Effective Field Theories

of

Partially Broken Supergravity

Richard Altendorfer

A dissertation submitted to The Johns Hopkins University in conformity with the

requirements for the degree of Doctor of Philosophy.

Baltimore, Maryland

2000

Copyright c© 2000 by Richard Altendorfer,

All rights reserved.

Abstract

In this dissertation, effective field theories of partially broken N = 2 supergravity are

constructed.

First, it is shown that the partial breaking of supersymmetry N = 2 → N = 1

in flat space can be accomplished using any one of three dual representations for the

massive N = 1 spin-3/2 multiplet. Each of the representations can be reparametrized

and coupled to gravity so that they give rise to a set of dual N = 2 supergravities

and supersymmetry algebras. The massive off-shell N = 1 spin-3/2 multiplet is also

examined as a starting point for partial supersymmetry breaking.

Second, it is shown that the partial breaking of supersymmetry in anti-de Sitter

space can be accomplished using two of four dual representations for the massive

OSp(1, 4) spin-3/2 multiplet. The procedure gives rise to a set of dual N = 2 super-

gravities and supersymmetry algebras.

Third, theories of partial supersymmetry breaking in four dimensions are derived

by coupling the N = 2 massless gravitino multiplet to N = 2 supergravity in five

dimensions and performing a generalized dimensional reduction on S1/Z2 with the

Scherk-Schwarz mechanism. These theories agree with results that were previously

derived from four dimensions.

Thesis advisor: Prof. Jonathan Bagger

ii

Acknowledgements

First and foremost, I am deeply grateful to my family for their understanding and

support in the past five years.

Upon arrival in Baltimore I had the privilege of sharing a two-bedroom apartment

with a graduate student I was then unacquainted with. Ralf became my permanent

roommate as well as my best friend. Without his friendship and support I would not

have realized my pursuits.

I am indebted to my advisor Prof. Jonathan Bagger for his guidance and invaluable

advice during the last five years. His constant encouragement and optimism was

indispensable for the completion of my dissertation. He also set and tried to teach

me the standards of how to present scientific results both in oral and written form.

A special thank to my friend and office mate George, with whom I shared most

of my academic life here in Baltimore. His warmth and broad knowledge of physics

are truly appreciated.

I would like to thank Prof. Adam Falk and Prof. Gordon Feldman for sharing

many physical insights with me and Prof. Gabor Domokos for a critical reading of

this dissertation. I also benefited greatly from many discussions with the faculty,

postdoctoral fellows, and graduate students of our particle theory group: Dmitry

Belyaev, Dr. Alexander Galperin, Dr. Francisco Gonzalez-Rey, Prof. Thomas Fulton,

Prof. Chung Kim, Prof. Susan Kovesi-Domokos, Adam Lewandowski, Dr. Edwin Lo,

Dr. Michael Mandelberg, Dr. Konstantin Matchev, Michael May, Dr. Tom Mehen,

Dr. Paul Mikulski, Dr. Samuel Osofsky, Rustem Ospanov, Dr. Alexey Petrov, Aaron

Roane, Dr. Yi-Yen Wu, Chi Xiong, and Dr. Ren-Jie Zhang.

iii

To my mother

iv

Contents

List of Figures vii

List of Tables viii

1 Introduction and Overview 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Phenomenology of global extended supersymmetry and its partial break-

ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.1 The N = 2 super-Poincare algebra . . . . . . . . . . . . . . . 41.2.2 Phenomenology of global extended supersymmetry . . . . . . 51.2.3 The partial breaking no-go theorem and its loophole . . . . . . 7

1.3 Overview of this dissertation . . . . . . . . . . . . . . . . . . . . . . . 9

2 Partial Breaking of Extended Supersymmetry in MinkowskiBackground 12

2.1 Dual on-shell theories of partial supersymmetry breaking . . . . . . . 122.1.1 SuperHiggs effect in partially broken supersymmetry . . . . . 12

2.1.1.1 Dual versions of massive N = 1 spin-3/2 multiplets . 122.1.1.2 UnHiggsing massive N = 1 spin-3/2 multiplets . . . 14

2.1.2 Dual algebras from partial supersymmetry breaking . . . . . 192.1.3 Multiplet structure in the massless limit . . . . . . . . . . . . 212.1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Towards an off-shell theory for partial supersymmetry breaking . . . 232.2.1 An off-shell multiplet for the massive N = 1 gravitino multiplet 232.2.2 Superspin analysis of the massive Ogievetsky-Sokatchev multiplet 262.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Partial Breaking of Extended Supersymmetry in Anti-de SitterBackground 30

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.2 Partially broken AdS supersymmetry . . . . . . . . . . . . . . . . . . 31

3.2.1 Dual versions of massive AdS spin-3/2 multiplets . . . . . . . 31

v

3.2.2 SuperHiggs effect for AdS spin-3/2 multiplets . . . . . . . . . 343.3 Dual AdS supersymmetry algebras . . . . . . . . . . . . . . . . . . . 413.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Partial Supersymmetry Breaking from Five Dimensions 474.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Generalized compactification of the massless N = 2 D = 5 gravitino

multiplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Generalized dimensional reduction of pure N = 4 D = 5 supergravity 554.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Summary and Outlook 63

Appendix 67A Minimal superHiggs effect of partially broken supersymmetry . . . . 67B Massive N = 1 spin-3/2 multiplet with two antisymmetric tensors . . 69C Poincare dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72D Superfield projectors for the spinor superfield . . . . . . . . . . . . . 74E Ogievetsky-Sokatchev multiplet . . . . . . . . . . . . . . . . . . . . . 76F De Wit-van Holten multiplet . . . . . . . . . . . . . . . . . . . . . . 78G Geometry of AdS space . . . . . . . . . . . . . . . . . . . . . . . . . 80H Massive AdS spin-1 multiplet . . . . . . . . . . . . . . . . . . . . . . 83I Conventions of five-dimensional supersymmetry . . . . . . . . . . . . 86

Bibliography 87

vi

List of Figures

2.1 The unHiggsed versions of the (a) traditional and (b) dual representa-tions of the N = 1 massive spin-3/2 multiplet. . . . . . . . . . . . . . 16

3.1 The degrees of freedom of the unHiggsed OSp(1, 4) massive spin-3/2multiplet coupled to gravity. The massive spin-1 field can be repre-sented by either a vector or an antisymmetric tensor. . . . . . . . . . 36

3.2 The manifold of partially broken N = 2 supergravity theories as afunction of Newton’s constant κ and the cosmological constant Λ. . . 43

vii

List of Tables

4.1 Fermionic parity assignment of D = 5 N = 2 gravitino multiplet interms of D = 4 Weyl spinors. . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Bosonic parity assignment of D = 5 N = 2 gravitino multiplet in termsof D = 4 fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Bosonic parity assignment of the dualized D = 5 N = 2 gravitinomultiplet in terms of D = 4 fields. . . . . . . . . . . . . . . . . . . . . 54

4.4 Fermionic fields and parities of D = 5 N = 4 supergravity in terms ofD = 4 Weyl spinors. . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.5 Bosonic fields and parities of D = 5 N = 4 supergravity in terms ofD = 4 fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.6 Corresponding fields of partially broken N = 2 D = 4 supergravityand pure N = 4 D = 5 supergravity. . . . . . . . . . . . . . . . . . . 60

A.1 Degree of freedom count of the minimal partial superHiggs effect. . . 68

viii

1

My inquiries into physics could perhaps be given the title: legacies.

For people do also bequeath trifles, after all.

Aphorism No. 14; Notebook L (translated by R. J. Hollingdale)

Georg Christoph Lichtenberg (1742 - 1799)

Chapter 1

Introduction and Overview

1.1 Motivation

The purpose of particle physics is to understand the micro-physical basis of the

everyday world. Particle physics describes nature at a fundamental level in terms of

four forces: the gravitational force, the electromagnetic force, the strong force and

the weak force. The last three forces can be successfully described by the “Standard

Model of Particle Physics.” The accuracy of this field theory in modeling nature has

been experimentally verified up to an energy range of ca. 100GeV with high precision

over the last 20 years at particle accelerators in Geneva (CERN), Hamburg (DESY)

and Chicago (Fermilab).

Despite the success of the Standard Model, it is known to be incomplete. Al-

though the Standard Model describes all physical phenomena in the currently acces-

sible energy range with high precision, it suffers from the so-called ‘technical hierarchy

2

problem’ [1, 2]. This problem arises from the mechanism that is responsible for the

generation of mass (electroweak symmetry breaking). In the Standard Model, a fun-

damental scalar is introduced — the Higgs boson, which provides an “ether” that

renders some particles massive. Unfortunately, quantum corrections drive the mass

of the Higgs boson up to an energy scale where the regime of the Standard Model

breaks down (e. g. the Planck scale MPl ∼ 1019GeV ), whereas unitarity restricts its

mass to be mh < 1TeV [3]. The Higgs mass can be made that light only by carefully

adjusting the regularization counterterms to at least one part in 1014 order by order

in perturbation theory. This is considered unnatural and theoretically unsatisfactory.

Moreover, the attempt to include the forth force, gravity, leads to mathematical

inconsistencies (see e. g. [4]). There is, however, a unique candidate for unifying

all four forces: superstring theory (see [5] and references therein). Superstring the-

ory requires two additional features not present in the Standard model: it describes

extended objects instead of point-like particles and it is based on a new symmetry:

supersymmetry.

This new symmetry was formulated in four space-time dimensions by Wess and

Zumino in 1974 [6]. It overcomes a fundamental asymmetry between particles of

different spin within the Standard Model, which assigns different roles to particles

having different spin. The spectrum of particles is divided into bosons, which medi-

ate forces, and fermions, which are the building blocks of matter. Supersymmetry

is a symmetry between fermions and bosons — it unifies matter and forces. It is

an inherently quantum mechanical symmetry and is the unique extension of special

relativity to relate bosons and fermions, as proven in the Haag-TLopuszanski-Sohnius

theorem [7]. By introduction of fermionic symmetries this theorem circumvents the

no-go theorem of Coleman and Mandula [8], which states that only the momentum

generator Pm does not commute with the Lorentz generator Mmn in the most general

bosonic Lie algebra of symmetries of the S-matrix. If true, supersymmetry explains

why fermions exist in nature.

In the simplest supersymmetric extension of the Standard Model all known parti-

cles are supplemented by superpartners with spins differing by ±12

(“Minimal Super-

symmetric Standard Model”, MSSM, for a review see e. g. [9]). However, supersym-

3

metry predicts the masses of the Standard Model particles and their superpartners

to be equal, in contradiction to experiment. Therefore supersymmetry which is in-

dispensable at high energies cannot hold at low energies: it must be broken. The

breaking of a symmetry is not a novel effect; the dynamics of the Standard Model is

in fact based on exact as well as broken symmetries. Understanding the breaking of

supersymmetry in going from high to low energies is one of the most important tasks

in particle physics.

The breaking of supersymmetry can shift the masses of the superpartners above

the experimental limit. A supersymmetric theory with superpartner masses in the

TeV -range would be the sought-for cure to the technical gauge hierarchy problem,

because the quantum corrections to the Higgs mass are “tamed” by contributions of

the superpartners. A further boon of a supersymmetrized Standard Model is the fact

that the breaking of supersymmetry at a high energy scale can trigger the breaking

of the electroweak symmetry at a low energy scale. Hence the mass generation is tied

to supersymmetry, whereas without supersymmetry, the Higgs mechanism seems to

be a rather contrived sector of the Standard Model.

Although supersymmetry is broken at low energies, there is already indirect evi-

dence for its existence. It is based on the fact that the strengths of the three forces

of the MSSM become equal at a certain energy scale (MGUT ) [10], thus signaling a

unification of the three forces to one force, whereas the strengths in the SM do not

unify.

The most general framework for investigating spontaneously broken symmetries

is the method of nonlinear realizations, introduced by Callan, Coleman, Wess, and

Zumino [11]. This formalism was first developed for internal symmetries and later

generalized to space-time symmetries [12], as needed in the case of supersymmetry.

Every spontaneously broken symmetry gives rise to a new particle - a Goldstone

particle. Goldstone particles are massless. This enables one to pursue a model inde-

pendent investigation of spontaneous symmetry breakdown. In going to sufficiently

low energies, all particles which become heavy due to the symmetry breakdown will

be above the energy threshold and only the light particles and the Goldstone particles

are present. Studying the residual symmetries of the dynamics of the light particles

4

and the Goldstone particles in general severely restricts the high energy behavior of

the complete theory.

The field theory that contains supersymmetry and general relativity (gravity)

is supergravity. A complete formulation of supergravity with one supersymmetry

(N = 1 supergravity) is known since the early 1980’s (see e. g. [13]). To overcome

the mathematical inconsistencies of a field theoretic description of gravity, it must

be embedded in superstring theory. In a string theoretical context, N-extended su-

persymmetric field theories (with N ≤ 8) emerge naturally after compactification of

the superfluous space co-ordinates.1 However, at intermediately low energies, only

an N = 1 supersymmetric theory is viable [2]. Hence there must be a cascade of

supersymmetry breaking:

N > 1 → N = 1 → N = 0 .

As a starting point, this dissertation focuses on a scenario where an N = 2 su-

pergravity theory is spontaneously broken to an N = 1 supergravity theory. The

model-independent framework of low-energy effective theories allows for a straight-

forward extension to N ≥ 2 supergravity theories spontaneously broken to N = 1.

So the knowledge of the N = 2 super-Poincare algebra and its N = 1 subalgebra is

indispensable for all further investigations of the phenomenology of partially broken

supersymmetry.

1.2 Phenomenology of global extended supersym-

metry and its partial breaking

1.2.1 The N = 2 super-Poincare algebra

The N = 2 super-Poincare algebra for linearly realized symmetries is

[Mab,Mcd] = −i(ηbcMad + ηadMbc − ηacMbd − ηbdMac)

1Supersymmetric string theories can only be consistently formulated in ten or eleven space-time dimensions. The additional dimensions are hidden from the four dimensional world, either bycompactification or by localization of our world on a four-dimensional manifold (see e. g. [14]).

5

[Mab, Pc] = −i(ηbcPa − ηacPb)

[Pa, Pb] = 0[X ij, any generator

]= 0

Qiα, Qjβ = 2σa

αβPaδ

ij

Qiα, Q

βj = δαβX ij[

Mab, Qi]

= −iσabQi[

Pa, Qi]

= 0 . (1.1)

Here, ηab = diag(−1, 1, 1, 1) is the metric of four-dimensional Minkowski space, where

a, b, c . . . ∈ 0, . . . , 3 denote vectorial Lorentz indices. The Mab are the six generators

of the Lorentz group, and Pa are the four generators of translations — together they

generate the Poincare group. The Weyl-spinors Qiα (α ∈ 1, 2) and the complex

central charge X ij are the additional generators of the N = 2 graded extension of

the Poincare group with i, j ∈ 1, 2. In curved space, a distinction must be made

between Lorentz indices a, b, c . . . and world indices m,n, o . . .. They are related by

the vierbein eam which reduces to δam in flat space.

The commutation relations (1.1) are not the most general ones; more bosonic gen-

erators can be added to enlarge the automorphism group of the superalgebra. The

generators Pa and X12 ∈ CI form a D = 6 vector under the six-dimensional Lorentz

group SO(5, 1), whereas the supercharges transform as a SU(2) doublet. So a larger

automorphism group is actually GN=2 = SO(5, 1) × SU(2) and not just SO(3, 1).

Upon restriction to N = 1, the central charge vanishes and the automorphism group

becomes HN=1 = SO(3, 1) × SO(2) × U(1) ⊂ GN=2. Although the enlarged auto-

morphism group is essential to the construction of theories for partially broken global

supersymmetry [15, 16], it will be less relevant in the local case.

1.2.2 Phenomenology of global extended supersymmetry

In general, one restricts oneself in four space-time dimensions to extensions of the

Standard Model with a maximum of N = 8 supersymmetries. This is based on the

widely-held belief that it is impossible to consistently couple massless particles of spin

6

52

and higher to other particles. Massless multiplets with N > 8 would always contain

those particles.

On the other hand, as already mentioned, phenomenologically realistic models are

possible only for N = 1 [2]: In the Standard Model, the massless fermions (before

electro-weak symmetry breaking) transform in “complex” representations of the re-

spective gauge group, i. e. the massless fermions of helicity 12

do not transform under

the gauge group SU(2)L the same way the helicity −12

fermions transform. For global

supermultiplets with N > 1, helicity 12

and helicity −12

fermions necessarily transform

identically. In N = 2 supersymmetry, for example, there are two massless supermul-

tiplets with helicity 12

particles: the hypermultiplet with helicities (12, 0,−1

2) and the

vector-multiplet with helicities (1, 12, 0). Moreover, all particles in a given multiplet of

global supersymmetry transform in the same way under a gauge symmetry, because

the supersymmetry charges commute with the group generators.2 The hypermulti-

plet relates fermions of helicity 12

and −12, which would have the same charge under

the internal symmetry. The vector-multiplet only contains one fermion. However,

massless bosons of helicity 1 are always gauge bosons, which transform in the (real)

adjoint representation. Therefore, there are helicity −1 bosons transforming in the

same way, which have helicity −12

superpartners.

The conclusion is that in N = 2 supersymmetry, the helicity 12

and −12

fermions

transform equivalently under internal symmetries. This argument also holds for N >

2, since higher N multiplets can be decomposed in terms of lower N multiplets.

Therefore only an N = 1 supersymmetric theory is phenomenologically acceptable

at intermediately high energies. Hence any higher-N four-dimensional supersymmet-

ric theory must be spontaneously broken to N = 1.

2In general, some generators of the automorphism algebra do not commute with Qiα; however,

the corresponding symmetry cannot be gauged without gauging supersymmetry.

7

1.2.3 The partial breaking no-go theorem and its loophole

There is a theorem stating that it is impossible to to partially break N > 1 to

N = 1 [2]. Start with the anticommutator

Qiα, Qαj = 2σa

αα Paδij

Hence the Hamiltonian of a supersymmetric theory is manifestly positive definite:

H = 14

∑2α=1Qi

α, Qαi with no sum over i. Let Qα = Q1α and its conjugate Qα = Qα1

denote the first, unbroken supersymmetry, and Sα = Q2α, Sα = Qα2 the second.

Suppose that one supersymmetry is not broken, so

Qα |0〉 = Qα |0〉 = 0 . (1.2)

Because of the supersymmetry algebra, this implies that the Hamiltonian H =

14(Q1Q1 + Q1Q1 + Q2Q2 + Q2Q2) = 1

4(S1S1 + S1S1 + S2S2 + S2S2) also annihilates

the vacuum,

H |0〉 = 0 . (1.3)

Then, according to the supersymmetry algebra,

(SαSα + SαSα) |0〉 = 0 . (1.4)

For a positive definite Hilbert space, this leads one to conclude that

Sα |0〉 = Sα |0〉 = 0 . (1.5)

This argument lacks mathematical rigor since the supercharge Sα for a sponta-

neously broken symmetry does not exist as an operator in Hilbert space: the integral∫d3xJ2

α0(&x, t) over the time-like component of the second supercurrent diverges due

to the presence of a massless fermion (goldstino) [17]. It can be made rigorous by

working in a finite periodic box (thereby giving up Lorentz invariance) or by using

the supersymmetric current algebra

limV→∞

QVαj, J i

αa(x) = 2σbαα Tabδ

ij(x) . (1.6)

Here, J iαa(x) are the supercurrents and Tab(x) is the stress-energy tensor. This ar-

gument can be evaded by two loopholes, namely by changing its assumptions. They

can be changed in two ways [18]:

8

i) If the Hilbert space of the theory is not positive definite, then Qiα |0〉 = 0 can be

consistent with 〈0|QαiQiα |0〉. This is what happens in covariant formulations

of supergravity, where the gravitino ψmα is a gauge field with negative-norm

components. It is similar to Gupta-Bleuler quantization of the electromagnetic

field, where the negative norm states of the photon are decoupled from the

positive norm states in the Hilbert space.

ii) The supersymmetry current algebra is modified. This happens in supergravity if

the local supersymmetry is non-covariantly gauge-fixed. Even in rigid super-

symmetry, the supersymmetry current algebra can be modified.

In the latter case the modification reads [19]

limV→∞

QVαj, J i

αa(x) = 2σbαα (v4ηabC

ij + Tabδ

ij(x)) . (1.7)

The additional term Cij is a constant, thus the supersymmetry algebra on local

operators is not modified by its presence. The algebra is still finite and Lorentz

covariant. Upon integration over an infinite volume, the new term in Eq. (1.7) is

infinite.

There are by now many examples of partial supersymmetry breaking which take

advantage the second loophole. The first was given by Hughes, Liu, and Polchinski

[20, 19], who showed that supersymmetry is partially broken on the world volume of

an N = 1 supersymmetric 3-brane propagating in six-dimensional superspace. Later,

Bagger and Galperin [15, 16, 21] used the techniques of Coleman, Wess, and Zumino

[11], and Volkov [12] to construct an effective field theory of partial supersymmetry

breaking, with the broken supersymmetry realized nonlinearly. They found that the

Goldstone fermion could belong to an N = 1 chiral or an N = 1 vector multiplet or

a linear multiplet. Antoniadis, Partouche and Taylor discovered another realization

in which the Goldstone fermion is contained in an N = 2 vector multiplet [22].

Each of these examples relies on the modified current algebra (1.7) which rewritten

9

for a partially broken N = 2 theory reads3

Qα, J1αm = 2σn

αα Tmn

Sα, J2αm = 2σn

αα (v4ηmn + Tmn) , (1.8)

The shift in the second stress-energy tensor in Eqs. (1.8) prevents the current algebra

from being integrated into a charge algebra, and circumvents the no-go theorem.

In gravity, however, a shift in the stress-energy tensor corresponds to a shift in the

vacuum energy. Moreover, there is only one stress-energy tensor that gravity couples

to, so gravity can distinguish between the right-hand sides of Eqs. (1.8). This suggests

that the mechanism of partial breaking might be different in supergravity theories.

Indeed, theories with partial breaking were constructed by Cecotti, Girardello, and

Porrati, and by Zinov’ev [23], starting from linearly realized N = 2 supergravity. (A

geometrical interpretation was given in Ref. [24].) These authors considered scenarios

with vector- and hypermultiplets and found that the gravitational couplings exploited

the second loophole. It is natural to ask whether their results apply more generally

in supergravity theories. The construction of such a model-independent framework

is the topic of this dissertation.

1.3 Overview of this dissertation

In the next chapter I will discuss partially broken supergravity using a model-

independent approach with a minimal field content motivated by the superHiggs-

effect. It will turn out that partial breaking in flat space can be accomplished using

three dual representations for the N = 1 massive spin-3/2 multiplet. When coupled

to gravity, the dual representations give rise to new N = 2 supergravities with new

N = 2 supersymmetry algebras (Ref. [25] in collaboration with Jonathan Bagger).

In each case, the technique will be as follows: I will start with the Lagrangian and

supersymmetry transformations for the massive N = 1 spin-3/2 multiplet. I shall then

3The supersymmetry algebra has an SU(N) symmetry that acts by unitary transformations onthe indices i and j; therefore the matrix Ci

j can be taken to be diagonal without loss of generality.

10

“unHiggs”4 the representation by adding appropriate Goldstone fields and coupling

it to gravity. The resulting effective field theories describe the physics of partial

supersymmetry breaking at a mass scale m ∼ κv2 v, where κ ≈ 10−19GeV −1

denotes Newton’s constant and v is the scale where the second supersymmetry is

broken.5 They are the result of the path integral

eiSN=1[φ] =∫

[dΦ]eiSN=2[φ,Φ] ,

where Φ denotes the set of all fields present in the N = 2 theory with masses M ≥v m. The effective action SN=1[φ] contains non-renormalizable interactions, which

are suppressed by powers of M . Their influence is negligible at energy scales m M .

This is the justification for the assumption that one can construct a meaningful theory

at a certain energy scale without complete knowledge of the theory at higher energies.

I will also try to extend this procedure to an off-shell description by starting

with the off-shell massive N = 1 spin-3/2 multiplet based on a spinor superfield.

Although it is possible to “unHiggs” the superfield by adding appropriate “Goldstone”

superfields, it cannot be done in such a way that all degrees of freedom are preserved

in the massless limit m → 0.

In the third chapter I will examine the partial breaking of supersymmetry in

anti-de Sitter space. It will turn out that partial breaking in AdS space can be

accomplished using two of four dual representations of the massive N = 1 spin-3/2

multiplet. During the course of this work, new N = 2 supergravities and new N = 2

supersymmetry algebras will emerge (Refs. [28, 29] in collaboration with Jonathan

Bagger). They are based on the semi-direct product OSp(2, 4) ×s U(1), where the

U(1) is always nonlinearly realized for finite Λ.

In the fourth chapter it is shown that partial supersymmetry breaking N = 2 →N = 1 in four dimensions can be easily reproduced by compactifying N = 4 D = 5 su-

4By “unHiggsing” I mean the reparametrization of the longitudinal degrees of freedom of amassive field in such a way that the Lagrangian and supersymmetry transformations are non-singularin the massless limit.

5At the mass scale m, N = 1 supersymmetry is linearly realized. Hence, m must be well abovethe breaking scale v1 for the remaining N = 1 supersymmetry, which is between v1 ∼ 1011GeV forgravity mediated supersymmetry breaking [26] and v1 ∼ 105GeV for gauge mediated supersymmetrybreaking [27].

11

pergravity on the orbifold S1/Z2 and using the Scherk-Schwarz mechanism [30]. This

means that compactification of N = 4 D = 5 supergravity on S1/Z2 automatically

leads to an N = 2 supersymmetric theory in four dimensions with a very particular

geometry and multiplet structure — thus allowing for partial supersymmetry break-

ing [24]. Although the derivation of partially broken theories in four dimensions from

five dimensions is considered here only as a convenient tool, it allows for straight-

forward extensions like matter couplings or embeddings in higher-N theories in five

dimensions.

The rest of this dissertation consists of a summary of the obtained results and a

discussion of evolving research perspectives. Side issues and technical details during

the course of this dissertation are relegated to the Appendix.

12

Chapter 2

Partial Breaking of Extended

Supersymmetry in Minkowski

Background

2.1 Dual on-shell theories of partial supersymme-

try breaking

2.1.1 SuperHiggs effect in partially broken supersymmetry

2.1.1.1 Dual versions of massive N = 1 spin-3/2 multiplets

The starting point for my investigation is the massive N = 1 spin-3/2 multiplet.

This multiplet contains six bosonic and six fermionic degrees of freedom, arranged in

states of the following spins,

32

1 1

12

. (2.1)

The traditional representation of this multiplet contains the following fields [31]: one

spin-3/2 fermion, one spin-1/2 fermion, and two spin-one vectors, each of mass m. The

dual representations have the same fermions, but one or two antisymmetric tensors

13

in place of one or two of the vectors. As one shall see, each representation gives rise

to a distinct N = 2 supersymmetry algebra.

The traditional representation is described by the following Lagrangian [31],

L = εmnρσψmσn∂ρψσ − iζσm∂mζ − 1

4AmnAmn

− 1

2m2 AmAm +

1

2mζζ +

1

2m ζζ

− mψmσmnψn − mψmσmnψn . (2.2)

Here ψm is a spin-3/2 Rarita-Schwinger field, ζ a spin-1/2 fermion, and Am = Am +

iBm a complex spin-one vector. This Lagrangian is invariant under the following

N = 1 supersymmetry transformations,

δηAm = 2ψmη − i2√3ζ σmη − 2√

3m∂m(ζη)

δηζ =1√3

Amnσmnη − i

m√3σmηAm

δηψm =1

3m∂m(Arsσ

rsη + 2imσnηAn) − i

2(H+mnσ

n +1

3H−mnσ

n)η

− 2

3m(σm

nAnη + Amη) , (2.3)

where H±mn = Amn ± i2εmnrsArs and Amn = ∂mAn − ∂nAm.

A dual Lagrangian and its supersymmetry transformations can be found by using

a Poincare duality which relates a massive vector field to a massive antisymmetric

tensor field of rank two (see Appendix C). This duality can be used to relate the vector

Bm to an antisymmetric tensor Bmn by Bmn = 1/m εmnrs∂rBs or Bm = vm/m, where

vm = 12εmnrs∂

nBrs is the field strength for the antisymmetric tensor Bmn. [32].

This dual representation is special in the sense that it can also be written in N = 1

superspace formulation (Appendix E). It has the following component Lagrangian,

L = εpqrsψpσq∂rψs − iζ σm∂mζ − 1

4AmnA

mn +1

2vmvm

− 1

2m2AmAm − 1

4m2BmnB

mn +1

2mζζ +

1

2m ζζ

− mψmσmnψn − mψmσmnψn , (2.4)

14

where Amn is the field strength associated with the real vector field Am. This La-

grangian is invariant under the following N = 1 supersymmetry transformations:1

δηAm = (ψmη + ψmη) +i√3

(ησmζ − ζ σmη) − 1√3m

∂m(ζη + ζ η)

δηBmn =2√3

(ησmnζ +

i

2m∂[mζ σn]η

)+ iησ[mψn] +

1

mηψmn + h.c.

δηζ =1√3Amnσ

mnη − im√3σmηAm − 1√

3mσmnηB

mn − 1√3vmσmη

δηψm =1

3m∂m (Arsσ

rsη + 2imσnηAn) − i

2(HA

+mnσn +

1

3HA

−mnσn)η

− 2

3m(σm

nAnη + Amη) +1

3m∂m (2vnσ

nη − mσrsηBrs)

− 2i

3(vm + σmnv

n)η − im

3(Bmnσ

nη + iεmnrsBnrσsη) . (2.5)

A third representation can be found by dualizing the remaining vector, Am (see

Appendix B).

Each of the three dual Lagrangians describe the dynamics of free massive spin-

3/2 and 1/2 fermions, together with their supersymmetric partners, massive spin-one

vector and tensor fields. They can be regarded as “unitary gauge” representations

of theories with additional symmetries: a fermionic gauge symmetry for the massive

spin-3/2 fermion, as well as additional gauge symmetries associated with the massive

gauge fields.

2.1.1.2 UnHiggsing massive N = 1 spin-3/2 multiplets

To study partial breaking, these Lagrangians must be unHiggsed by including

appropriate gauge and Goldstone fields. In each case one has to add a Goldstone

fermion and Goldstone bosons and then gauge the full N = 2 supersymmetry. In this

way one can construct theories with N = 2 supersymmetry nonlinearly realized, and

N = 1 represented linearly on the fields. The resulting effective field theories describe

the physics of partial supersymmetry breaking at a mass scale m ∼ κv2 v.

In what follows I will focus on the first two cases presented above; the example

with two antisymmetric tensors can be worked out in a similar fashion (Appendix B).

1Here and hereafter, the square brackets denote antisymmetrization, without a factor of 1/2.

15

In each case I introduce Goldstone fields by a Stuckelberg redefinition. The complex

massive vector is unHiggsed Am by replacing

Am → Am −√

2

m∂mφ ; (2.6)

for the dual representation, I take

Am → Am − 1

m∂mφ

Bmn → Bmn − 1

m∂[mBn] . (2.7)

The introduction of the Goldstino ν requires an additional shift

ψm → ψm − 1√6m

(2∂mν + imσmν) (2.8)

to obtain a proper kinetic term for ν.

In Fig. 2.1(a) the physical fields of the traditional representation for the massive

spin-3/2 multiplet are arranged in terms of massless N = 1 multiplets. The lowest

superspins form an N = 1 chiral and an N = 1 vector multiplet. These fields may be

thought of as N = 1 “matter.” The remaining fields are the gauge fields of N = 2

supergravity. In unitary gauge, the two vectors eat the two scalars, while the Rarita-

Schwinger field eats one linear combination of the spin-1/2 fermions. This leaves the

massive N = 1 multiplet coupled to N = 1 supergravity.

A natural question to ask is whether there are partial breaking cases where there

are no additional matter fields in the unHiggsed phase, so that the degrees of freedom

of the lower N supergravity multiplet together with the appropriate number of massive

spin-3/2 multiplets add up to the degrees of freedom of the higher N supergravity

multiplet (minimal superHiggs effect). This is discussed in Appendix A, where it is

shown that in four space-time dimensions only for N = 8 → N = 6 breaking no

matter fields are present.

It will become clear later that Fig. 2.1 only illustrates the field content; it does

not describe the N = 1 multiplet structure of the unHiggsed theory.

16

a) 232

3

2

1

11

2

1

2

0 0

| z N=2 supergravity

| z N=1 matter

b) 232

3

2

1

11

2

0B@01

2

0

1CA

| z N=2 supergravity

| z N=1 matter

Figure 2.1: The unHiggsed versions of the (a) traditional and (b) dual representationsof the N = 1 massive spin-3/2 multiplet.

The resulting Lagrangian is as follows,

e−1L =

− 1

2κ2R + εmnrsψmiσnDrψ

is − iχ σmDmχ − iλσmDmλ − DmφDmφ

− 1

4AmnAmn −

( 1√2mψ2

mσmλ + imψ2mσmχ +

√2imλχ +

1

2mχχ

+ mψ2mσmnψ2

n +κ

4εijψ

imψj

nHmn+ +

κ√2χσmσnψ1

mDnφ

+κ

2√

2λσmψ1

nHmn− +

κ√2εmnrsψm2σnψ

1rDsφ + h.c.

), (2.9)

where m = κv2, and Dm is the covariant derivative. The supercovariant derivatives

take the form

Dmφ = ∂mφ − κ√2ψ1

mχ − 1√2κv2Am

Amn = Amn + κψ2[mψ1

n] − κ√2λσ[nψ

1m] . (2.10)

This Lagrangian is invariant under two independent abelian gauge symmetries, as

well as the following supersymmetry transformations,

δeam = iκ(ηiσaψmi + ηiσaψmi)

δψim =

2

κDmηi

+(

− i

2H+mnσ

nη1 +√

2Dmφη1 − κψ1m(χη1) + iv2σmη2

)δ2

i

17

δAm = 2εijψimηj +

√2λσmη1

δλ =i√2

Amnσmnη1 − i

√2v2η2

δχ = i√

2σmDmφη1 + 2v2η2

δφ =√

2χη1 , (2.11)

for i = 1, 2. This result holds to leading order, that is, up to and including terms

in the transformations that are linear in the fields. Note that this representation is

irreducible in the sense that there are no subsets of fields that transform only into

themselves under the supersymmetry transformations.

Let me now consider the dual case with one massive tensor. The degree of freedom

counting is shown in Fig. 2.1(b). This time, however, the “matter” fields include an

N = 1 vector multiplet together with an N = 1 linear multiplet. In unitary gauge,

one vector eats one scalar, while the antisymmetric tensor eats the other vector.

These are the minimal set of fields that arise when coupling the alternative spin-3/2

multiplet to N = 2 supergravity.

The Lagrangian and supersymmetry transformations for this system can be worked

out following the same procedures described above. They can also be derived by du-

alizing first the scalar φB and then the vector Bm using the method2 described in Ref.

[33]. The order is crucial since the bare Bm-terms disappear only after the Lagrange

multiplier of the φB dualization has been eliminated by its equation of motion. As

κ → 0, the dualities relating a massless antisymmetric tensor Bmn to a massless scalar

φ and a massless vector Am to another vector Bm reduce to the simple expressions

vm = −∂mφ and FBmn = 1/2 εmnrsF

Ars (see Appendix C).

The Lagrangian is as follows,

e−1L =

− 1

2κ2R + εpqrsψpiσqDrψ

is − iχσmDmχ − iλσmDmλ − 1

2DmφDmφ

− 1

4FA

mnFAmn − 1

4FB

mnFBmn +1

2vmvm −

( 1√2mψ2

mσmλ + miψ2mσmχ

2The transformations (2.14) do not appear to be dual to Eq. (2.11), because the vectors Am andBm in Eq. (2.14) have been rotated to simplify the transformations.

18

+√

2miλχ +1

2mχχ + mψ2

mσmnψ2n +

κ

2√

2εijψ

imψj

nFAmn−

+κ

2χσmσnψ1

mDnφ +κ

2λσmψ1

nFBmn+ +

κ

2εpqrsψ2

pσqψ1rDsφ

−iκ

2χσmσnψ1

mvn − iκ

2εpqrsψ2

pσqψ1rvs + h.c.

)(2.12)

where, as before, m = κv2, and

Dmφ = ∂mφ − m√2

(Am + Bm)

FAmn = ∂[mAn] +

m√2Bmn

FBmn = ∂[mBn] − m√

2Bmn . (2.13)

This Lagrangian is invariant under an ordinary abelian gauge symmetry, an antisym-

metric tensor gauge symmetry, as well as the following two supersymmetries,

δηeam = iκ(ηiσaψmi + ηiσaψmi)

δηψ1m =

2

κDmη1

δηAm =√

2εij(ψimηj + ψi

mηj)

δηBm = η1σmλ + λσmη1

δηBmn = 2η1σmnχ + i η1σ[mψ2n] + i η2σ[mψ1

n] + h.c.

δηλ = i FBmnσ

mnη1 − i√

2v2η2

δηχ = iσmη1Dmφ − vmσmη1 + 2v2η2

δηψ2m =

2

κDmη2 + iv2σmη2 − i√

2FA

+mnσnη1

+ Dmφη1 + κ((ψ1

mχ)η1 − (χη1)ψ1m

)− i vmη1

δηφ = χη1 + χη1 (2.14)

up to linear order in the fields. The supercovariant derivatives are given by

Dmφ = Dmφ − κ

2(ψ1

mχ + ψ1mχ)

FAmn = FA

mn +κ√2

(ψ2[mψ1

n] + ψ2[mψ1

n])

FBmn = FB

mn − κ

2(λσ[nψ

1m] + ψ1

[mσn]λ)

vm = vm +(

iκψ1nσm

nχ − iκ

2εm

nrsψ1nσrψ

2s + h.c.

). (2.15)

19

These fields form an irreducible representation of the N = 2 algebra.

In both cases, the commutator of two first supersymmetries ξ and η induces a first

and a second supersymmetry transformation with parameters proportional to κ, as

expected from a local supersymmetry:

[δξ, δη] = −2i(ξσmη − ησmξ)Dm + (gauge transformation)

+δ(2. supersymmetry)((ξσnη−ησnξ)(iκψ2

n+κ2σnχ))

+ δ(1. supersymmetry)((ξσnη−ησnξ)(iκψ1

n))(2.16)

Each of the two Lagrangians has a full N = 2 supersymmetry (up to the appro-

priate order). The first supersymmetry is realized linearly. The second is realized

nonlinearly: it is spontaneously broken. In each case, the transformations imply that

ζ =1√3

(χ − i√

2λ) (2.17)

does not shift, while

ν =1√3

(√

2χ + iλ) (2.18)

does. Therefore ν is the Goldstone fermion for N = 2 supersymmetry, spontaneously

broken to N = 1.

2.1.2 Dual algebras from partial supersymmetry breaking

Now that one has explicit realizations of partial supersymmetry breaking, one can

see how they avoid the no-go argument presented in the introduction. I first compute

the second supercurrent. In each case it turns out to be

J2mα = v2 (

√6 iσααmνα + 4σαβmnψ

2nβ) , (2.19)

plus higher-order terms. The commutator of the second supercharge with the second

supercurrent is then

Sα, J2mα = 0 + terms at least linear in the fields . (2.20)

From this one can see that the stress-energy tensors in the current algebra (1.8) do

not differ by a constant shift. The supergravity couplings must exploit the second

loophole to the no-go theorem.

20

To check this assertion, note that the operators J iαm and Tmn contain contributions

from all of the fields, including the second gravitino. When covariantly-quantized,

the second gravitino gives rise to states of negative norm. Indeed, one finds

(SS + SS) |0〉 = 0 , (2.21)

even though3

S |0〉 = |0′〉 = 0 S |0〉 = |0′′〉 = 0 . (2.22)

To elucidate the role of the bosonic symmetries associated with partial supersym-

metry breaking, let me now compute the closure of the first and second supersym-

metry transformations to zeroth order in the fields. In this way one can identify the

Goldstone fields associated with any spontaneously broken bosonic symmetries.

For the traditional representation, (Fig. 2.1(a)), I find

[ δη1 , δη2 ] φ = 2√

2 v2 η1η2

[ δη1 , δη2 ] Am =4

κ∂m (η1η2) . (2.23)

This shows that the complex scalar φ is indeed the Goldstone boson for a gauged

central charge. Moreover, in unitary gauge, where

φ = ν = 0 , (2.24)

this Lagrangian reduces to the usual representation for a massive N = 1 spin-3/2

multiplet [31].

For the dual representation (Fig. 2.1(b)), one has

[ δη1 , δη2 ] φ = 2 v2 (η1η2 + η1η2)

[ δη2 , δη1 ] Am =2√

2

κ∂m(η1η2 + η1η2) −

√2 i v2 (η2σmη1 − η1σmη2)

[ δη2 , δη1 ] Bm =√

2 i v2 (η2σmη1 − η1σmη2)

[ δη2 , δη1 ] Bmn =2 i

κD[m(η2σn]η

1 − η1σn]η2) . (2.25)

3This intuitive picture that the generator of a spontaneously broken symmetry relates degener-ate vacua is only correct at the heuristic level; as mentioned in Sec. 1.2.3 the state S |0〉 is notnormalizable. If S |0〉 can be normalized, S must needs annihilate the vacuum |0〉 [34].

21

The real vector −(Am−Bm)/√

2 is the Goldstone boson for a gauged vectorial central

extension of the N = 2 algebra. In addition, the real scalar φ is the Goldstone boson

associated with a single real gauged central charge. In unitary gauge, with

− 1√2

(Am − Bm) = φ = ν = 0 , (2.26)

this Lagrangian reduces to the dual representation for the massive N = 1 spin-3/2

multiplet [35].

Finally, for the case with two tensors Amn = Amn + iBmn and two Goldstone

vectors Am = Am + iBm, the algebra is

[δη2 , δη1 ] Am =4

κDm(η1η2) − 4iv2η2σmη1

[δη2 , δη1 ] Amn = −4i

κD[m(η2σn]η

1),

This case requires two vectorial central extensions of the supersymmetry algebra.

2.1.3 Multiplet structure in the massless limit

Each of these theories gives rise to different N = 1 multiplet structures in the

limit κ → 0. For the traditional representation, one finds a massless chiral multiplet,

(χ, φ), together with a pair of “twisted” massless N = 1 multiplets, (ψ2m, Am, λ).

The twisted multiplets transform irreducibly into each other under the first, unbroken

supersymmetry. They can be untwisted with the help of a second unbroken super-

symmetry which appears in this limit.4 The second supersymmetry transformations

are obtained from Eq. (2.11) (in the κ → 0 limit) by Am → Am, λ → −λ. One can

see that the twisted multiplet is actually a massless N = 2 multiplet.

In the case of the dual representation, the N = 1 transformations (2.14) reduce,

in the κ → 0 limit, to those of a massless vector multiplet, (Bm, λ), a linear multiplet,

(χ, Bmn, φ), and a massless spin-3/2 multiplet, (ψ2m, Am).5

The multiplet structure of the dual theory with two antisymmetric tensors consists

of the N = 2 representation, (ψ2m, Am + iBm = Am, λ), as well as a linear multiplet

4I am indebted to W. Siegel for pointing this out.5The transformations that mix the gravitino and the antisymmetric tensor are physically irrele-

vant because the transformations of the corresponding field strengths vanish on-shell.

22

with two antisymmetric tensors, (χ, Amn+iBmn = Amn). The argument that prevents

the coupling of this multiplet to supergravity (see Ref. [36] and references therein)

does not apply here since the “non-closure” terms in the supersymmetry algebra are

cancelled by terms from the variation of ψ2m.

2.1.4 Discussion

In this section I have examined the partial breaking of supersymmetry in flat

space. It was shown that partial breaking can be accomplished using either of three

representations of the massive N = 1 spin-3/2 multiplet. I unHiggsed the represen-

tations, and found a new N = 2 supergravity and a new N = 2 supersymmetry

algebra.

The Lagrangian for the traditional representation is a truncation of the supergrav-

ity coupling found by Cecotti, Girardello, and Porrati, and by Zinov’ev [23]. Their

results were based on N = 2 supersymmetry with complete N = 2 multiplets; they in-

volved at least one N = 2 vector-multiplet and one hypermultiplet. The Lagrangians

for the dual cases are new. They contain new realizations of N = 2 supergravity.

In each case, the couplings presented here are minimal and model-independent.

They describe the superHiggs effect in the on-shell low-energy effective theories that

arise from partial supersymmetry breaking. However one would like to have an off-

shell description as well in order to facilitate matter couplings to this minimal theory

and to gain more theoretical insight into the superHiggs mechanism. An approach

towards an off-shell theory for partial supersymmetry breaking is discussed in the

next section. As before, the starting point is the massive massive N = 1 gravitino

multiplet.

23

2.2 Towards an off-shell theory for partial super-

symmetry breaking

2.2.1 An off-shell multiplet for the massive N = 1 gravitino

multiplet

In order to describe the massive gravitino multiplet one needs a superfield that

contains a spin-32

component field and no component fields of spin-2 and higher. There

are three such superfields: the spinor superfield Ψα, the chiral vector-superfield Ψm

with DαΨm = 0, and the chiral antisymmetric tensor-superfield Ψ[mn] with DαΨ[mn] =

0. However, by dimensional analysis the only possible kinetic term of the latter two

superfields is of the form ΨΨ|θ2θ2 (indices suppressed), which has no gauge invariance

[37]. Therefore no additional fields can be introduced by a gauge transformation to

unHiggs those multiplets. For simplicity, no extra auxiliary superfields are considered,

which would make the representation non-minimal.

Now that an appropriate superfield has been identified, a Lagrangian for the spinor

superfield Ψα which expanded in component fields reads

Ψ = ψ +√

2((U1 + iU2)θ − iσmθ(Um3 + iUm

4 ) − 2σmnθUmn5 )

+1

2θ2ψ3 +

1

2θ2ψ4 + θσmθψm − i√

2σmθθ2(um

3 + ium4 )

+1√2θ2(θ(u1 + iu2) − 2σmnθu

mn5 ) +

1

4θ2θ2ψ7 (2.27)

must be found. The Grassmann coefficient of the gravitino ψm determines the dimen-

sion of Ψ: [Ψ] = 12. Hence the kinetic term for Ψ must be bilinear in Ψ/Ψ and Dα/Dα.

The procedure to construct the kinetic term is due to Ogievetsky and Sokatchev [35]:

Analogous to the Proca equation for a massive vector vm

vm − ∂m∂nvn + m2vm = 0 (2.28)

which contains the localized form of the spin-1 projector ηmn − ∂m∂n

, the superfield

equation of motion for Ψ should contain the localized form of the superspin-1 pro-

jector Π1 (see Appendix D). Depending on the dimension of this localized superspin

24

projector, the root of it has to be taken to make it compatible with a superspin

equation of motion that can be derived from a Lagrangian (similar to the derivation

of the Dirac equation from the Klein-Gordon equation). For the spinor superfield Ψ

the kinetic operator that projects onto superspin-1 is π⊥ =√

Π1 and the superfield

Lagrangian reads

L =(−1

2(ΨΨ)π⊥

(Ψ

Ψ

)︸ ︷︷ ︸

L⊥

+1

2m(ΨΨ + ΨΨ)

)|θ2θ2 (2.29)

= −1

2

(DβΨαDαΨβ +

1

4DβΨαDβΨα +

1

4DαΨβDαΨβ − 1

4(DαΨα + DβΨβ)2

−m(ΨΨ + ΨΨ))

|θ2θ2 (2.30)

The supermultiplet based on L⊥ is often referred to as the Ogievetsky-Sokatchev

multiplet. Expanding this Lagrangian in terms of the components of Ψ (2.27), the

resulting Lagrangian is not diagonal in the fields. Only after the field redefinitions

Um4 → Um

4 − 1

m∂nU

nm5

Umn5 → Umn

5 +1

2m(∂mUn

3 − ∂nUm3 )

u1 → u1 + ∂mUm3

u2 → u2 − ∂mUm4 − mU2

um3 → um

3 − ∂mU1 − 2∂nUnm5 + 2mUm

3 − 1

m(Um

3 − ∂m∂nU3n)

um4 → um

4 + ∂mU2 + ∂nUnm5

umn5 → umn

5 +1

2εmn

rs(∂rU s

4 − 1

m∂r∂tU

ts5 ) +

1

2(∂mUn

3 − ∂nUm3 ) − 2mUmn

5

ψ → ψ

ψ3 → ψ3 − σmψm +4

3(iσm∂mψ + mψ)

ψ4 → ψ4 − σmψm − 2

3(iσm∂mψ + mψ) − 1

2ψ3

ψm → ψm − 1

2σmψ4 − 1

4σmψ3 +

1

3(i∂mψ − mσmψ)

ψ7 → ψ7 − iσm∂mψ4 + ψ +i

2σm∂mψ3

25

−2

3i(∂m + σnσm∂n)ψm − 2

3m(iσm∂mψ + 4mψ) − mψ3

have been performed, is it possible to identify the the physical fields (ψm, ψ, Um3 , Umn

5 )

and the auxiliary fields (ψ3, ψ4, ψ7, u1, u2, um3 , um

4 , umn5 , U1, U2, U4m):

L → 1

2εmnrsψmσn∂rψs − m

2(ψmσmnψn + h.c.)

−2

3m2(iψσm∂mψ +

m

2(ψψ + h.c.))

− 3

16(ψ7ψ3 + h.c.) − m

4(ψ4ψ4 + h.c.)

−U3mnUmn3 − 2m2Um

3 U3m

+2∂nUnm5 ∂rU5rm − 4m2Umn

5 U5mn

−u22 +

1

2um

3 u3m + umn5 u5mn + 2mu1U1 + m2U2

2 + 2mum4 U4m

From this Lagrangian it can be seen that the massive Ogievetsky-Sokatchev multiplet

contains 32 Bose and 32 Fermi off-shell degrees of freedom, which reduce to 6 + 6 on-

shell degrees of freedom.

So contrary to the on-shell theories of partial supersymmetry breaking discussed

in Sec. 2.1, where three equivalent on-shell versions for the massive spin-32

multiplet

could be used, this off-shell approach restricts one to a bosonic field content of one

vector and one antisymmetric tensor.

A first step in unHiggsing that multiplet would be to find those superfield redefi-

nitions that reduce on-shell to the Stuckelberg redefinitions

U3m → U3m − 1

m∂mφ

U5mn → U5mn − 1

m∂[mAn]

ψm → ψm − 1√6m

(2∂mν + imσmν)

in order to obtain proper kinetic terms for the longitudinal components (φ,An, ν) of

the massive spin-1 and spin-32

fields, respectively.

This minimal consistency requirement is independent of supersymmetry trans-

formations and the closure of the supersymmetry algebra and can be investigated

without considering the N = 1 supergravity multiplet (which eventually has to be

included).

26

2.2.2 Superspin analysis of the massive Ogievetsky-Sokatchev

multiplet

With the knowledge of the off-shell formulation of the massive gravitino multi-

plet, one can attempt to promote the on-shell formulation of the superHiggs effect

discussed in Sec. 2.1 to an off-shell one. This requires an extension of the Stuckelberg

redefinition to superfields, where the superfield Ψα will be redefined in terms of su-

perfields with component fields of lower highest Poincare spin (i. e. superspin, see

Appendix D). In order to gain a feeling for the structure of these lower superspin

fields, the unHiggsing of the gravitino field in terms of Poincare spin projectors is

considered first.

The spin-vector ψmα contains three Poincare irreducible representations: one spin-

32

and two spin-12

fields. Unlike in the superspin case, the kinetic operator is not√P3/2 since the corresponding Lagrangian is not hermitian. A possible square root

(there is a one parameter group) is obtained from the standard Rarita-Schwinger

expression by the shift ψm → ψm + 13σmσnψn in the field equation. The most general

Rarita-Schwinger action is obtained by such a shift in the action. A convenient set of

spin projectors (PIij)mn with (PI

ij)mn(PJkl)no = δIJδjk(PJ

il)mo andP3/2+P1/211 +P

1/222 = 1

is given in [13]. In terms of those projectors6, the free massive Lagrangian can be

written as:

L = −iψmσr∂r(P3/2−2P

1/211 )mnψn−m

2(ψm(P3/2−2P

1/211 −

√3(P

1/212 +P

1/221 ))mnψn +h.c.)

Obviously, the kinetic term is invariant under δψm = ∂m∂n

χn = P

1/222 mnχ

n with χn

arbitrary. This is a gauge transformation. The redefinition to introduce the Goldstino

(i. e. with a proper kinetic term) is

ψm → ψm − 1√6m

(2∂mν + imσmν) (2.31)

where the shift σmν is an eigenvector of P1/211 +P

1/222 . The gravitino kinetic term is not

6The explicit expressions for the projectors are: (P3/2)mn

= δmn + 1

3σmσn + 13

∂m∂r

σnσr +

13

∂n∂r

σrσm, (P1/2

11 )mn

= − 13 (σmσn + ∂m∂r

σnσr + ∂n∂r

σrσm + 3∂m∂n

), (P1/2

22 )mn

= ∂m∂n

,

(P1/212 )m

n= − 1√

3(σmσr∂r∂

n + ∂m∂n), (P1/221 )m

n= − 1√

3(σrσn∂r∂m + ∂m∂n).

27

invariant under this shift. Hence, to obtain the correct kinetic term for the Goldstino,

the gauge field ν must be put in both lower spins.

This procedure must be reformulated in superfield notation. The superfield Ψα

contains superspins 0, 12, and 1: Ψα = 0⊕ 1

2

−r ⊕ 12

−i ⊕ 12

+r ⊕ 12

+i ⊕1 (see Appendix D).

The kinetic piece of the Lagrangian (2.29) possesses the gauge invariance (Appendix

E)

δΨ = DV + iW

= D(φ + φ) + DVWZ + iW

= 0 ⊕ 1

2

+r

⊕ 1

2

−i

.

The remaining lower superspins 12

+i ⊕ 12

−rare purely auxiliary; they can be made

explicit by the introduction of compensating fields [37].

In analogy to the unHiggsing of the gravitino (2.31) the set of superfields in

0 ⊕ 12

+r ⊕ 12

−i(corresponding to P

1/222 ) should also be put in 1

2

+i ⊕ 12

−r(corresponding

to P1/211 ), such that

i) the Lagrangian is nonsingular in the massless limit;

ii) three superspins (1, 12, 1

2) have proper kinetic terms to get the correct physical

component fields, namely those of a massless gravitino multiplet, a massless

vector multiplet, and a massless linear multiplet;

iii) the remaining three superspins are auxiliary but survive the massless limit.

However, since the Goldstino is partly in superspin-0, a kinetic term for a chiral

scalar superfield should be generated - in contradiction to the naive degree of freedom

counting of the physical fields. Moreover, since the operator π⊥ is invariant under

superspin-0, the kinetic term for the chiral field must come from the mass term.

The only possible shift of ψα under superspin-0 is δψα ∼ 1mDαφ with φ chiral. This

introduces 1/m-pieces in the Lagrangian and prevents a smooth massless limit.

It may be worth pointing out that it is possible to define a redefinition of ψα such

that the massive Lagrangian (2.29) decomposes in the massless limit into a massless

28

gravitino ψα, a massless real vector V , and a massless linear multiplet Lα (Wα and L

are the corresponding field strengths):

Ψα → Ψα − iDαV + Lα +2i

mWα +

1

4mDαL (2.32)

L → L⊥ + 2W αDαV − 1

8L2 +

1

2m((Ψα − iDαV + Lα)2 + h.c.) ,

The correct multiplet structure as described in Sec. 2.1.1.2 is obtained in the limit

m → 0. However, the auxiliary superspin-0 is lost and one expects 1/m-singularities

in the supersymmetry transformations.

2.2.3 Discussion

Because of the failure to even construct a Lagrangian with the correct kinetic

terms for the massless gravitino and two superspin-12

multiplets while retaining all

auxiliary fields, one has to conclude that the massive Ogievetsky-Sokatchev multiplet

cannot be used to promote the superHiggs effect discussed in Sec. 2.1 to an off-shell

formulation. However, one might try to construct massive multiplets based on the

spinor superfield in a way that is different from the projection technique discussed

above. Indeed, there is another expression for a massless Lagrangian for Ψα [38, 39]

(sometimes called the de Wit-van Holten multiplet) and one might be tempted to

simply add a mass term to it. However, its kinetic term is based on the operator

π‖ =√

(Π0 + Π1) which leads to a reducible representation of supersymmetry when

coupled to a mass term. Also, the component field Lagrangian does not have correct

kinetic terms for the physical fields, since it contains ghosts.

The massless de Wit-van Holten multiplet (see Appendix F) is related to the mass-

less Ogievetsky-Sokatchev multiplet (see Appendix E) by a Legendre transformation

[40], where an auxiliary chiral multiplet is dualized to an auxiliary linear multiplet.

Other, non-minimal massless Lagrangians based on the spinor superfield Ψα can be

constructed by introducing compensating fields and performing Legendre transfor-

mations on them. For an overview see [41]. The investigation whether any of those

could be used for an off-shell formulation of the superHiggs effect was not pursued

further.

29

In this chapter, the discussion of the superHiggs effect was restricted to theories

in a Minkowski background. However, it is interesting to investigate the superHiggs

effect in an anti-de Sitter background, too. In anti-de Sitter space, fields of a super-

multiplet do not have a uniform mass — they are split by the cosmological constant.

Likewise the superalgebra is changed, having new contributions proportional to the

cosmological constant. Therefore the superHiggs effect in a Minkowski background re-

quiring scalar and vectorial extensions of the superalgebra cannot be readily extended

to AdS space.

30

Chapter 3

Partial Breaking of Extended

Supersymmetry in Anti-de Sitter

Background

3.1 Introduction

The Minkowski-space theories from Sec. 2.1 were based on N = 2 super-Poincare

algebras with certain central extensions. In anti-de Sitter (AdS) space, however, the

N = 2 supersymmetry algebra is different. The algebra is known as OSp(2, 4); the

relevant parts are (see Appendix G)

Qiα, Qjβ = 2σa

αβRaδ

ij

Qiα, Q

βj = 2iΛσabαβMabδ

ij + 2iδαβT ij (3.1)[

T ij, Qk]

= iΛ(δjkQi − δikQj) .

In this expression, the Qiα (i ∈ 1, 2) denote the two supercharges, while Mab and Ra

are the generators of SO(3, 2). The antisymmetric matrix T ij is the single Hermitian

generator of an additional SO(2). As the cosmological constant Λ goes to zero, the

algebra contracts to the usual N = 2 Poincare supersymmetry algebra with at most

one real central charge. (The generator Ra contracts to the momentum generator

31

Pa, while T ij contracts to zero or to a single real central charge, depending on the

rescaling of the operators.)

In Minkowski space, partial supersymmetry breaking was found to require super-

Poincare algebras with two central extensions (Sec. 2.1). The OSp(2, 4) algebra

contracts to a super-Poincare algebra with at most one central charge. This suggests

that if partial breaking is to occur, the AdS algebra must be modified.

In this chapter I will study this question using the same approach as in Sec.

2.1. One will see that partial breaking in AdS space occurs for two of four dual

representations of the OSp(1, 4) massive spin-3/2 multiplet. One will find that the two

dual representations give rise to new AdS supergravities with appropriately modified

OSp(2, 4) supersymmetry algebras.1 As the cosmological constant Λ goes to zero, the

new algebras contract to the N = 2 Poincare algebras with the required set of central

extensions.

3.2 Partially broken AdS supersymmetry

3.2.1 Dual versions of massive AdS spin-3/2 multiplets

The starting point for my investigation is the massive OSp(1, 4) spin-3/2 multiplet.

This multiplet contains six bosonic and six fermionic degrees of freedom, arranged in

states of the following spins,

32

1 1

12

. (3.2)

It contains the following AdS representations (see e.g. [43] and references therein):

D(E + 12, 3

2) ⊕ D(E, 1) ⊕ D(E + 1, 1) ⊕ D(E + 1

2, 1

2) (3.3)

where D(E, s) is labeled by the eigenvalues of the diagonal operators of the maximal

compact subgroup SO(2) × SU(2) ⊂ SO(3, 2) and unitarity requires E ≥ 2. [The

1A theory exhibiting partial supersymmetry breaking in AdS space was derived in [42]. However,this construction contains more fields because it has complete N = 2 multiplets.

32

eigenvalue E is the AdS generalization of a representation’s rest-frame energy. As

E → 2, the first two representations in Eq. (3.3) become “massless,” with eigenvalues

(s + 1, s). The massless representations are short representations of OSp(1, 4).]

As in Minkowski space, a massive spin-1 field can be represented by a vector or

by an antisymmetric tensor (see Appendix C). For the case at hand, there are four

possibilities. The Lagrangian with two vectors is given by

e−1L = e−1εmnrsψmσn∇rψs − iζσm∇mζ − 1

4AmnA

mn − 1

4BmnB

mn

− 1

2(m2 − mΛ)AmAm − 1

2(m2 + mΛ)BmBm

+1

2mζζ +

1

2m ζζ − mψmσmnψn − mψmσmnψn (3.4)

where Λ ≥ 0 and ∇m is the AdS covariant derivative (see Appendix G). Here ψm is

a spin-3/2 Rarita-Schwinger field, ζ a spin-1/2 fermion, and Amn and Bmn are the

field strengths of the real vectors Am and Bm. This Lagrangian is invariant under the

following supersymmetry transformations:2

δηAm =√

1 + ε(ψmη + ψmη)

+1√

1 − ε

(i

1√3

(1 − ε)(ησmζ − ζ σmη) − 1√3m

∂m(ζη + ζ η)

)

δηBm =√

1 − ε(−iψmη + iψmη)

+1√

1 + ε

(− 1√

3(1 + ε)(ησmζ + ζ σmη) +

i√3m

∂m(ζη − ζ η)

)

δηζ =√

1 − ε

(1√3Amnσ

mnη − im√

3σmηAm

)

+√

1 + ε

(− i√

3Bmnσ

mnη +m√

3σmηBm

)

δηψm =1√

1 + ε

(1

3m∇m(Arsσ

rsη + 2imσnηAn) − i

2

(HA

+mnσn +

1

3HA

−mnσn)η

− 2

3m(σm

nAnη + Amη) − i

2εHA

+mnσnη − εmAmη

)

+1√

1 − ε

( −i

3m∇m(Brsσ

rsη − 2imσnηBn) +1

2(HB

+mnσn

2Here, and in all subsequent rigid supersymmetry transformations, the parameter η is covariantlyconstant but x-dependent (see Eq. (G.3) in Appendix G).

33

+1

3HB

−mnσn)η +

2

3im(σm

nBnη + Bmη) − 1

2εHB

+mnσnη − iεmBmη

),

(3.5)

where HA±mn = Amn ± i

2εmnrsA

rs and ε = Λ/m.

These transformations were derived by demanding that the AdS transformations

are a perturbation in Λ of the corresponding flat space transformations (2.3). Since

the transformations are valid on-shell only, they must satisfy the corresponding equa-

tions of motion. This requirement already constrains many coefficients, which are

completely determined by invariance of the action and closure of the algebra. Note

that the “mass” m is defined to be m = (E − 1)Λ. This definition is consistent with

the AdS representations in Eq. (3.3). The fact that E ≥ 2 implies that 0 ≤ ε ≤ 1.

In Minkowski space, other field representations of the massive spin-3/2 multiplet

can be derived using a Poincare duality which relates massive vector fields to massive

antisymmetric tensor fields of rank two. The same duality also holds in AdS space

where, for example, the vector Bm can be replaced by an antisymmetric tensor Bmn

(see Appendix C). The Lagrangian for the dual theory is then

e−1L = e−1εmnrsψmσn∇rψs − iζσm∇mζ − 1

4AmnA

mn +1

2vBmvB

m

−1

2(m2 − mΛ)AmAm − 1

4(m2 + mΛ)BmnB

mn

+1

2mζζ +

1

2m ζζ − mψmσmnψn − mψmσmnψn (3.6)

where Amn is the field strength associated with the real vector field Am and vm is the

field strength for the antisymmetric tensor Bmn. This Lagrangian is invariant under

the following supersymmetry transformations:

δηAm =√

1 + ε(ψmη + ψmη)

+1√

1 − ε

(i

1√3

(1 − ε)(ησmζ − ζ σmη) − 1√3m

∂m(ζη + ζ η)

)

δηBmn =

√1 − ε

1 + ε

(− 1

m∇[m(ηψn]) − iησ[mψn]

)− 2√

3

(ησmnζ +

i

2m∇[m(ζ σn]η)

)+ h.c.

δηζ =√

1 − ε

(1√3Amnσ

mnη − im√

3σmηAm

)

34

+m√

3(1 + ε)Bmnσ

mnη +1√3σmηvB

m

δηψm =1√

1 + ε

(1

3m∇m(Arsσ

rsη + 2imσnηAn) − i

2

(HA

+mnσn +

1

3HA

−mnσn)η

− 2

3m(σm

nAnη + Amη) − i

2εHA

+mnσnη − εmAmη

)

+1√

1 − ε

(1

3m∇m

(m

√1 + εBrsσ

rsη − 21√

1 + εσnηvB

n

)

+im√

1 + ε[(

1

3− ε

2

)Bmnσ

nη + i(

1

3− ε

4

)εmnrsB

nrσsη]

+2

3

i√1 + ε

(σmnvB

n η + vBmη) − i

ε√1 + ε

vBmη

).

Two more representations can be found by dualizing the vector Am. The deriva-

tions are straightforward, so I will not write the Lagrangians and transformations

here. Each of the four dual Lagrangians describe the dynamics of free massive spin-

3/2 and 1/2 fermions, together with their supersymmetric partners, massive spin-one

vector and tensor fields.

In what follows one shall see that the first two representations are special because

they can be regarded as “unitary gauge” descriptions of theories with a set of addi-

tional symmetries: a fermionic gauge symmetry for the massive spin-3/2 fermion, as

well as additional gauge symmetries associated with the massive gauge fields.

3.2.2 SuperHiggs effect for AdS spin-3/2 multiplets

To exhibit the superHiggs effect, I will first introduce a Goldstone fermion and

its superpartners. I will then gauge the full N = 2 supersymmetry. In this way I

will construct theories with a local N = 2 supersymmetry nonlinearly realized, but

with N = 1 represented linearly on the fields. The resulting Lagrangians describe the

physics of partial supersymmetry breaking well below the scale v where the second

supersymmetry is broken.

In flat space, the Goldstone fields become physical degrees of freedom in the mass-

less limit of the unHiggsed Lagrangian as in Sec. 2.1. In AdS space, the “massless”

limit corresponds to E → 2. In this limit the massive spin-3/2 multiplet splits into

35

a massless spin-3/2 multiplet, plus a massive vector/tensor multiplet of spin one (see

also Appendix H):

massive spin 3

2multiplet

z |

D(E; 1)D(E +1

2;3

2)D(E +

1

2;1

2)D(E + 1; 1)

E ! 2

XXXXXXXXXXXXXXXXXXXXz

XXXXXXXXXXXXXXXXXXXXz

D(2; 1)D(5

2;3

2)

| z

massless spin 3

2mult:

and D(5

2;1

2)D(3; 1)D(3; 0)D(

7

2;1

2)

| z

massive spin1 multiplet (E=5=2)

The spin-one multiplet with E = 5/2 cannot itself be unHiggsed because that would

require E → 1 (the normalization of E differs for different multiplets; see Appendix

H and Ref. [43].). For the case at hand, this would spoil the unitarity of the spin-3/2

field.

In Fig. 3.1 the physical fields of the massive spin-3/2 multiplet coupled to gravity

are arranged in terms of N = 1 multiplets. The fields of lowest spin form a mas-

sive N = 1 vector/tensor multiplet. They may be thought of as N = 1 “matter.”

The remaining fields are the gauge fields of N = 2 supergravity. In unitary gauge,

the massless vector eats the scalar, while the Rarita-Schwinger field eats one linear

combination of the spin-1/2 fermions. This leaves the massive N = 1 spin-3/2 mul-

tiplet coupled to N = 1 supergravity. In contrast to the superHiggs effect in a flat

background, where an N = 2 multiplet emerges in the massless limit (κ → 0) of the

unHiggsed theory (Sec. 2.1), the equivalent E → 2 limit in AdS space gives rise to

N = 1 multiplets only.

To find the Lagrangian, let me introduce a set of Goldstone fields by the following

Stuckelberg redefinitions. For the case with two vectors, I include Goldstone fields

by replacing

Am → Am − 1√1 − εm

∂mφA

Bm → Bm − 1√1 + εm

∂mφB . (3.7)

36

0B@23

2

1CA 0B@3

2

1

1CA

| z N=2 supergravity

0BBBBB@

11

2

1

2

0

1CCCCCA

| z N=1 matter

(massive)

Figure 3.1: The degrees of freedom of the unHiggsed OSp(1, 4) massive spin-3/2multiplet coupled to gravity. The massive spin-1 field can be represented by either avector or an antisymmetric tensor.

For the dual representation, one takes

Am → Am − 1√1 − εm

∂mφ

Bmn → Bmn − 1√1 + εm

∂[mBn] . (3.8)

In each case, the introduction of the Goldstino ν requires an additional shift

ψm → ψm − 1√6√

1 − ε2m(2∇mν + imσmν) (3.9)

to obtain a proper kinetic term for ν.

For the case with two vectors, the Lagrangian is as follows,

e−1L =

− 1

2κ2R + εmnrsψimσnDrψ

is − iλσmDmλ − iχσmDmχ

− 1

4AmnA

mn − 1

4BmnB

mn − 1

2DmφADmφA − 1

2DmφBDmφB

−( 1√

2m

√1 − ε2ψ2

mσmλ + m√

1 − ε2iψ2mσmχ

+√

2miλχ +1

2mχχ + mψ2

mσmnψ2n + εmψ1

mσmnψ1n

+κ

4εijψ

imψj

n(√

1 + εHmnA− − i

√1 − εHmn

B−)

37

+κ

2χσmσnψ1

m(DnφA − iDnφB)

+κ

2√

2λσmψ1

n(√

1 − εHmnA+ − i

√1 + εHmn

B+)

+κ

2εmnrs

√1 − ε

1 + εψm2σnψ

1r(∂sφA − i∂sφB)

− κ

2mεmnrsψm2σnψ

1r(

√1 + εAs − i

√1 − εBs)

− 2κεm

√1 − ε

1 + εψm2σ

mnψn1φA +κεm√

2λσmψ1

mφA

+ iκεmχσmψ1mφA + h.c.

)+ 3

ε2m2

κ2. (3.10)

In this expression, κ denotes Newton’s constant, m =√

Λ2 + κ2v4 and Dm is the

full covariant derivative. The scalar-field gauge-invariant derivatives are as follows,

DmφA = ∂mφA − m√

1 − εAm

DmφB = ∂mφB − m√

1 + εBm , (3.11)

while the supercovariant derivatives take the form

DmφA = ∂mφA − m√

1 − εAm − κ

2(ψ1

mχ + ψ1mχ)

DmφB = ∂mφB − m√

1 + εBm + iκ

2(ψ1

mχ − ψ1mχ)

Amn = Amn +κ

2

√1 + ε(ψ2

[mψ1n] + ψ2

[mψ1n])

−√1 − ε

κ

2√

2(λσ[nψ

1m] + ψ1

[mσn]λ)

Bmn = Bmn − iκ

2

√1 − ε(ψ2

[mψ1n] − ψ2

[mψ1n])

+i√

1 + εκ

2√

2(λσ[nψ

1m] − ψ1

[mσn]λ) . (3.12)

This Lagrangian is invariant (to lowest order in the fields) under the following

supersymmetry transformations,

δηeam = iκηiσaψmi + iκηiσ

aψim

δηψ1m =

2

κDmη1 + i

εm

κσmη1

δηAm =√

1 + εεij(ψimηj + ψi

mηj) +√

1 − ε1√2

(η1σmλ + λσmη1)

38

δηBm =√

1 − εεij(−iψimηj + i ψi

mηj) +√

1 + εi√2

(η1σmλ − λσmη1)

δηλ = i√

1 − ε1√2Amnσ

mnη1 +√

1 + ε1√2Bmnσ

mnη1

+√

2 i εmφAη1 − i

√2m

κ

√1 − ε2η2

δηχ = iσmη1DmφA − σmη1DmφB − 2 εmφAη1 + 2

m

κ

√1 − ε2η2

δηψ2m =

2

κDmη2 + i

m

κσmη2 − i

2

√1 + εHA

+mnσnη1 − m

√1 + εAmη1

+1

2

√1 − εHB

+mnσnη1 +

√1 − ε

1 + ε(∂mφA − i DmφB)η1

−κ

2

√1 − ε

1 + εψ1

m(δη1φA − i δη1φB) − i εm

√1 − ε

1 + εφAσmη1

δηφA = χη1 + χη1

δηφB = −iχη1 + i χη1 . (3.13)

This result holds to leading order, that is, up to and including terms in the trans-

formations that are linear in the fields. Note that this representation is irreducible in

the sense that there are no subsets of fields that transform only into themselves under

the supersymmetry transformations. The Lagrangian (3.10) describes the sponta-

neous breaking of N = 2 supersymmetry in AdS space. It has N = 2 supersymmetry

and a local U(1) gauge symmetry. In unitary gauge, it reduces to the massive N = 1

Lagrangian of Eq. (3.4).

Let me now consider the dual case with one massive tensor. The degree of freedom

counting is as in Fig. 3.1. Note that the massive N = 1 “vector” multiplet now

contains a massive antisymmetric tensor.

The Lagrangian and supersymmetry transformations for this system can be worked

out following the procedures described above. They can also be derived by dualizing

first the scalar φB and then the vector Bm using the method described in [33] (see

also Appendix C). The Lagrangian is given by

e−1L =

− 1

2κ2R + εmnrsψimσnDrψ

is − iλσmDmλ − iχσmDmχ

− 1

4AmnA

mn − 1

4FB

mnFBmn − 1

2DmφADmφA +

1

2vBmvB

m

39

−( 1√

2m

√1 − ε2ψ2

mσmλ + m√

1 − ε2iψ2mσmχ

+√

2miλχ +1

2mχχ + mψ2

mσmnψ2n + εmψ1

mσmnψ1n

+κ

4εijψ

imψj

n(√

1 + εHmnA− +

√1 − εFBmn

− )

+κ

2χσmσnψ1

m(DnφA + ivBn )

+κ

2√

2λσmψ1

n(√

1 − εHmnA+ − √

1 + εFBmn+ )

+κ

2εmnrs

√1 − ε

1 + εψm2σnψ

1r(∂sφA + ivB

s )

− κ

2mεmnrsψm2σnψ

1r

√1 + εAs

− 2κεm

√1 − ε

1 + εψm2σ

mnψn1φA +κεm√

2λσmψ1

mφA

+ iκεmχσmψ1mφA + h.c.

)+ 3

ε2m2

κ2(3.14)

where

DmφA = ∂mφA − m√

1 − εAm

FBmn = ∂[mBn] − m

√1 + εBmn (3.15)

and

DmφA = ∂mφA − m√

1 − εAm − κ

2(ψ1

mχ + ψ1mχ)

vm = vm − iκψ1

nσmnχ − iκ

2

√1 − ε

1 + εεm

nrsψ1nσrψ

2s + h.c.

Amn = Amn +κ

2

√1 + ε(ψ2

[mψ1n] + ψ2

[mψ1n])

−√1 − ε

κ

2√

2(λσ[nψ

1m] + ψ1

[mσn]λ)

FBmn = FB

mn +κ

2

√1 − ε(ψ2

[mψ1n] + ψ2

[mψ1n])

+√

1 + εκ

2√

2(λσ[nψ

1m] + ψ1

[mσn]λ) . (3.16)

The supersymmetry transformations are as follows:

δηeam = iκηiσaψmi + iκηiσ

aψim

40

δηψ1m =

2

κDmη1 + i

εm

κσmη1

δηAm =√

1 + εεij(ψimηj + ψi

mηj) +√

1 − ε1√2

(η1σmλ + λσmη1)

δηBm =√

1 − εεij(ψimηj + ψi

mηj) − √1 + ε

1√2

(η1σmλ + λσmη1)

δηBmn = −2η1σmnχ −√

1 − ε

1 + ε(i η1σ[mψ2

n] + i η2σ[mψ1n]) + h.c.

δηλ = i√

1 − ε1√2Amnσ

mnη1 − √1 + ε

i√2

FBmnσ

mnη1

+√

2 i εmφAη1 − i

√2m

κ

√1 − ε2η2

δηχ = iσmη1DmφA + vmσmη1 − 2 εmφAη1 + 2

m

κ

√1 − ε2η2

δηψ2m =

2

κDmη2 + i

m

κσmη2 − i

2

√1 + εHA

+mnσnη1 − m

√1 + εAmη1

+

√1 − ε

1 + ε∂mφAη

1

−κ

2

√1 − ε

1 + εψ1

mδη1φA − i εm

√1 − ε

1 + εφAσmη1

− i

2

√1 − εFB

+mnσnη1 + i

√1 − ε

1 + εvmη1

δηφA = χη1 + χη1 . (3.17)

These fields form an irreducible representation of the N = 2 algebra.

In both cases, the commutator of two first supersymmetries ξ and η induces a first

and a second supersymmetry transformation with parameters proportional to κ like

in Eq. (2.16); there is an additional piece coming from the Lorentz generator in the

AdS algebra (3.1):

[δξ, δη] = −2i(ξσnη − ησnξ)Dn + (gauge transformation)

+δ(2. supersymmetry)

((ξσnη−ησnξ)(iκψ2n+κ

2

√1−ε1+ε

σν χ))+ δ

(1. supersymmetry)((ξσnη−ησnξ)(iκψ1

n))

+δ(Lorentz transformation)(2iεΛξσabη) (3.18)

Each of the two Lagrangians has a full N = 2 supersymmetry (up to the appro-

priate order). The first supersymmetry is realized linearly.3 The second is realized

3In AdS supergravity, the gravitinos undergo a shift even for linearly realized supersymmetry

41

nonlinearly: it is spontaneously broken. In each case, the transformations imply that

ζ =1√3

(χ − i√

2λ) (3.19)

does not shift, while

ν =1√3

(√

2χ + iλ) (3.20)

does. Therefore ν is the Goldstone fermion for N = 2 supersymmetry, spontaneously

broken to N = 1.

I do not know how to unHiggs the other two representations of the massive spin-

3/2 multiplet. If one uses Stuckelberg redefinitions as in Eqs. (3.8), (3.9), the super-

symmetry transformations are singular as ε → 1. If one tries to dualize the above

representations, the procedure is thwarted by the bare φA fields in the Lagrangians

and transformation laws.

3.3 Dual AdS supersymmetry algebras

To find the supersymmetry algebras, one can compute the closure of the first and

second supersymmetry transformations to zeroth order in the fields. This will allow

me to identify the Goldstone fields associated with any spontaneously broken bosonic

symmetries.

In the case with two scalars (3.13), the algebra is as follows:

[δη2 , δη1 ]φA = 2m

κ

√1 − ε2(η1η2 + η1η2)

[δη2 , δη1 ]Am =√

1 + ε2

κ∂m(η1η2 + η1η2)

[δη2 , δη1 ]φB = −2im

κ

√1 − ε2(η1η2 − η1η2)

[δη2 , δη1 ]Bm = −i√

1 − ε2

κ∂m(η1η2 − η1η2) . (3.21)

From these expressions one can see that φA and φB are Goldstone bosons associated

with nonlinearly realized U(1) symmetries that are gauged by the vectors Am and

Bm.

[44]; see Eqs. (3.13), (3.17) and Eq. (G.4) in Appendix G.

42

In the case with one scalar and one antisymmetric tensor, Eq. (3.17), the last two

lines in Eq. (3.21) are replaced by

[ δη2 , δη1 ] Bm =2

κ

√1 − ε∂m(η1η2 + η1η2) + 2i

m

κ

√1 − ε(η1σmη2 − η2σmη1)

[ δη2 , δη1 ] Bmn = −2 i

κ

√1 − ε

1 + εD[m(η2σn]η

1 − η1σn]η2) . (3.22)

In this case φA and Bm are the Goldstone bosons of nonlinearly realized U(1)’s gauged

by Am and Bmn.

To find the symmetry algebra, let me consider these algebras in the limit κ → 0,

with fixed v2 = 0, Λ = 0. This limit corresponds to a fixed AdS background, in which

central charges can be identified. For the case with two scalars, I find

[ δη2 , δη1 ]φA = 2v2(η1η2 + η1η2)

[ δη2 , δη1 ]Am = 0 (3.23)

[ δη2 , δη1 ]φB = −2iv2(η1η2 − η1η2)

[ δη2 , δη1 ]Bm = −√

2iv2 ∂m

Λ(η1η2 − η1η2) . (3.24)

For the case with one scalar and one antisymmetric tensor, the last two lines are

replaced by

[ δη2 , δη1 ] Bm = 2iv2(η1σmη2 − η2σmη1)

[ δη2 , δη1 ] Bmn = −√

2iv2

Λ∇[m(η2σn]η

1 − η1σn]η2) . (3.25)

Equation (3.23) implies that the real scalar φA is the Goldstone boson associated

with the U(1) generator of the AdS algebra. (It is this generator which contracts to

a real central charge in flat space.) Equation (3.24) [(3.25)] indicates that the scalar

φB (vector Bm) is the Goldstone boson associated with a spontaneously-broken U(1)

symmetry, one which is gauged by the vector field Bm (tensor field Bmn).

These results imply that when v = 0 and Λ = 0, the full current algebra is actually

OSp(2, 4) ×s U(1), nonlinearly realized. The symbol ×s is a semi-direct product;

it is appropriate because the supersymmetry generators close into the local U(1)

symmetry. This construction evades the AdS generalization of the Coleman-Mandula

43

0

0

1

κ Λ

ε(κ,Λ)

Figure 3.2: The manifold of partially broken N = 2 supergravity theories as a functionof Newton’s constant κ and the cosmological constant Λ.

[8] and Haag-TLopuszanski-Sohnius [7] theorems because the broken supercharges do

not exist. The OSp(2, 4) ×s U(1) symmetry only exists at the level of the current

algebra; the U(1) symmetry is always spontaneously broken.

The supergravity theories that I have found depend on three dimensionfull pa-

rameters: κ, Λ, and v2. Since I am interested in partial supersymmetry breaking, I

shall keep v2 = 0. I then consider the Lagrangians (3.10), (3.14) as a function of κ

and Λ only. The dimensionless variable ε = Λ/√

Λ2 + κ2v4 is a particularly useful

parameter, because the limit ε → 0 corresponds to the case of partially broken N = 2

supergravity in Minkowski space, while ε → 1 approaches the “massless” limit of par-

tially broken supersymmetry in a fixed AdS background. The full manifold of N = 2

supergravities, described by the parameter ε, is plotted in Fig. 3.2. The center region

corresponds to the new AdS supergravities described above.

A prominent feature in Fig. 3.2 is the vertical line at (κ = 0, Λ = 0). This line

connects theories in a Minkowski background (ε = 0, Λ = 0) with the “massless” limit

of theories in a fixed AdS background (κ = 0, ε = 1). The line suggests that there

should be a family of globally supersymmetric theories in Minkowski space, only one

representative of which (ε = 0) can be deformed to a partially broken supergravity

theory in a Minkowski background. In contrast, a continuum of theories (0 < ε < 1)

can be deformed to partially broken supergravity theories in an AdS background.

Indeed, let me consider the limit κ → 0, Λ → 0 such that ε remains finite. If one

44

writes ε = sin(2θ), one finds the following N = 1 transformations for the case with

two scalars (3.13):

δηψ2m = − i

2cos θH−mnσ

nη1 − i

2sin θH+mnσ

nη1 +√

2∂mφη1

δηAm = 2 cos θψm2η1 + 2 sin θψ2

mη1

+√

2 sin θλσmη1 +√

2 cos θλσmη1

δηχ = i√

2σm∂mφη1

δηλ =i

2√

2sin θ(H−mnσ

mn)η1 − i

2√

2cos θ(H+mnσ

mn)η1

δηφ =√

2χη1 . (3.26)

Here, φ = (φA + iφB)/√

2 and Am = Am + iBm; Amn is its corresponding field

strength. (The case with one scalar and one antisymmetric tensor can be obtained

by dualization of φB and Bm; it is not presented here.)

The angle θ can be interpreted in terms of models with a full N = 2 multiplet

structure [23, 24]. In these models, a necessary ingredient for partial supersymmetry

breaking seems to be presence of at least one vector- and one hyper-multiplet, as

well as the non-existence of a prepotential F for the special Kahler manifold [24]

parametrized by the complex scalars zi of the i vector multiplets. It was shown in

[45] that such models can always be obtained by a symplectic transformation from a

model with a prepotential.

In [24] the symplectic vector Ω = (XΣ, FΣ) for the special Kahler manifold

SU(1, 1)/U(1) with one complex scalar z1 = z takes the form

Ω =

−12

i2

iz

z

. (3.27)

Here, no prepotential F exists such that ∂F∂XΣ = FΣ. If one assumes that the scalar

z acquires a vacuum expectation value 〈z〉 ∼ κ−1, and one expands the supersym-

metry transformations [46] around this vacuum expectation value, one finds that the

angle θ parametrizes the symplectic transformation S that maps this model with no

45

prepotential continuously to the case of the so-called “minimal coupling models” [47]:

Ω = SΩ =

cos θ 0 0 −12

sin θ

0 cos θ −12

sin θ 0

0 2 sin θ cos θ 0

2 sin θ 0 0 cos θ

−12

i2

iz

z

. (3.28)

Of course, this identification of the angle θ only holds to linear order in the fields; at

higher order, the model in [24] cannot be consistently truncated to my field content.

3.4 Discussion

In this chapter I have examined the partial breaking of supersymmetry in anti-de

Sitter space. One could see that partial breaking in AdS space can be accomplished

using two of four dual representations of the massive N = 1 spin-3/2 multiplet.

During the course of this work, I found new N = 2 supergravities and new N = 2

supersymmetry algebras based on the semi-direct product OSp(2, 4) ×s U(1), where

the U(1) is always nonlinearly realized for finite Λ.

The minimal theories for partial supersymmetry breaking in flat space in Sec. 2.1

were observed to be a consistent truncation of partially broken N = 2 supergravities

with complete N = 2 supermultiplets, i. e. the N = 2 supergravity multiplet, one

vector-, and one hypermultiplet. The nontrivial flat, global limit of the minimal the-

ory for partial supersymmetry breaking in AdS space in Sec. 3.3 provided a further

link to the special, symplectic geometry of those supergravities. In N = 1 language,

they contain two more massless chiral multiplets. Therefore the minimal theories from

Sec. 2.1 cannot be obtained from those supergravities by integrating out the addi-

tional chiral superfields. This makes one wonder whether the minimal theories from

Sec. 2.1 can be expanded to higher orders in the fields (by the Noether-procedure)

without adding those two chiral multiplets. Such a cumbersome undertaking was not

attempted.

Instead, there are other hints that the minimal theories and the method of their

derivation are only consistent to lowest order in the fields. The method was to start

46

with a massive gravitino multiplet in unitary gauge with global supersymmetry, per-

form the Stuckelberg redefinitions (2.6, 2.7, 2.8) and couple it to N = 1 supergravity.

One could imagine starting with the massive gravitino multiplet with local super-

symmetry, i. e. it is already coupled to N = 1 supergravity. Even in a supergravity

background, the 1/m-terms in the covariantized Stuckelberg redefinitions for a vector

Am → Am − 1m∂mφ and for an antisymmetric tensor Bmn → Bmn − 1

mD[mBn] do not

induce 1/m-singularities in the corresponding field strengths Amn and vm because

of the symmetry properties of the Christoffel-symbols. However, the covariantized

redefinition ψm → ψm − 1√6m

(2Dmν + imσmν) in the kinetic term 12εmnrsψmσnψrs

entails a term

1

2εmnrsψmσnD[rDs](− 2√

6mν) ∼ 1

mεmnrsψmσnRrsabσ

abν = 0 . (3.29)

Also the gravitino mass term induces a 1/m-singularity. Hence the method from Sec.

2.1 to unHiggs the gravitino multiplet carried out in a supergravity background does

not have a massless limit.

It is therefore desirable to find another starting point for the derivation of partially

broken supergravity theories that does not require singular Stuckelberg redefinitions.

This is accomplished in the next chapter.

47

Chapter 4

Partial Supersymmetry Breaking

from Five Dimensions

4.1 Introduction

Low energy effective theories with N = 1 supersymmetry are assumed to be the

low energy limit of a more fundamental theory like string theory or M-theory. Those

fundamental theories are formulated in higher space-time dimensions and possess an

extended number of supersymmetries which have to be spontaneously broken. Hence

dimensional reduction and partial supersymmetry breaking are both present in this

scenario. A technical tool that satisfies both features simultaneously is provided by

the Scherk-Schwarz mechanism [48] (also called generalized dimensional reduction),

where nontrivial boundary conditions for the fields along the compactified dimensions

can lead to spontaneously broken gauge and space-time symmetries. In this chap-

ter the simplest setup to investigate partial supersymmetry breaking by generalized

dimensional reduction is used, namely the partial breaking of N = 2 → N = 1 in

four dimensions coming from a theory in five dimensions. The focus is on the super-

Higgs effect, where Goldstone fermions from the broken supersymmetries become the

longitudinal components of massive gravitinos.

The physics that underlies this superHiggs effect for the spontaneous breaking of

N = 2 supersymmetry to N = 1 was investigated in a flat Minkowski background

48

in Sec. 2.1, where theories were constructed that describe partial supersymmetry

breaking in a model-independent approach with a minimal field content motivated by

the superHiggs effect.

In this chapter these theories of partial N = 2 → N = 1 supersymmetry breaking

in four dimensions will be reproduced by compactifying the massless N = 2 D = 5

massless gravitino multiplet1 on the orbifold S1/Z2 and using the Scherk-Schwarz

mechanism [48] to introduce a symmetry-breaking mass parameter.2 In order to

exhibit the superHiggs effect, this procedure must be carried out in a supergravity

background.

This derivation serves several purposes: The starting point of the investigation

is the massless N = 2 D = 5 gravitino multiplet Noether-coupled to N = 2 D = 5

supergravity which leads to pure N = 4 D = 5 supergravity (without matter). This

is a rather simple theory that allows for easy extensions such as embedding into a

higher N theory or introducing matter couplings. Second, the theories obtained from

five dimensions are already out of unitary gauge, so that the symmetry breaking mass

parameter m can be set to zero with impunity. In particular, no singular Stuckelberg

redefinitions (containing 1/m-terms as in Eq. 2.8) are required.

What other motivation apart from simplicity is there to look at a theory in five

dimensions? It seems natural to derive a massive 4-dimensional theory from a massless

theory in five dimensions, because the degrees of freedom of a massless D = 5 vector

field [3] and the degrees of freedom of a massless symplectic D = 5 Majorana spinor

[4] match those of their massive counterparts in D = 4. This transition can be easily

implemented by a Scherk-Schwarz compactification on S1, where it can be seen that

the fifth components of a symplectic gravitino ΨiαM and a complex vector AM , Ψi

α4

and A4, provide the longitudinal components of the massive 4-dimensional fields Ψiαm

and Am.3

1Here, N = 2 refers to the lowest supersymmetry in five dimensions, corresponding to twosymplectic Majorana spinors, which are equivalent to one Dirac spinor.

2In Ref. [49] such a combination of orbifold projection and Scherk-Schwarz compactification wasintroduced to derive string theories with spontaneously broken supersymmetry.

3In this chapter, i, j... denote symplectic indices, M,N ... are 5-dimensional world indices andm,n... are 4-dimensional world indices. A dotted numerical index stands for a world index, to bedistinguished from undotted Lorentz indices.

49

A natural candidate for partial supersymmetry breaking in four dimensions ef-

fected by Scherk-Schwarz compactification on S1 would be pure N = 2 D = 5 su-

pergravity [50, 51], in particular since the degrees of freedom exactly match those of

one of the minimal cases (with two vectors) considered in Sec. 2.1. Unfortunately,

Scherk-Schwarz compactification applied to this theory breaks both supersymme-

tries. In fact, any Scherk-Schwarz compactification of N -extended supergravity on

S1 breaks an even number of supersymmetries, since the Scherk-Schwarz generator

must be a generator of USp(N) [52]. This can be circumvented by projecting out half

of the states by compactifying on the orbifold S1/Z2. In what follows, the length of

the interval S1/Z2 is assumed to be of the order of the Planck length; hence only the

massless Kaluza-Klein modes (possibly lifted by the Scherk-Schwarz mass parameter)

are retained.

The outline of the chapter is as follows: In the second section the massless N = 2

D = 5 gravitino multiplet will be compactified on S1/Z2, yielding a massive N = 1

gravitino multiplet in four dimensions together with a spontaneously broken fermionic

gauge symmetry. In the third section, the massless N = 2 D = 5 gravitino multiplet

will be Noether-coupled to N = 2 D = 5 supergravity thus yielding N = 4 supergrav-

ity before the dimensional reduction is carried out. The corresponding 4-dimensional

theory is shown to coincide with theories exhibiting partial supersymmetry breaking

which were derived from four dimensions [23, 53, 24].4

4.2 Generalized compactification of the massless

N = 2 D = 5 gravitino multiplet

The massless N = 2 D = 5 gravitino multiplet contains two symplectic Majo-

rana gravitinos ΨiM , two symplectic Majorana spinors Λi and four real vector fields.

Its Lagrangian and supersymmetry transformations can be conveniently obtained by

truncation of the N = 4 D = 5 supergravity constructed in Ref. [57]. The conventions

4The geometric construction of [24] could be generalized to accommodate an arbitrary N = 2matter and gauge content [54] and to give masses to light mirror fermions [55]. String modelsexhibiting the mass spectrum of partially broken supersymmetry were constructed in Refs. [56].

50

for the 5-dimensional Dirac algebra and the symplectic geometry of the automorphism

group are presented in Appendix I.

The five abelian vector fields AijM i, j ∈ 1, ..., 4 in the 5-dimensional representa-

tion of USp(4) which are part of the spectrum of N = 4 D = 5 supergravity satisfy

the reality condition [50]

AijM = (AMij)

∗ ,

and are parametrized by

AMij =

0 AM BM CM

−AM 0 −CM BM

−BM CM 0 −AM

−CM −BM AM 0

, (4.1)

where AM is real and BM and CM are complex.

Truncation of the N = 4 D = 5 supergravity to the massless N = 2 gravitino

multiplet requires AM = 0. The remaining two complex vectors can be written as

AMia =

( −BM CM

−CM −BM

)a, i ∈ 1, 2 ,

which corresponds to a decomposition USp(4) → USp(2)⊗USp(2) ∼= SU(2)⊗SU(2).

The field AM now carries two different symplectic indices i and a, indicating the two

independent SU(2)’s that it is charged under. Here,

Ωij = Ωab =

(0 1

−1 0

).

The Lagrangian is given by

L =i

2ΨiMΓMNO∂NΨi

O − i

2ΛiΓ

M∂MΛi +1

8FMN

iaF

MNai , (4.2)

and the supersymmetry transformations are

δΞΨiM =

1

6FNO

ia(ΓM

NO + 2δ[NM ΓO])Ξa

δΞAMai = − 2i√

3ΞaΓMΛi − 2iΞaΨi

M

δΞΛi = − 1

2√

3FMN

iaΓMNΞa , (4.3)

51

λi ψim ψi

4 ξa i/aParity 1 -1 1 1 1

-1 1 -1 -1 2

Table 4.1: Fermionic parity assignment of D = 5 N = 2 gravitino multiplet in termsof D = 4 Weyl spinors.

Bm B4 Cm C4

Parity -1 1 1 -1

Table 4.2: Bosonic parity assignment of D = 5 N = 2 gravitino multiplet in terms ofD = 4 fields.

where FMNia = ∂[MAN ]

ia. In order to implement the Scherk-Schwarz mechanism on

the orbifold S1/Z2, the Z2 transformation and the generator T of a global symmetry

of the 5-dimensional theory used for the Scherk-Schwarz mechanism must satisfy [58]

Z2eimTx4

= eimT (−x4)Z2 ⇔ Z2, T = 0 . (4.4)

In addition, the parity operation must be part of the discrete symmetries of the 5-

dimensional theory. The effect of the parity operator Z2 = ±(

1 0

0 −1

)⊗ iΓ4 on

the fields is listed in Tables 4.1 and 4.2. With the choice T = σ2 ∈ su(2) and the

assignment5

Φi → (eimσ2x4

)ijΦj

for a generic field Φi having a symplectic i (not a) index, the generalized dimensional

reduction of the massless gravitino multiplet is completely specified.

The limit l → 0 of the length l of the interval S1/Z2 is implicitly understood, so

that only the lowest even Fourier-modes of the fields which depend only on the first

four space-time co-ordinates x0, ..., x3 (zero-modes) need to be retained.

To simplify the identification of the physical fields in the 4-dimensional theory,

the following redefinitions are necessary to diagonalize and normalize the fermionic

kinetic terms:

Ψim → Ψi

m +1

2ΓmΓ4Ψi

4 , (4.5)

5The additional x4-dependent SU(2) group element promotes Φi to a non-trivial fiber-bundle onS1/Z2.

52

and νi =√

3/2ψi4. Note that in (4.5) no terms ∼ 1

m∂mΨi

4 as in Eq. (3.9) occur;

in a gravitational background, the covariantized form of this term would not leave

the gravitino kinetic term invariant and would therefore induce terms ∼ 1m

in the

Lagrangian, as discussed in Sec. 3.4.

With additional “chiral” redefinitions Λi → −Γ4Λi and Ξa → Γ4Ξa the Lagrangian

becomes

L4 = εmnpqψ2mσn∂pψ2q − iλ1σ

m∂mλ1 − iν1σm∂mν1

−1

4CmnC

mn − 1

2DmB4DmB4

−√

3

2m(iψ2

mσmν1 + h.c.) − m(ν1ν1 + h.c.)

+1

2mλ1λ1 − mψ2

mσmnψ2n + h.c. (4.6)

with supersymmetry transformations

δξCm = i2√3ξ1σmλ1 + 2ξ1ψ2

m − i

√2

3ξ1σmν1

δξB4 =2√3ξ1λ1 +

2√

2√3ξ1ν1

δξλ1 =

1√3Cmnσ

mnξ1 +i√3

DmB4σmξ1

δξψ2m = − i

2C+mnσ

nξ1 + DmB4ξ1

δξν1 = − 1√

6Cmnσ

mnξ1 + i

√2

3DmB4σ

mξ1 ,

where DmB4 = ∂mB4 − mCm and C+mn = ∂[mCn] + i2εmnrs∂

[rCs]. The expression for

the gauge invariant derivative Dm shows that the complex scalar B4 is the Goldstone

boson of a spontaneously broken abelian gauge symmetry mediated by the vector

Cm. The spinor ν1 is the longitudinal component of the massive gravitino ψ2m. The

massless five-dimensional theory also contains a fermionic gauge symmetry:

δΘΨiM =

2

κ∂MΘi .

A generalized Scherk-Schwarz compactification leads to a spontaneously broken fermionic

53

symmetry:

δθψ2m = i

m

κσmθ2

δθν1 =

√6m

κθ2 . (4.7)

Therefore, ν1 can also be interpreted as the Goldstino for this broken symmetry.

The Lagrangian for the massless gravitino multiplet (4.2) is not unique. In flat

5-dimensional space-time a massless vector BM is dual6 (see Appendix C) to an

antisymmetric tensor GMN

∂[MBN ] =1

2εMNOPQ∂

OGPQ + · · · , (4.8)

where the dots stand for contributions from interaction terms. So all four real vectors

of the massless N = 2 gravitino multiplet can be dualized and then dimensionally

reduced to four dimensions. The implementation of the Scherk-Schwarz mechanism

on S1/Z2 requires that fields of the same index structure come in pairs, so that

one of them can be projected out by the Z2 reflection whereas the other becomes

a massive field in the 4-dimensional theory. If both BM and CM are dualized, the

field strength FMNia is simply replaced by its dual in the Lagrangian and the su-

persymmetry transformations and the automorphism group decomposes as before as

USp(4) → SU(2) ⊗ SU(2).

The other possibility is the dualization of only two vectors, which must be the

real or imaginary parts of the vectors BM = BRM + iBI

M and CM = CRM + iCI

M . Other-

wise, either the complex vector or the complex antisymmetric tensor are completely

projected out. Here, the imaginary parts are chosen to be dualized: BIM → BI

MN and

CIM → CI

MN . This corresponds to the decomposition

FMNia = Re(FMN

ia) + iIm(FMN

ia)

→ FRMN

i

a + ivMNia ,

where vMNia = 1/2εMNOPQ∂

OGPQia and GMN

ia =

( −CIMN BI

MN

BIMN CI

MN

). To conserve

this structure, the SU(2) ⊗ SU(2) transformations must be restricted to those which

6This is not true for a 5-dimensional AdS background, where antisymmetric tensors satisfy ad-ditional self-duality conditions [59].

54

BRm BR

4 CRm CR

4 BImn BI

m4 CImn CI

m4

Parity -1 1 1 -1 1 -1 -1 1

Table 4.3: Bosonic parity assignment of the dualized D = 5 N = 2 gravitino multipletin terms of D = 4 fields.

do not mix real and imaginary parts. Therefore only the generator σ2 is allowed,

which corresponds to a decomposition SU(2) ⊗ SU(2) → U(1) ⊗ U(1).

Dualization of the imaginary parts of the complex vectors BM and CM yields the

Lagrangian

L =i

2ΨiMΓMNO∂NΨi

O − i

2χiΓ

M∂Mχi

+1

8FR

MN

i

aFRMNa

i +1

8vMN

iav

MNai , (4.9)

and the supersymmetry transformations are

δΞΨiM =

1

6FR

NO

i

a(ΓMNO + 2δ

[NM ΓO])Ξa +

i

6vNO

ia(ΓM

NO + 2δ[NM ΓO])Ξa

δΞARMa

i= − i√

3ΞaΓMΛi − iΞaΨi

M + h.c.

δΞGMNai =

1√3

ΞaΓMNχi − ΞaΓ[MΨiN ] + h.c.

δΞΛi = − 1

2√

3FR

MN

i

aΓMNΞa − i

2√

3vMN

iaΓMNΞa . (4.10)

The parities of the antisymmetric tensors in terms of 4-dimensional fields are deter-

mined by the parities of the dual vectors and the dualization relation (4.8) and are

listed in Table 4.3. Performing the same procedure and redefinitions (4.5) as before,

the following Lagrangian is obtained:

L4 = εmnpqψ2mσn∂pψ2q − iλ1σ

m∂mλ1 − iν1σm∂mν1

−1

4CR

mnCRmn − 1

4FCI

4mnFCI

4mn +1

4vBI

m vBIm − 1

2DmBR

4 DmBR4

−√

3

2m(iψ2

mσmν1 + h.c.) − m(ν1ν1 + h.c.)

+1

2mλ1λ1 − mψ2

mσmnψ2n + h.c. (4.11)

with supersymmetry transformations

δξCRm = i

1√3ξ1σmλ1 + ξ1ψ2

m − i1√6ξ1σmν1 + h.c.

55

δξBR4 =

1√3ξ1λ1 +

√2

3ξ1ν1 + h.c.

δξBImn = − 2√

3ξ1σmnλ

1 − 2

√2

3ξ1σmnν

1 − iξ1σ[mψ2n] + h.c.

δξCIm4 = − i√

3ξ1σmλ1 +

i√6ξ1σmν1 + ξ1ψ2

m + h.c.

δξλ1 =

1√3CR

mnσmnξ1 +

i√3

DmBR4 σ

mξ1 − 1√3

FCI4

mnσmnξ1 +

1√3vBI

m σmξ1

δξν1 = − 1√

6CR

mnσmnξ1 + i

√2

3DmBR

4 σmξ1 +

1√6

FCI4

mnσmnξ1 +

√2

3vBI

m σmξ1

δξψ2m = − i

2CR

+mnσnξ1 + DmBR

4 ξ1 − i

2FCI

4+mnσ

nξ1 + ivBI

m ξ1 ,

where DmBR4 = ∂mBR

4 − mCRm, FCI

4mn = ∂[mCI

n]4 − mBImn and vBI

m = 1/2εmnop∂nBIop.

In the dual case, the vector CIm4 becomes the Goldstone boson of the spontaneously

broken gauge symmetry mediated by the antisymmetric tensor BImn. This is the

dual Higgs mechanism investigated in Ref. [60]. As before, the 5-dimensional theory

has a fermionic gauge symmetry which is spontaneously broken upon generalized

dimensional reduction (4.7). The generalized dimensional reduction of the massless

gravitino multiplet with four antisymmetric tensors is analogous to the case illustrated

above and will not be presented here.

These 4-dimensional theories correspond exactly, up to field relabelings, to the

dual versions of the massive N = 1 gravitino multiplet (out of unitary gauge) before

being coupled to gravity (Sec. 2.1.1.1). In order to exhibit the superHiggs effect for

partial supersymmetry breaking, the above multiplets must be coupled to gravity and

the fermionic gauge symmetry must be promoted to a local supersymmetry. This will

be addressed in the next section.

4.3 Generalized dimensional reduction of pure N =

4 D = 5 supergravity

The complete theory of partially broken supergravity in four dimensions to all or-

ders in the fields should now be obtainable by Noether-coupling the massless gravitino

56

λi ψim ψi

4ξi i

Parity -1 1 -1 1 11 -1 1 -1 21 -1 1 -1 3-1 1 -1 1 4

Table 4.4: Fermionic fields and parities of D = 5 N = 4 supergravity in terms ofD = 4 Weyl spinors.

eam e4m e4

4 Gm G4 σ Am A4 Bm B4 Cm C4

Parity 1 -1 1 -1 1 1 -1 1 -1 1 1 -1N = 2 supergravity vector mult. gravitino mult.

Table 4.5: Bosonic fields and parities of D = 5 N = 4 supergravity in terms of D = 4fields.

multiplet to pure N = 2 D = 5 supergravity [50, 51, 61] and performing a generalized

dimensional reduction. Since consistency requires that the fermionic gauge symme-

tries of the massless gravitinos become local supersymmetries, the resulting theory

must be N = 4 supersymmetric. Therefore, the Scherk-Schwarz mechanism should

be applied to N = 4 D = 5 supergravity. The pure five-dimensional N = 4 super-

gravity was shown to be derivable as a consistent truncation of the maximal N = 8

supergravity [50]. The explicit form of the N = 4 supergravity Lagrangian and its

supersymmetry transformations was given in Ref. [57].

For the sake of clarity representative parts of the result of [57] are reproduced7.

N = 4 supergravity has a USp(4) symmetry inherited from the automorphism group

of the supersymmetry algebra and its field content together with a consistent parity

assignment for inversion of the fifth co-ordinate is given in Tables 4.4 and 4.5. The

Lagrangian is given up to terms of order O(κ)O(fermions)8 and the supersymmetry

transformations are given up to three-fermion terms. The Lagrangian reads

7Here, the vector in the singlet representation of USp(4), Bµ, from Ref. [57] has been renamedto GM and the vector fields in the 5-dimensional representation of USp(4) are parametrized as in(4.1). Also, the 5-dimensional spinors χi have been renamed to Λi. The following redefinitions wereperformed: ΓA → −iΓA, AMij → 1/2AMij , and Ξi → 2Ξi.

8The Lagrangian derived in Ref. [57] includes such terms up to four-fermion terms, but they arenot illustrative in the context of partial supersymmetry breaking.

57

κe−1L = − 1

2κ2R +

i

2ΨiMΓMNODNΨi

O − i

2ΛiΓ

MDMΛi

− 1

16e

2√3κσF ij

MNFMNij − 1

4e− 4√

3κσGMNGMN − 1

2∂Mσ∂Mσ

+κ

16√

2e−1εMNOPQF ij

MNFOPijGQ + O(κ)O(fermions) , (4.12)

and the supersymmetry transformations are

δΞΨMi =2

κDMΞi − 1

6e

1√3κσFNOij(ΓM

NO + 2δ[NM ΓO])Ξj

− 1

6√

2e− 2√

3κσGNO(ΓM

NO + 2δ[NM ΓO])Ξi + (3-fermion terms)

δΞAMij =

i√3e− 1√

3κσ

(2Ξ[iΓMΛj] − ΩijΞkΓMΛk)

−ie− 1√

3κσ

(2Ξ[iΨj]M − ΩijΞkΨk

M)

δΞΛi = −ΓMΞi∂Mσ − 1

2√

3e

1√3κσFMNijΓ

MNΞj

+1√6e

2√3κσGMNΓMNΞi + (3-fermion terms)

δΞGM = i

√2

3e

2√3κσ

ΞiΓMΛi +i√2e

2√3κσ

ΞiΨiM

δΞσ = −iΞiΛi

δΞeAM = iκΞiΓ

AΨiM . (4.13)

The fields and supersymmetry parameters that are enclosed in double lines in Tables

4.4 and 4.5 are the ones coming from the gravitino multiplet. This field content shows

that a consistent Noether-coupling of the gravitino multiplet to N = 2 supergravity

necessitates the inclusion of a vector multiplet with fields (λ1, λ2, σ, Am, A4).

From the parities in Tables 4.4 and 4.5 it is clear that the four dimensional theory

contains in addition to the fields of N = 2 D = 4 supergravity and the massive N = 1

gravitino multiplet out of unitary gauge [25] two chiral multiplets (λ2, ψ24, P,G4, σ, A4)

where eP = e44. This is the field content of one vector- and one hyper-multiplet

coupled to N = 2 supergravity.

The rest of this section is now devoted to showing that a generalized dimensional

reduction of N = 4 D = 5 supergravity on S1/Z2 with appropriately chosen USp(4)-

generator for the Scherk-Schwarz mechanism indeed leads to partially broken N = 2

58

supergravity in four dimensions. First, those USp(4)-generators which satisfy the

condition (4.4) with Z2 = ±diag(1,−1,−1, 1) ⊗ iΓ4 must be singled out. Out of Tr

r ∈ 1, ..., 10 (see Appendix I), the generators Ts with s ∈ 1, 2, 4, 5, 8, 10 satisfy

(4.4).

From the supersymmetry transformation of the gravitinos (4.13)

δΞΨiM =2

κDMΞi + · · ·

→ δΞ(eimTsx4

)ijΨjM =

2

κDM

((eimTsx4

)ijΞj

)+ · · · (4.14)

it is obvious that the fifth component Ψi4 of the gravitinos can pick up a constant

shift ∼ m. With the redefinition (4.5) the corresponding 4-dimensional gravitino Ψim

will also pick up a shift, thus identifying the broken supersymmetries.

The choice Ts with s ∈ 8, 10 would break all supersymmetries. In order to match

the Scherk-Schwarz mechanism in Sec. 4.2, the generator T5 =

(0 0

0 σ2

)will be used

to implement the partial breaking of supersymmetry in the 4-dimensional theory.

With T5, both ψ34

and ψ44

acquire a shift under supersymmetry transformations

δξ

(ψ3

4

ψ44

)=

2

κm

(0 1

−1 0

) (ξ3

ξ4

)+ . . .

Since ψ44

is projected out under the Z2 reflection, only ψ34

will be the Goldstino of one

spontaneously broken supersymmetry in the 4-dimensional theory.

The dimensional reduction from M5 → M4 × S1/Z2 requires Weyl rescalings and

field redefinitions described in Refs. [51, 62] in order to obtain canonically normalized

diagonal kinetic terms. They are simplified, however, because the vector e4m from

the funfbein used for the Weyl rescaling is odd under the Z2 reflection. Explicitly the

rescalings and redefinitions are: eam → e−P/2eam, ξi → e−P/4ξi, λi → −ieP/4λi, ψ1m →

e−P/4ψ1m + i/2σmψ2,4e

−3P/2, ψ4m → e−P/4ψ4

m − i/2σmψ3,4e−3P/2, ψi

4→ e5P/4

√2/3 νi.

The resulting 4-dimensional Lagrangian up to terms of order O(κ)O(fermions) in

the conventions of [63] reads

e−1L = − 1

2κ2R + εpqrsψipσqDrψ

is − iχiσ

mDmχi − iΩiσmDmΩi

− 1

4e√

2κϕAmnAmn− 1

2∂mϕ∂mϕ − 1

2e−2

√2κϕ∂mπ∂mπ

59

−1

2∂mϕ∂mϕ − 1

2e2κϕDmπaDmπa

−meκ(ϕ− 1√

2ϕ)

(− i√

2ψ2

mσmΩ1 + iψ2mσmχ1 −

√2Ω1χ1

+1

2χ1χ1 + ψ2

mσmnψ2n + h.c.

)+

κ

4√

2πAmnAmn + O(κ)O(fermions) , (4.15)

and the supersymmetry transformations up to three-fermion terms are

δξeam = −iκψi

mσaξi + h.c.

δξψim =

2

κDmξi − 1

2εije

κ ϕ√2 A+mnσ

nξj +i√2e−

√2κϕ∂mπξi

−ieκϕDmπaσaijξ

j + im

κδi2e

κ(ϕ− 1√2ϕ)σmξ2

+(3-fermion terms)

δξAm = e−κ ϕ√

2 (−2iεijψimξj −

√2Ωiσmξi)

δξΩi = − i√

2eκ ϕ√

2 Amnσmnξi − iεijσmξj(∂mϕ − ie−

√2κϕ∂mπ)

−√

2m

κεi2e

κ(ϕ− 1√2ϕ)ξ2 + (3-fermion terms)

δξχi = −iεijσmηj∂mϕ − eκϕεijσa

jkDmπaσmηk

+2m

κεi2e

κ(ϕ− 1√2ϕ)ξ2 + (3-fermion terms)

δξπ = e√

2κϕ(iεijΩiξj + h.c.)

δξϕ = εijΩiξj + h.c.

δξϕ = εijχiξj + h.c.

δξπa = e−κϕ(−iχiσa

ijεjkξ

k + h.c.) , (4.16)

where Dmπaσaij = ∂mπaσa

ij − m(σ1

ijAm + σ2

ijBm), σai

j = εilεjkσalk, Am = Am +

iBm, and Amn = εmnopAop. Here, several field redefinitions have been performed to

facilitate the identification of 4-dimensional N = 2 multiplets. With the rescaling

P →√

2/3P the chart of fields in Eqs. (4.15, 4.16) corresponding to the fields in Eqs.

(4.12, 4.13) is given in Table 4.6. In terms of N = 2 multiplets, the fields (eam, ψ1m,

ψ2m, Am) constitute the N = 2 supergravity multiplet, (ϕ, π, Ω1, Ω2, Bm) constitute

an N = 2 vector multiplet, and (ϕ, π1, π2, π3, χ1, χ2) form a hyper-multiplet.

60

part. broken N = 2 D = 4 SUGRA N = 4 D = 5 SUGRA

ϕ (√

2σ + P )/√

3

ϕ (σ − √2P )/

√3

π G4

π1 + iπ2 B4

−π3 A4

Am Cm

eam eamψ1

m ψ1m

ψ2m ψ4

m

for i ∈ 2, 3 χ4−i

Ω4−i

(λi +√

2νi)/√

3(√

2λi − νi)/√

3

Table 4.6: Corresponding fields of partially broken N = 2 D = 4 supergravity andpure N = 4 D = 5 supergravity.

From the Lagrangian (4.15) and the supersymmetry transformations (4.16) it

is obvious that ψ2m is the gauge field for the spontaneously broken supersymmetry,

whereas a linear combination of Ω1 and χ1 is the associated Goldstino. Its supersym-

metric partner, the scalar π1 + iπ2 is the Goldstone boson of a spontaneously broken

central charge gauged by the vector Am. That can be seen from the closure of the

first and second supersymmetry algebra on π1 + iπ2 and Am as in Sec. 2.1.2.

This result should be compared to partially broken 4-dimensional N = 2 super-

gravities with complete N = 2 multiplets constructed in Refs. [23, 53, 24]. Albeit

equivalent up to field redefinitions, the non-singular parametrization of the scalar

fields in Ref. [53] is closest to the parametrization of the scalars as obtained by the

above-described dimensional reduction, so the field labels from [53] are used in Ta-

ble 4.6. In Ref. [53] the breaking of the ith supersymmetry is parametrized by the

quantities µi, i ∈ 1, 2. In this context, the second supersymmetry is chosen to be

broken, so µ1 = 0 and µ2 = m. The more general supersymmetry breaking scenario

in Ref. [53] can be obtained from five dimensions by the choice µ1T2 + µ2T5 for the

Scherk-Schwarz generator. One finds complete agreement of (4.15) and (4.16) up to

trivial phase factors with the corresponding expressions in Ref. [53].

61

4.4 Discussion

In this chapter it was shown that partial supersymmetry breaking N = 2 → N = 1

in four dimensions can be easily reproduced by compactifying N = 4 D = 5 super-

gravity on the orbifold S1/Z2 and using the Scherk-Schwarz mechanism. This means

that compactification of N = 4 D = 5 supergravity on S1/Z2 automatically leads

to an N = 2 supersymmetric theory in four dimensions in which no prepotential

exists for the vector multiplet — thus allowing for partial supersymmetry breaking

[24]. Although the derivation of partially broken theories in four dimensions from five

dimensions is considered here only as a convenient tool, it allows for straightforward

extensions like matter couplings or embeddings in higher-N theories in five dimen-

sions. This is possible because the starting point — N = 4 D = 5 supergravity —

is a linearly realized massless theory with complete multiplets. Therefore it is not

surprising that complete N = 2 supermultiplets as in Refs. [23, 53, 24] persist in four

dimensions in the nonlinearly realized broken phase.

The Poincare dualities exploited in Sec. 2.1 for massless vectors and scalars in

four dimensions can now be seen to be consequences of a duality in five dimension

relating a massless vector to a massless antisymmetric tensor (see Appendix C). Dual

formulations of the theories in Refs. [23, 53, 24] can be found by dualizing first the

Goldstone scalars to antisymmetric tensors and then dualizing the vectors according

to the method described in Ref. [33]. These theories correspond to lowest order in

the fields to the effective theories derived in Sec. 2.1.

In Sec. 3 the partial breaking of extended supersymmetry in four dimensions

with a minimal field content as dictated by the superHiggs effect was extended to an

anti-de Sitter background. There it was found that only one of the two Goldstone

scalars in the theory out of unitary gauge could be dualized to an antisymmetric

tensor, the other one could not be dualized because it occurred in the theory without

derivatives acting upon it. If one were to derive this theory by a compactification

AdS5 → AdS4 × S1/Z2, the resulting 4-dimensional theory would as well contain

one vector and one antisymmetric tensor, since the bosonic group structure of the

N = 4 AdS5 automorphism group requires two of the five vectors, which reside in the

62

5-dimensional representation of USp(4) in a Minkowski background, to be replaced

by antisymmetric tensors [64]. The gauge group then becomes SU(2) ⊗U(1) and the

Z2 reflection would project out two of the vectors in SU(2) and one antisymmetric

tensor in U(1).

63

I believe that, just as the adherents of Herr Kant always accuse their

opponents of not understanding him, so there are many who believe

Herr Kant is right because they do understand him. His mode of ex-

position is novel and differs greatly from the usual one, and once we

have finally succeeded in understanding it there is a great temptation

to regard it as true, especially as he has so many zealous adherents; we

ought always to remember, however, that the fact that we understand

it is in fact no reason for regarding it as true. I believe that delight

at having understood a very abstract and obscure system leads most

people to believe in the truth of what it demonstrates.

Aphorism No. 77; Notebook J (translated by R. J. Hollingdale)

Georg Christoph Lichtenberg (1742 - 1799)

Chapter 5

Summary and Outlook

In the first part of this dissertation, low-energy effective field theories of partially

broken N = 2 supergravities with a minimal field content as dictated by the super-

Higgs effect were derived in Minkowski and AdS space. In both background geome-

tries, different theories were obtained by performing Poincare duality transformations

on vectors and scalars. The dualized theories possess different (dual) supersymmetry

algebras, which give rise to central scalar as well as vectorial extensions.

This must be contrasted to the situation in global N = 2 supersymmetry, where

the algebra only admits scalar central charges (in Minkowski space). Partially broken

64

global N = 2 supersymmetries were derived for two, one, and no central charge

[15, 16, 21]. But even the partial breaking case with two central charges, where the

Goldstino multiplet is a chiral multiplet, cannot be matched to the massless limit of

the unHiggsed theory with two scalars in Sec. 2.1 — there the Goldstino is a linear

combination of λ (the spinor from a real multiplet) and χ (the spinor from a chiral

multiplet) and not just χ. This discrepancy stems from the fact that the superHiggs

effect requires more degrees of freedom than just the Goldstone multiplet — the

whole massive gravitino multiplet is needed. The additional degrees of freedom alter

expressions only based on the Goldstone fields; e. g. the shift of the second stress-

energy tensor in Eq. (1.8), which is realized by the Goldstone multiplet for global

partially broken supersymmetry [15] is canceled by the contribution from the second

gravitino (Eq. (2.19)).

There is no literature yet on partially broken global supersymmetry in AdS space,

so the κ → 0 limit of the unHiggsed theory in Sec. 3 cannot be compared to other

results. However it is interesting to note that the OSP (2, 4) algebra does not have

a central charge, since the generator T ij does not commute with Qi (Eq. (3.1));

nevertheless the unHiggsed theory contains central extensions. Therefore the same

observation as in Minkowski space holds, namely that the algebra of the local theory

can be different form the one of the global theory.

Having obtained those theories with a minimal field content, a logical extension is

the coupling of matter to the minimal partial breaking sector. A more fundamental

problem is the question whether the minimal theories can be completed to all orders

in κ and v2. If there were a relation to non-linear realizations, one would expect

1/v2 terms in the Lagrangian and the supersymmetry variations. The possibility of

O(1 = v2/v2)-terms makes the Noether-method completion rather cumbersome. On

the other hand, the unHiggsed theory in a Minkowski background with two scalars

(Sec. 2.1) can be shown to be a consistent truncation of partially broken supergravities

derived in [23, 53, 24]. The latter contain two additional massless chiral multiplets,

which cannot be integrated out, whereas the supersymmetry breaking parameter only

occurs in powers of v2. Hence the question whether there exists a partially broken

supergravity theory to all orders with a minimal field content is still open.

65

The derivation of a partially broken supergravity with complete N = 2 multiplets

was a highly nontrivial task; it was not before 1986 [23] that an ad-hoc model had

been found. It took another ten years to elucidate the geometric framework that can

embrace partially broken supergravity [24]. However, there are other ways to derive

four-dimensional supergravity theories, namely by Scherk-Schwarz compactification

[48] (see Chapt. 4). Although this mechanism was known for a long time, it did not

seem powerful enough to yield partially broken N = 2 supergravity. With hindsight,

this failure was due to the restriction of the compactified space to n-tori, as opposed

to more general orbifolds. The orbifold compactification M5 → S1/Z2 ×M4 can arise

from string theories [65] and has been considered for theoretical and phenomenological

investigations [66, 67].

In Chapt. 4 I obtained a partially broken supergravity theory by Scherk-Schwarz

compactification on M5 → S1/Z2 × M4. The advantage of this derivation lies in its

simplicity: no knowledge of complicated matter couplings with exceptional geome-

tries is required; the pure N = 4 D = 5 supergravity suffices to incorporate partial

breaking. The additional space-like dimension is still required to be small, as in the

ordinary Kaluza-Klein approach to higher dimensions, because the particles in the

low-energy effective theory in four dimensions are the zero-modes of genuinely five-

dimensional fields. However, the stringent bounds on the compactification radius can

be relaxed by assuming that at least the standard model particles are constrained

to live on a four-dimensional submanifold1 (e. g. the fixed points of S1/Z2), with-

out the tower of massive Kaluza-Klein modes2 [68]. The trapping mechanism can

be provided by topological defects [67] or by interpreting the 3-brane as a Dirichlet-

brane (D-brane) on which open strings can end [69]. Advocated as a solution to the

hierarchy problem between the Planck scale and the weak scale, the brane scenario

with a large extra dimension actually shifts this hierarchy to a hierarchy between the

compactification radius and the weak scale.

An alternative approach was recently proposed by Randall and Sundrum [14, 70].

1Also referred to as a 3-brane for three spatial dimensions; here, is has a more general meaningthan in string theory.

2Since gravity is so intimately tied to space-time, it would be hard to conceive of gravity notbeing present in all the extra dimensions.

66

Here, the five-dimensional anti-de Sitter metric

ds2 = e−2krcφηmndxmdxn + r2

c (dφ)2 with 0 ≤ φ ≤ π (5.1)

does not factorize and the hierarchy between the Planck and the weak scale arises

from the exponential factor in the metric; a large radius rc is not required.3 The

boundaries of the fifth co-ordinate are the positions of two 3-branes. Randall and

Sundrum’s original construction was restricted to the four-dimensional graviton zero-

mode propagating in a Minkowski background. The absence of an effective four-

dimensional cosmological constant necessitated the introduction of a positive and a

negative cosmological constant on the 3-branes at r = 0 and r = πrc, respectively.

An urgent problem is the supersymmetrization of this scenario, which was recently

completed at the level of pure N = 1 supergravity by Bagger, Nemeschansky, and

myself [71]. Here, the starting point was pure five-dimensional N = 2 AdS supergrav-

ity [61] in the presence of two opposite-tension branes as in [14]. Upon Z2 projection,

the “zero-modes”4 which admit four-dimensional flat Killing spinors are identified as

a N = 1 supergravity multiplet and a chiral multiplet. The chiral multiplet contains

a scalar field T — the “radion” — which parametrizes the proper distance between

the two branes; its vacuum expectation value is given by rc. Truncation of the chi-

ral multiplet yields pure N = 1 supergravity. This can be regarded as a first step

towards a brane realization of partial supergravity breaking in the spirit of the very

first realization of partially broken global supersymmetry — a 3-brane propagating

in higher dimensions [20].

3In Ref. [70] however, it is demonstrated that the radius rc can be taken to infinity with impunity.4They are not zero-modes in the usual sense since they are constructed from a linear combination

of all Fourier modes [71].

67

Appendix

A Minimal superHiggs effect of partially broken

supersymmetry

In Sec. 2.1 the minimal superHiggs effect was referred to as a scenario where the

degrees of freedom of the lower N supergravity multiplet together with the appropriate

number of massive spin-3/2 multiplets add up to the degrees of freedom of the higher

N supergravity multiplet. This would be desirable in the sense that the partial

supersymmetry breaking were purely contained in the gravitational sector of the

higher N theory. The following discussion is restricted to four space-time dimensions.

To start with, one can observe that the higher-N must satisfy N ≥ 4, because any

massive gravitino multiplet contains a massive vector which contains a longitudinal

scalar. The lowest N for which the pure gravity multiplet has scalars is N = 4 (For

a classification of supersymmetry multiplets see Ref. [72] and references therein.).

On the other hand the highest N which has massive multiplets with highest spin 32

is N = 3. However, due to multiplet shortening in the presence of r central charges,

massive complex multiplets of (N − r) extended supersymmetry5 without central

charges can be present in an N -extended theory (with the restriction r ≤[N2

]). By

exhaustion of all possibilities, the only surviving partial breaking scenario is that of

N = 7/8 → N = 6 with one complex massive N = 3 multiplet with 3 central charges

5The real degrees of freedom of such multiplets are twice those of a massive (N − r) extendedmultiplet, since the central charge transforms under a CPT transformation and the Clifford vacuumupon which the creation and annihilation act must be doubled [73].

68

(upon CPT-completion of the states of the N = 7 supergravity multiplet, the N = 7

and N = 8 supergravities are identical).

s N = 8 sugra mult. N = 6 sugra mult. compl. mass. N = 3 mult.2 1 1

3/2 8 6 2 · 11 28 = 16 + 2 · 6

1/2 56 26 2 · (14+1)0 70 30 2 · (14+6)

Table A.1: Degree of freedom count of the minimal partial superHiggs effect.

The additional degrees of freedom in the column of the massive multiplet are

the longitudinal components of the gravitinos and vectors, respectively. This case

actually must be minimal in the sense defined above, because there are no other

matter multiplets in N = 6 supergravity (excluding massive spin-2 multiplets and

massless gravitino multiplets - it is expected that their Noether coupling leads to a

higher N > 6).

69

B Massive N = 1 spin-3/2 multiplet with two an-

tisymmetric tensors

The massive N = 1 spin-3/2 multiplet with two antisymmetric tensors is obtained

by another massive duality transformation (C.2) (Am → Amn). The Lagrangian is

L = εpqrsψpσq∂rψs − iζ σm∂mζ +1

2vAmvAm

−1

4m2AmnAmn

+1

2mζζ + h.c.

−mψmσmnψn + h.c.

where Amn = Amn + iBmn and vAm = 12εmnrs∂

nArs. The global supersymmetry trans-

formations are

δηAmn = − 4i√3ησmnζ +

2√3m

∂[m(ζ σn]η)

+2ησ[mψn] − 2i

m∂[m(ψn]η)

δηζ =im√

3σrsηArs − i√

3vAmσmη

δηψm =1

3m∂m(2ivAn σ

nη + imσrsηArs) − 2

3(vAm + σmnv

An)η

+m

3(Amnσ

nη + iεmnrsAnrσsη)

This Lagrangian can be unHiggsed following the method described in Sec. 2.1.1.2:

e−1L =

− 1

2κ2R + εpqrsψpiσqDrψ

is − iχσmDmχ − iλσmDmλ

+1

2vmvm − 1

4FA

mnFAmn

−(1√2mψ2

mσmλ + miψ2mσmχ

+√

2miλχ +1

2mχχ + mψ2

mσmnψ2n + h.c. + first order Noether-coupling )

with

FAmn = ∂[mAn] − mAmn .

70

The supersymmetry transformations become

δηeam = iκηiσaψmi + iκηiσ

aψim

δηψ1m =

2

κDmη1

δηAm = −√

2η1σmλ + 2εijψimηj

δηAmn = −4η1σmnχ

+2iψ2[nσm]η

1 − 2iη2σ[mψ1n]

δηλ = − i√2

FAmnσ

mnη1 − i√

2v2η2

δηχ = ˆvmσmη1 + 2v2η2

δηψ2m =

2

κDmη2 + iv2σmη2

− i

2ˆFA−mnσ

nη1 + ivmη1 . (B.1)

The N = 1 multiplet structure can be obtained by restricting the transformations to

N = 1 supersymmetry and taking the massless limit κ → 0. A massless gravitino

multiplet, a massless vector multiplet and a dualized linear multiplet where the real

scalar has been dualized to another antisymmetric tensor emerges. This multiplet is

only known on-shell. Its Lagrangian reads

L = −iχσm∂mχ +1

2vmvm , (B.2)

and the supersymmetry transformations are

δηAmn = −4ησmnχ

δηχ = vmσmη .

This Lagrangian has an additional invariance: δAmn = iλ[mvn] with λm = const ∈ RI 4.

It is only present for complex vm. Hence, the algebra on Amn is expected to close

into this symmetry, too:

[δξ, δη] Amn = −2i(ξσdη − ησdξ)(∂dAmn + ∂[mAn]d) + 2(ξσ[mη − ησ[mξ)vn] .

71

In local supersymmetry, this additional invariance is lost [36]. Hence the additional

term in the algebra must either be canceled by another field (in Eq. (B.1) it is

canceled by the variation of the second gravitino) or the complex two-form must be

coupled in a gauge invariant way to a three-form.

72

C Poincare dualities

In field theory, bosons of the same spin (helicity) can sometimes be described by

fields with a different Lorentz-index structure. This is motivated by the fact that

the degrees of freedom of massive and massless antisymmetric tensor fields can be

expressed as a binomial coefficient [60]:

- degrees of freedom of a massive rank-p antisymmetric tensor field in D space-time

dimensions:(

D − 1

p

)=

(D − 1

D − 1 − p

);

- degrees of freedom of a massless rank-p antisymmetric tensor field in D space-time

dimensions (gauge-fixed):(

D − 2

p

)=

(D − 2

D − 2 − p

).

The Stuckelberg redefinition of a massive antisymmetric tensor field in terms of in-

teracting massless antisymmetric tensor fields

T[m1...mp] → T[m1...mp] − 1

m∂[m1Tm2...mp] (C.1)

is of course consistent with that degree of freedom count:(

D − 1

p

)=

(D − 2

p

)+

(D − 2

p − 1

).

For example in D = 4, massive p = 1 and p = 2 antisymmetric tensors with( 3

1

)= 3 degrees of freedom, and massless p = 0 and p = 2 antisymmetric tensors with( 2

0

)= 1 degrees of freedom, and massless p = 1 antisymmetric tensors with

( 2

1

)= 2

degrees of freedom are dual. In D = 5, massless p = 1 and p = 2 antisymmetric

tensors with( 3

1

)= 3 degrees of freedom are dual. It can be shown that the little

group of these particles agree, i. e. they describe particles of the same spin (helicity).

In the case of noninteracting fields, this duality can be made explicit by writing

down a Lagrangian containing one field and one Lagrange multiplier. For a massive

vector Am/antisymmetric tensor Bmn the Lagrangian reads

L = −1

4m2BmnBmn − 1

2m2AmAm +

1

2mεmnrsBmn∂rAs . (C.2)

The equations of motions can be used to express one field in terms of the other:

Bmn = 1mεmnrs∂

rAs and Am = 1mvm (vm = 1

2εmnrs∂

nBrs) [32]. This procedure can be

readily extended to interacting fields.

73

For massless vectors Am/Bm with field strengths Fmn/Gmn the Lagrangian is

L = −1

4FmnFmn +

1

4FmnGmn (C.3)

with the equations of motion Fmn = 12Gmn or Fmn = 0, whereas for a massless

antisymmetric tensor Bmn/scalar φ the Lagrangian is

L =1

2vmvm + vm∂mφ (C.4)

with the equations of motion vm = −∂mφ or ∂mvm = 0. Here, the equations of

motion only contain the field strengths; therefore one might think that one can-

not unambiguously extract (on-shell) transformation laws of the dual fields from the

transformations of the dualized fields. A trick how to solve this problem is described

in Ref. [33].

The above discussion of Poincare dualities also holds for antisymmetric tensors in

a four-dimensional AdS background.6 In particular Eqs. (C.2,C.3,C.4) are the same

in AdS space upon substitution of the AdS covariant derivative ∂m → ∇m (see (G.2)).

6It is not true in odd-dimensional AdS spaces, where antisymmetric tensors obey additionalself-duality conditions [59].

74

D Superfield projectors for the spinor superfield

The N = 1 supersymmetry algebra

Qα, Qβ = 2σaαβPa

Qα, Qβ = 0

[Mab, Q] = −iσabQ (D.1)

is realized on a general superfield Φi(x, θ, θ). Here i is an external Lorentz index. The

superfield Φi has external spin j if it obeys the irreducibility condition for Poincare

spin j with respect to the index i (e. g. ∂aΦa = 0 for external spin 1).

Superfields are in general reducible (except chiral superfields). The irreducible

massive representations of the algebra (D.1) are labeled by the eigenvalues of the

Casimir operator C2 = −2m4Y (Y + 1), where C2 is a generalization of the square of

the Pauli-Lubanski vector [74] and Y is an integer or half-integer called superspin. A

representation with superspin Y contains four Poincare spins J :

J ∈ Y − 1

2, Y, Y, Y +

1

2 .

The most common irreducible superfield representations of N = 1 supersymmetry

are the chiral superfield with superspin 0 and the real superfield with superspin 12.

Since the gravitino has Poincare spin 32, the corresponding superfield must have

superspin 1. For the construction of the Lagrangian in Sec. 2.2.1, the square root

of the localized superspin-1 projector Π1 must be taken. The lower superspins corre-

spond to auxiliary multiplets or gauge invariances. The superspin decomposition of

Ψα reads [75]:

Ψα = 0 ⊕ 1

2⊕ 1

= 0 ⊕ 1

2

−⊕ 1

2

+

⊕ 1

= − 1

16(−DαD

2DβΨβ − D2D2Ψα − D2D2Ψα + DβD2D(αΨβ)) .

Following [76], the two odd superspins 12

−and 1

2

+can be further decomposed into

their “real” and “imaginary” parts 12

±rand 1

2

±i:

Ψα = 0 ⊕ 1

2

−r

⊕ 1

2

−i

⊕ 1

2

+r

⊕ 1

2

+i

⊕ 1

75

= − 1

16(−DαD

2DβΨβ

+D2Dα(DβΨβ + DβΨβ) + D2Dα(DβΨβ − DβΨβ)

−1

2D2Dβ(DβΨα − 2DαΨβ) − 1

2D2Dβ(DβΨα + 2DαΨβ)

+DβD2D(αΨβ))

= (Π0 + Π 12

−r + Π 12

−i + Π 12

+r + Π 12

+i + Π1)Ψα .

76

E Ogievetsky-Sokatchev multiplet

The massless Lagrangian of the Ogievetsky-Sokatchev multiplet from Sec. 2.2.1

reads [35]

L⊥ = −1

2

(DβΨαDαΨβ +

1

4DβΨαDβΨα +

1

4DαΨβDαΨβ − 1

4(DαΨα + DβΨβ)2

)|θ2θ2

= −1

2(ΨΨ)π⊥

(Ψ

Ψ

)|θ2θ2

where π⊥ =√

Π1 and

Ψ = ψ +√

2((U1 + iU2)θ − iσmθ(Um3 + iUm

4 ) − 2σmnθUmn5 )

+1

2θ2ψ3 +

1

2θ2ψ4 + θσmθψm − i√

2σmθθ2(um

3 + ium4 )

+1√2θ2(θ(u1 + iu2) − 2σmnθu

mn5 ) +

1

4θ2θ2ψ7 . (E.1)

The following redefinitions are necessary to diagonalize the kinetic terms:

u1 → u1 + arbitrary

u2 → u2 − ∂mUm4

um3 → um

3 − ∂mU1 − 2∂nUnm5

um4 → um

4

umn5 → umn

5 +1

2εmn

rs∂rU s

4 +3

2(∂mUn

3 − ∂nUm3 )

ψ → ψ

ψ3 → ψ3 − σmψm + iσm∂mψ

ψ4 → ψ4 − σmψm − iσm∂mψ − 1

2ψ3

ψm → ψm − 1

2σmψ4 − 1

4σmψ3

ψ7 → ψ7 − iσm∂mψ4 + ψ +i

2σm∂mψ3 − 2

3i(∂m + σnσm∂n)ψm .

These expressions differ from those in the original paper [35]. Then the Lagrangian

becomes

L =1

2εmnrsψmσn∂rψs − 3

16(ψ7ψ3 + h.c.)

77

−2U3mnUmn3 − 2∂mu4nU

mn5 − u2

2 +1

2um

3 u3m + umn5 u5mn .

So ψm and U3m are physical fields, the rest are auxiliary fields. The auxiliary fields

u4n and Umn5 possess gauge invariances. This multiplet has 20 + 20 off-shell degrees

of freedom, which reduce to 2 + 2 on-shell degrees of freedom.

In superfield language, the Lagrangian has the invariance

δΨα = DαV + iWα

= Dα(φ + φ) + DαVWZ + iWα

= 0 ⊕ 1

2

+r

⊕ 1

2

−i

,

where V is a real superfield and Wα is the field strength of another real superfield V .

The superfield V can be further decomposed into a chiral and anti-chiral superfield

and a real superfield in Wess-Zumino gauge VWZ in order to exhibit the superspin

content. Alternatively, the invariance can be expressed as

δΨ =1

mσmααD

α∂mV +1

mDαL

= (0 ⊕ 1

2

−i

) ⊕ 1

2

+r

,

where L = DαLα + DαLα is the field strength of a chiral spinor superfield Lα.

78

F De Wit-van Holten multiplet

The Lagrangian of the de Wit-van Holten multiplet reads [39]

L‖ = −1

2(DβΨαDαΨβ +

1

4DβΨαDβΨα +

1

4DαΨβDαΨβ)|θ2θ2

= −1

2(ΨΨ)π‖

(Ψ

Ψ

)|θ2θ2

where π‖ =√

(Π0 + Π1).

The following redefinitions are necessary to diagonalize the kinetic terms:

u1 → u1 + ∂mUm3

u2 → u2 − 1

3∂mUm

4

um3 → um

3 + ∂mU1 − 2∂nUnm5

um4 → um

4 + ∂mU2 − ∂nUnm5

umn5 → umn

5 +1

2εmn

rs∂rU s

4 +3

2(∂mUn

3 − ∂nUm3 )

ψ → ψ

ψ3 → ψ3 + arbitrary

ψ4 → ψ4 − σmψm − iσm∂mψ

ψm → ψm

ψ7 → ψ7 − iσm∂mψ4 + ψ − 2i(∂m + σnσm∂n)ψm .

Then the Lagrangian becomes

L =1

2εmnrsψmσn∂rψs +

1

8(ψ7ψ4 + h.c.)

−2U3mnUmn3 − 4

3∂mUm

4 ∂nUn4 − u2

1 − 3

2u2

2 +1

2um

3 u3m +1

2um

4 u4m + umn5 u5mn .

Here, ψm and U3m are physical fields, the rest are auxiliary fields. The combination

∂mUm4 is to be interpreted as an auxiliary scalar. This multiplet has 20 + 20 off-shell

degrees of freedom, which reduce to 2 + 2 on-shell degrees of freedom.

79

In superfield language, the Lagrangian has the invariance

δΨα = DβDβΛα + iσm

αα∂mW α

=1

2

−⊕ 1

2

+r

,

where Λα is an unconstrained spinor superfield and W α is the anti-chiral field strength

of a real superfield.

80

G Geometry of AdS space

Anti-de Sitter space is the space of constant curvature R < 0 [77]. It is the unique

maximally symmetric curved space-time that admits supersymmetry.7 It has the

topology S1 × R3 and can be represented as the hyperboloid

xAxBηAB = − 1

Λ2with ηAB = diag(−1, 1, 1, 1,−1)

in flat five-dimensional space. It contains closed time-like curves; therefore its uni-

versal covering space is taken as the physical space.

In the spirit of nonlinear realizations, the coset spaces of anti-de Sitter space and

OSp(1, 4) are parametrized by the coset elements [78]

g(z) = O(3, 2)/O(3, 1) = e−izmRm

G(z, θ, θ) = O(3, 2)/O(3, 1) · OSp(1, 4)/O(3, 2) = g(z)ei(1− 13Λ(θθ+θθ))(θQO(3,2)+θQO(3,2))

with m ∈ 0, ..., 3. Subsequent expressions are facilitated by a transformation to

new bosonic co-ordinates xm

xm = zmtanh(1

2Λ

√|zmzm|)

12Λ

√|zmzm|

.

In these co-ordinates, the vierbein takes the form

eam = a(x)δam with a(x) =1

1 − Λ2

4x2

,

and the spin connection is

ωmab = a(x)Λ2xneanebm . (G.1)

The AdS covariant derivative ∇m acting on a field φ with Lorentz index b is therefore

∇aφb = eam(∂mφb − ia(x)

Λ2

2xn(Jnm)b

cφc) , (G.2)

where (Jmn)bc is the matrix part of the Lorentz generator (Mmn)b

c = −ix[m∂n]δbc +

(Jmn)bc. Explicitly the expressions for (Jmn)b

c are Jmn = 0 acting on a scalar φ,

7Like Minkowski space, (Anti-) de Sitter space possesses ten Killing vectors. In de Sitter space(R > 0), Majorana spinors that are necessary for supersymmetry representations cannot be defined.

81

(Jmn)αβ = i(σmn)α

β acting on a spinor φα, and (Jmn)bc = iηb[nδ

cm] acting on a vector

φb.

With the above choice of the fermionic co-ordinate system, the co-ordinates θ

transform like a Lorentz-spinor and not like an O(3, 2) spinor. Therefore, the com-

ponents of OSp(1, 4) superfields do not transform like O(3, 2) fields.

The generators QO(3,2), however, are O(3, 2) spinors. They can be transformed

into O(3, 1) spinors Q by shifting the factor8 Λ(x) to the transformation parameter

ε:

eiεQO(3,2)

= eiηQ .

Hence the supersymmetry transformation parameter η becomes x-dependent [78, 79]:

∇mη(x) = −iΛ

2σmη(x) . (G.3)

The supersymmetry transformations of the gravitinos [see Eqs. (3.13), (3.17)] are

also modified by the transition from O(3, 2) to O(3, 1) spinors:

δψO(3,2)m = ∂mεs(x)

δ(Λ(x)ψm) = ∂m(Λ(x)η(x))

δψm = ∇mη(x) + Λ−1(x)(∇mΛ(x))η(x)

= ∇mη(x) + iΛ

2σmη(x) . (G.4)

The algebra of OSp(2, 4) (i, j ∈ 1, 2) reads

[Mab,Mbc] = −i(ηbcMad + ηadMbc − ηacMbd − ηbdMac)

[Mab, Rc] = −i(ηbcRa − ηacRb)

[Ra, Rb] = −iΛ2Mab[T ij, Qk

]= iΛ(δjkQi − δikQj)

Qiα, Qjβ = 2σa

αβRaδ

ij

Qiα, Q

βj = 2iΛσabαβMabδ

ij + 2iδαβT ij (G.5)[

Mab, Qi]

= −iσabQi

[Ra, Q

i]

=1

2ΛσaQi .

8Here, Λ(x) is the group-theoretical factor that maps O(3,1)-spinors to O(3,2)-spinors [78]; it isnot the cosmological constant.

82

Here T ij ∼ σ2 is the hermitian generator of SO(2). The N = 2 Minkowski-algebra

is recovered in the limit Λ → 0 (Inonu-Wigner contraction). The N = 2 Poincare

algebra with one central charge is recovered in the limit ΛT ij = X ij, with Λ → 0.

The term proportional to Λ in (G.5) is also reflected in the representation of the

OSP(2,4) algebra on a field φ with Lorentz index b (ξ and η parametrize the first or

the second supersymmetry):

[δξ, δη]φb = −2i(ξσaη − ησaξ)∇aφb + 2iΛ(ξσcdη − ξσcdη)(Jcd)baφa (G.6)

In particular the closure of the algebra on scalars, spinors, vectors, and gravitinos

reads

[δξ, δη]φ = −2i(ξσaη − ησaξ)∂aφ

[δξ, δη]λ = −2i(ξσaη − ησaξ)∇aλ + 2Λξ(ηλ) − 2Λη(ξλ)

[δξ, δη]Ab = −2i(ξσaη − ησaξ)∇aAb + 4ΛξσbaηAa + 4ΛξσbaηA

a

[δξ, δη]ψb = −2i(ξσaη − ησaξ)∇aψb

+4Λξσbaηψa + 4Λξσbaηψ

a + 2Λξ(ηψb) − 2Λη(ξψb) .

If the algebra is evaluated on a field in a Wess-Zumino-type gauge, the algebra con-

tains an additional gauge transformation, which combines together with the contri-

butions from the vector part of Jab and the Ra part of the algebra to a field strength.

83

H Massive AdS spin-1 multiplet

In Sec. 3.2.2, the unHiggsing of the massive spin-32

multiplet led to the appearance

of a massive spin-1 multiplet with E = 5/2. Here, the Lagrangian and transformations

for general E will be presented.

The massive spin-1 multiplet contains the following AdS representations (see [43]):

D(E,

1

2

)⊕ D

(E +

1

2, 1

)⊕ D

(E +

1

2, 0

)⊕ D

(E + 1,

1

2

)with E ≥ 3

2.

The corresponding Lagrangian is

L = −1

4vmnv

mn − 1

2∂mC∂mC

−iλσm∇mλ − iχσm∇mχ

−1

2DmφDmφ − 1

2m2(1 − ε)(1 + 2ε)C2

−(

1

2mλλ +

1

2m(1 + ε)χχ + h.c.

)

where m = (E − 32)Λ ≥ 0 and ε = Λ/m. The Stuckelberg redefinition Dmφ =

∂mφ − m√

1 + εvm has already been performed.

This Lagrangian is invariant under the supersymmetry transformations:

δηvm =1√

1 + ε2

1√2

(ησmχ + χσmη)

+

√√√√ 1 + ε

1 + ε2

i√2

(ησmλ − λσmη)

δηλ =

√√√√ 1 + ε

1 + ε2

1√2

(vmnσ

mnη − i1√

1 + εDmφσmη

)

+1√

1 + ε2

1√2

(∂mCσmη − im(1 − ε)ηC)

δηχ =1√

1 + ε2

i√2

(vmnσmnη + i

√1 + εDmφσmη)

−√√√√ 1 + ε

1 + ε2

i√2

(∂mCσmη + im(1 + 2ε)ηC)

δηC = −√√√√ 1 + ε

1 + ε2

1√2

(ηχ + ηχ) +1√

1 + ε2

i√2

(ηλ − ηλ)

84

δηφ = −√√√√ 1 + ε

1 + ε2

i√2

(ηχ − ηχ) − 1√1 + ε

2

1√2

(ηλ + ηλ) .

In the limit E → 32

(m → 0) this Lagrangian reduces to that of a massless spin-1

multiplet and a chiral multiplet [78]:

massive spin1 multipletz |

D(E;1

2)D(E +

1

2; 1)D(E +

1

2; 0)D(E + 1;

1

2)

E ! 3

2

PPPPPPPPPPPPPPPPq

D(3

2;1

2)D(2; 1)

| z

massless spin1 mult:

and D(2; 0)D(5

2;1

2)D(3; 0)

| z

chiral multiplet (E=2)

It is only this chiral multiplet with E = 2 that can be dualized to a linear multiplet

in AdS; other chiral multiplets with E = 2 have bare φ-terms (without derivatives)

in its transformations and cannot be dualized. For completeness the Lagrangian and

the transformations of the chiral multiplet containing the representations

D(E, 0) ⊕ D(E +1

2,1

2) ⊕ D(E + 1, 0) with E >

1

2

are listed [78]. The Lagrangian is

L = −1

2∂mC∂mC − 1

2∂mφ∂mφ − iχσm∇mχ

−1

2Λ2(E + 1)(E − 2)φ2 − 1

2Λ2E(E − 3)C2

−1

2Λ(E − 1)(χχ + h.c.) , (H.1)

and the supersymmetry transformations are:

δηφ = −iχη + iχη

δηC = −χη − χη

δηχ = −σmη∂mφ − iσmη∂mC + ΛECη + iΛ(E − 2)φη .

85

The Lagrangian of the dual linear multiplet with E = 2 is (∂mφ = −vm, see Appendix

C)

L = −1

2∂mC∂mC +

1

2vmvm − iχσm∇mχ

−1

2Λχχ − 1

2Λχχ + Λ2C2 (H.2)

and its transformations are given by

δηBmn = −2ησmnχ − 2ησmnχ

δηC = −χη − χη

δηψ = −iσmη∂mC + 2ΛCη + σmηvm

where vm = 12εmnrs∂

nBrs.

86

I Conventions of five-dimensional supersymmetry

The five-dimensional Dirac algebra with the space-time metric ηAB = diag(−1, 1, 1, 1, 1)

reads

ΓA,ΓB = −2ηAB ,

where ΓA ∈ γ0, γ1, γ2, γ3, γ5. The matrices γa and γ5 are defined as in Ref. [63] and

the antisymmetric combinations ΓA1...An = 1n!

Γ[A1 · · · ΓAn] are defined with strength

one. The 5-dimensional epsilon-tensor is defined by ε01234 = 1.

The real symplectic metric to lower USp(2N)-indices is chosen to be Ωij = 1N ⊗iσ2. It is used to raise and lower symplectic indices of the vectors V i and Vi according

to V i = ΩijVj and Vi = ΩijVj.

A symplectic Majorana spinor in five dimensions (denoted by upper case Greek

letters) is written in terms of Weyl spinors in four dimensions (denoted by lower case

Greek letters) as

Ψi =

(ψi

α

−Ωijψαj

), Ψi = (Ωijψ

jα ψiα) ,

where ψαi = εαβ(ψi

β)∗. “Symplectic” indices on Weyl spinors in four dimensions are

merely labels — they are not covariant indices.

The massless superalgebra in five dimensions has a USp(N) automorphism group.

USp(N) with N even is the compact Lie group defined by the set of complex N ×N

matrices that are both unitary and symplectic. The generators of the corresponding

Lie algebra usp(N) form a set of N(N + 1)/2 hermitian matrices Tr that satisfy the

symplectic condition TrΩ..+Ω..Tr = 0 ∀r ∈ 1, ..., N(N+1)/2. The representation

of the basis elements of usp(4) used in this dissertation is

Tr ∈(

σj 0

0 0

),

(0 0

0 σj

),

(0 σ1

σ1 0

),

(0 −σ2

−σ2 0

),

(0 σ3

σ3 0

),

(0 i

−i 0

)

with r ∈ 1, ..., 10.

87

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93

Vita

I, Richard Eugen Altendorfer, was born on the 16th of March 1969 in Rosenheim

(Germany). From 1989 until 1992 I was enrolled as a student of physics at the Ludwig-

Maximilians-Universitat Munchen. In 1992 I took a leave of absence and joined the

Centre for Particle Theory of the University of Durham in England, from where I

graduated with a Master of Science degree in 1993. Then I returned to the Ludwig-

Maximilians-Universitat Munchen and finished my studies under the supervision of

Professor Jan Louis and Professor Julius Wess in 1995 with a Physik-Diplom. Since

1995 I have been a PhD student in the Department of Physics and Astronomy of the

Johns Hopkins University in Baltimore under the supervision of Professor Jonathan

Bagger. In 1997 I was awarded a Master of Arts degree.