhD. Eric, thank you
for giving me the opportunity to join your group, you widened my vision of physics
through many interesting projects and encouraged me to attend international
conferences, schools and workshops. I cannot truly express my appreciation for
your permanent leadership, guidance, advices and enthusiasm, and for teaching
me how to become a better and mature researcher. David, I am very thankful for
the solid formation you gave me during my first two years, your experience and
insights on many research topics inspired me to continue reading and learning.
I wish to thank you for encouraging me to travel to the Netherlands, this PhD
would not have been possible without the initial and final momentum you gave
me.
I also wish to express my gratitude to the members of my PhD commit-
tee, Rocío Jaúregui and Antonio García for your help and advice trough all the
PhD process. To my dutch reading committee Ana Achúcarro, Silvia Penati and
Mees de Roo, and to my mexican reading committee, Myriam Mondragón, Hugo
Morales and Mariano Chernicoff, I would like to thank you for carefully reading
the manuscript and for the valuable comments you gave me to improve it. I
am also very grateful to Quim Gomis, for the interesting discussions and clear
explanations during the research projects we had together and for hosting me for
a week in Barcelona in order to finish the Carroll project.
The van Swinderen Institute is a place full of incredible talented people start-
ing from its faculty members, Elisabetta Pallante, Kyriakos Papadodimas and
Diederik Roest, thank you for the interesting conversations on many aspects of
life and physics. I would also like to thank my collaborators Marija Kovačević,
Jan Rossell, Yihao Yin and Thomas Zojer, I learned a ton of things thanks to all
of you.
Jan Willem, thank you for your kindness, friendship and for giving me help
whenever I needed it. I already miss the coffee/tea/chocolate breaks. To my
favorite latin people, thank you Víctor, Lucy and Julián, for your friendship, for
all the conversations and the laughs we had. Lucy, thank you for your sweetness
and all your support.
Outside the university, there were many people whose friendship I treasure.
I would like to express my heartfelt appreciation towards Suruchi, Kshama, and
197
198 Acknowledgments
José for all the nice moments we spend together.
In the course of this journey, I would never had imagine that I will find two
sisters in the Netherlands, I am very lucky and fortunate because of that. Thank
you Ana Huerta and Ana Cunha. Anita, hermanita, thank you for building a
home with me, for sharing all those laughs and tears, for encouraging me to
become the best and joyful version of myself and for all the support and love you
gave me. Anita, my dreamer, thank you for taking care of me, for those tasty
dinners, for agreeing to disagree with me in all those interesting conversations we
had and for the kindness and love you gave me. I love both of you and I am sure
that it does not matter in which corner of the world we are, we will always stay
friends.
A mis queridos amigos del MT, Ángeles, Angélica, Itzel, Kenji, Lucero, Mar-
iana, Marianel, Paola y Víctor, gracias por todas esas risas y ese cariño que
atravesaba el mundo. Alberto, muchas gracias por tus cartas, me alegrabas la
tarde cada vez que encontraba una de ellas esperándome en casa. Lilian, gracias
por tu apoyo y tu amistad.
A mi hermano Gabriel (a.k.a. Gabo), por tu cariño y aliento, sobretodo en
los momentos más oscuros y difíciles. Gracias por siempre respaldarme y gracias
por llevarme a Emporio contigo, hiciste que mi regreso a México se hiciera más
colorido.
Quiero agradecer sobre todo a mi mamá Josefina por todo el cariño que me
da, por llenar mi vida de música, libros y ciencia, por impulsarme a seguir mis
sueños, por enseñarme a nunca rendirme y por alentarme a volar alto y sin límites.
Mi querida mamá, gracias por toda tu ternura, por cuidarme y por enseñarme a
ser valiente.
Edward, mi travesía habría sido mucho más difícil sin tu mano sosteniendo la
mía, gracias por creer en mí, por alegrarme los días, por ser parte de esa fuerza
que me impulsa a seguir adelante y por enseñarme que cuando el amor es mucho,
cualquier distancia se hace corta.