Conformal Killing Spinors in Supergravity and Related Aspects of Spin Geometry

Conformal Killing spinors in supergravity and related aspects of spin geometry

Hannu Rajaniemi

Doctor of Philosophy University of Edinburgh 2006 Äidille ja isälle Abstract

The aims of this thesis are two-fold: ﬁnding a geometric realisation for Nahm’s conformal superalgebras and generalising the concept of a conformal Killing spinor to supergravity, in particular M-theory.

We introduce the necessary tools of conformal geometry and construct a conformal

Killing superalgebra (that turns out not to be a Lie superalgebra in general) out of the conformal Killing vectors and the conformal Killing spinors of a semi-Riemannian spin manifold and investigate a natural deﬁnition of the spinorial Lie derivative that differs from the more commonly used Kosmann-Schwarzbach Lie derivative. We then attempt to generalise the deﬁnition of conformal Killing spinors to M-theory and characterise M-theory backgrounds admitting such spinors. We also construct a M-theory analogue of the conformal Killing superalgebra. We show that further examples can be constructed in type IIA and in the massive IIA theory of Howe,

Lambert and West via Kaluza-Klein reduction. We also comment on a curious identity involving the Penrose operator in type IIB supergravity.

Finally — building on known results about the relationship between the dimension of the space of conformal Killing spinors on a non-simply connected manifold and the choice of spin structure — we explore the importance of the choice of spin structure in determining the amount of supersymmetry preserved by a symmetric M- theory background constructed by quotienting a supersymmetric Hpp-wave with a discrete subgroup in the centraliser of its isometry group. Acknowledgements

A lion’s share of thanks for this thesis must go to José Figueroa-O’Farrill, who has been the very model of a modern supervisor, offering ideas, advice, patience, guid- ance and encouragement at every turn. I also owe much to discussions I have had with numerous other people, including Helga Baum, David Calderbank, Sunil Gad- hia, Emily Hackett-Jones, Felipe Leitner, George Moutsopoulos and in particular Si- mon Philip — thank you! Any misunderstandings and mistakes here are, of course, my own.

I wish to thank the Erwin Schrödinger Institute for generous hospitality during the

ﬁnal stages of this work, and gratefully acknowledge ﬁnancial support from the Academy of Finland and the Vaisala Fund, without which this work would have not been possible.

This thesis is built on the support of all those who have shared my life in this city for the past four years. You know who you are, but just to name a few — Darren

Brierton, Chrissy de Chaves, Felicity King-Evans, Nana Lehtinen, Jelena Meznaric-

Broadbent, Rebecca Smith and the whole East Coast Writers’ Group: Jack Deighton,

Stephen Christian, Andrew Ferguson, Gavin Inglis, Jane McKie, Martin Page, Stefan

Pearson, Charlie Stross and Big Andrew (and Lorna and James) Wilson — see you around.

I also want to thank to my colleagues and friends at ThinkTank Mathematics — Sam

Halliday, Angela Mathis and Dan Winterstein — for embarking upon a strange jour- ney that has just gotten started.

There are a lot of important people in Finland and elsewhere I don’t see nearly often enough: the Helminen family (Antti, Katri, Jaakko, Olli and Lauri), Esa Hilli, Anni

Siitonen and Antti Mäki, Panu Pasanen and Ismo Puustinen — thank you and see you soon. I want to thank all my fellow students at the University of Oulu in the good old days, for their friendship and all they taught me — especially Antti-Jussi Mattila, Petri

“Ketku” Mutka, Ossi Raivio and Janne Riihijärvi. Special thanks go to Professor Petri

Mähönen for support, advice and encouragement, to Alli Huovinen for showing me how to study mathematics, and ﬁnally to St. Jude, the patron saint of lost causes and desperate situations.

No words can thank Isabelle and the wild animals for all their love and patience.

Isälle ja äidille: kiitos, että toivoitte, luotitte ja uskoitte.

2 Declaration

I declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text. Much of the material contained in Chapters 2 – 7 has been developed in collaboration with José Figueroa-

O’Farrill and will be reported in a forthcoming publication.

Hannu Rajaniemi Table of Contents

Chapter 1 Introduction 4

Chapter 2 Preliminaries 9

2.1 Algebraic preliminaries ...... 9

2.1.1 Natural properties of vector spaces with an inner product . . . . 9

2.1.2 The Möbius Lie algebra of V ...... 12

2.1.3 The Clifford algebras of V and Vˆ ...... 13

2.1.4 Spinor representations of the Möbius algebra ...... 14

2.2 Geometric preliminaries ...... 15

2.2.1 The Riemann curvature tensor and its relatives ...... 16

Chapter 3 The Lie algebra of conformal Killing vectors 19

3.1 Conformal Killing transport ...... 19

3.2 The conformal Lie algebra ...... 22

3.3 Conformal changes of the metric ...... 24

3.4 Weyl connections ...... 25

3.5 Weyl-invariant conformal Killing transport ...... 27

3.6 The conformal Lie algebra of a Weyl structure ...... 29

3.7 Normal conformal Killing vectors ...... 29

Chapter 4 Conformal Killing spinors 31

4.1 Spinorial conformal Killing transport ...... 33

4.2 Conformal covariance of the Penrose operator ...... 35

4.3 Weyl-invariant spinorial conformal Killing transport ...... 37

1 Chapter 5 Conformal Killing superalgebras 39

5.1 From conformal Killing spinors to conformal Killing vectors ...... 39

5.2 Spinorial Lie derivatives ...... 40

5.3 The Kosmann-Schwarzbach Lie derivative ...... 45

5.4 The Jacobi identities ...... 49

5.5 The Minkowski conformal Killing superalgebra ...... 50

Chapter 6 Conformal Killing spinors in M-theory 53

6.1 M-theory backgrounds ...... 53

6.2 The M-theory Penrose operator ...... 55

6.3 Supercovariant conformal Killing transport ...... 56

6.4 M-theory backgrounds admitting conformal Killing spinors ...... 60

6.5 Conformal Killing spinors of Hpp-waves ...... 63

Chapter 7 Conformal Killing superalgebras in M-theory 66

7.1 M-theory conformal Killing spinors ...... 66

7.2 Supernormal conformal Killing vectors ...... 67

7.3 Jacobi identities in the M-theory conformal Killing superalgebra . . . . 69

Chapter 8 Conformal Killing spinors in type IIA and IIB supergravities 71

8.1 The Kaluza-Klein ansatz ...... 72

8.2 Kaluza-Klein reduction of the conformal Killing spinor equation . . . . 74

8.3 Kaluza-Klein reduction and conformal Killing superalgebras ...... 76

8.4 Conformal Killing spinors and Kaluza-Klein reductions of ﬂat space . . 81

8.5 Conformal superalgebras of nullbranes ...... 83

8.5.1 The nullbrane ...... 84

8.5.2 Interpolating solutions ...... 86

8.6 Conformal Killing spinors in HLW massive IIA supergravity ...... 87

8.7 A HLW massive IIA conformal Killing superalgebra ...... 89

8.8 Conformal Killing spinors in type IIB supergravity ...... 92

2 Chapter 9 Killing spinors, discrete quotients and spin structures 94

9.1 Discrete quotients and spin structures ...... 95

9.2 Hpp-waves in M-theory ...... 95

9.3 Symmetric discrete quotients of Hpp-waves ...... 103

9.3.1 The maximally supersymmetric case ...... 105

9.3.2 The four-parameter case ...... 106

9.3.3 The seven-parameter case ...... 109

9.4 A conjecture ...... 110

Chapter 10 Conclusions 112

Bibliography 116

Appendix A Metrics of pp-waves with supernumerary supersymmetries 122

3 Chapter 1

Introduction

Pitkän harkinnan jälkeen tultiin oikeaan tulokseen Neliulotteista ei voi ihmisaivoin visualisoida Riemannin metrinen tensori on Meidän Braillemme todellisuudelle

It took a lot of thought to conclude The human brain is blind to the the four-dimensional The Riemann metric tensor is Our Braille for reality

— A. W. Yrjänä, Tesserakti

It is widely acknowledged that one of the key components in any theory that at-

tempts to formulate physics beyond the Standard Model must be supersymmetry.

During the last three decades the study of supersymmetric ﬁeld theories and superstring theories has grown into a vast subject. Supersymmetry has appeared in many guises both in pure mathematics (inspiring intense study of Calabi-Yau manifolds and manifolds of exceptional holonomy) and theoretical physics, and the hope of

ﬁnding supersymmetric partners of known particles continues to drive experimen- tal particle physics as well.

In 1975, Haag, Lopuszan´ski, and Sohnius [1] showed that under relatively weak assumptions, the only possible symmetries of the S-matrix of a quantum ﬁeld theory in addition to the standard Poincaré symmetries and “internal” symmetries related to conserved quantum numbers are those which mix bosonic symmetries with 4 fermionic ones. Soon afterwards, Nahm [2] gave a classiﬁcation of possible super-

symmetry algebras, based on Kac’s classiﬁcation of simple Lie superalgebras [3].

In the mid-70s, study of supersymmetric ﬁeld theories led to the realisation that

in addition to global supersymmetry invariance, it is possible to construct theo-

ries with local supersymmetries, that is, theories with supersymmetry invariance

where the fermionic generators are allowed to depend on spacetime coordinates.

This led to the discovery of the supergravity zoo — a bewildering number of theo-

ries in spacetime dimensions ranging up to eleven, all incorporating Einstein’s grav-

ity and a variety of other bosonic and fermionic ﬁelds. Many of these theories were

constructed by “gauging” [4] one of Nahm’s supersymmetry algebras — basically by

requiring that the ground state of the theory should naturally admit a given symme-

try algebra.

A particularly interesting discovery was the eleven-dimensional supergravity in 1976

[5], which was found to be essentially unique: for a short while in the early 80s (to-

gether with its Kaluza-Klein compactiﬁcations) it was even a contender for the cov-

eted title of Theory of Everything [6]. However, it was soon overtaken by its younger

and hungrier siblings, so-called superstring theories [7, 8]. Superstring theories ap-

proach the problem of quantum gravity by quantising a one-dimensional extended

object — the string — instead of a pointlike particle. They have attracted an enor-

mous amount of attention from the mid-80s onwards, experiencing an explosive

renaissance during the past decade or so.

It was during this recent burst of activity that eleven-dimensional supergravity again

rose into prominence. The ﬁve consistent superstring theories — Type I, Type IIA,

Type IIB, Heterotic E E and Heterotic Spin/Z — all feature spacetime local su- 8 × 8 2 persymmetry in ten dimensions. In fact, their low energy limits correspond to known

ten-dimensional supergravity theories. It is a long-standing conjecture that there is

an eleven-dimensional quantum theory, tentatively called M-theory that underlies

all the known string theories and relates them to each other via a complex web of

dualities. The low-energy limit of M-theory is believed to be no other than eleven-

5 dimensional supergravity, and these days the terms are often used interchangeably in the literature.

Since the study of non-perturbative sectors of these theories is at present extremely difﬁcult, much of recent research has focused on studying supersymmetric solutions of eleven-dimensional supergravity and lower-dimensional supergravities. Since the bosonic sectors of supergravity theories resemble Einstein’s gravity coupled to

Maxwell-like p-form ﬁelds, they can be studied using classical tools of differential

geometry and spin geometry. In particular, it turns out that supersymmetry of a

supergravity background can be characterised geometrically: it corresponds to the

existence of so-called supergravity Killing spinors, spinors which are parallel with

respect to a special connection induced from the supersymmetry variation of the

gravitino. In this sense, supergravity Killing spinors are a supergravity generalisation

of geometric parallel spinors1. Parallel spinors have, of course, also played a crucial

role in string theory in the context of realistic compactiﬁcations of string theories

and M-theory to four dimensions due to their relationship with manifolds of special

holonomy.

There are many parallels between the study of supergravity Killing spinors and the

study of special spinors (that is, spinors annihilated by some natural differential op-

erator) in spin geometry. Parallel spinor ﬁelds are in fact special cases of a more

general class of objects called conformal Killing spinors or twistor spinors. Con-

formal Killing spinors were originally introduced by Penrose in the context of gen-

eral relativity [9] and appeared in pure mathematics as integrability conditions for

the complex structure of a four-dimensional Riemannian manifold [10]. In the late

80s Lichnerowicz started a systematic investigation of conformal Killing spinors on

Riemannian spin manifolds in the context of conformal differential geometry [11],

and since then, a body of strong structure results, examples and a partial classiﬁca-

tion has developed, both in the Riemannian and in the Lorentzian setting (see e.g.

[12, 13, 14, 15] and references therein).

1A better name might be “superparallel spinors”, to avoid confusion with geometric Killing spinors.

6 The space of parallel vector fields and parallel spinors on a semi-Riemannian spin manifold can be given the structure of a Lie superalgebra, with the Lie bracket between a vector field and a spinor given by the spinorial Lie derivative (which is actually trivial in this case). The odd bracket between two spinors can be defined as the so-called Dirac current associated to the spinors, which is also parallel if the spinors

are. Similarly, one natural object one can associate to a supergravity background is

its supersymmetry superalgebra. This is a Lie superalgebra constructed out of the

Killing spinors of the background and the Killing vectors of the background’s metric

which also preserve the p-form ﬁelds: guaranteeing the closure of these algebras

also usually requires imposing the equations of motion of the theory. The super-

symmetry superalgebras of many supergravity backgrounds have been computed

[16, 17, 18, 19] and it has been found that many of them correspond to Lie superal-

gebras on Nahm’s list. Thus, many of the supersymmetry algebras have a manifest

geometric origin.

The space of conformal Killing spinors and conformal Killing vectors on a semi-

Riemannian manifold also admits a natural algebraic structure, ﬁrst investigated

by Habermann[20], which we call a conformal Killing superalgebra. It would thus

be natural to be able to ﬁll in the question marks in the lower right corner of the

following diagram

parallel vectors Killing vectors ψ 0 / Dψ 0 ∇ = =

Pψ 0 / ?? l P = conformal Killing vectors ?? where is the Levi-Cività connection, P is the so-called Penrose operator whose ∇ kernel deﬁnes conformal Killing spinors, and D is the supercovariant connection.

The primary goal of this thesis is to construct the supergravity analogue of confor-

mal Killing spinors in eleven-dimensional supergravity and type IIA and IIB super-

gravities and to see if supergravity backgrounds admitting such spinors can be char-

acterised geometrically. A secondary — although not unrelated — goal is more al- 7 gebraic in ﬂavor. Many of the superalgebras on Nahm’s list incorporate conformal symmetry algebras in their even part. Although some of these superconformal al-

gebras have been realised as symmetry superalgebras of supergravity backgrounds

with an AdS factor, others still lack a manifest geometric origin. We will construct

conformal Killing superalgebras both in geometric and supergravity context and try

to see if these objects could provide a geometric origin for known Lie superalgebras

involving conformal symmetries.

Along the way we are naturally led to reconsider the deﬁnition of the spinorial Lie

derivative and introduce other machinery such as Weyl connections and Kaluza-

Klein reductions: we use the latter as a tool to construct a number of explicit exam-

ples of conformal Killing superalgebras.

8 Chapter 2

Preliminaries

In this chapter we introduce the notation and some basic algebraic and geometric tools that we will need in the following chapters. The material herein is mostly standard, although presented with a view towards our applications and utilising some non-standard but useful notation.

2.1 Algebraic preliminaries

2.1.1 Natural properties of vector spaces with an inner product

Let V be a n-dimensional vector space equipped with an inner product , . There 〈− −〉

are natural “musical” isomorphisms ! : V V ∗ and " : V ∗ V relating V and its → → dual, given by

X !(Y ) X ,Y = 〈 〉 α",Y α(Y ) . 〈 〉 =

where α V ∗ and X ,Y V . The isomorphisms also induce an inner product on V ∗, ∈ ∈ which we will similarly denote , . Any endomorphism of V also acts naturally 〈− −〉

on V ∗. Given A End(V ) and β V ∗, Aβ β A, or ∈ ∈ = − ◦

(Aβ)(Y ) β(AY ) . (2.1) = −

Now let so(V ) be the skew-symmetric endomorphisms of V with respect to the inner product. If X V is a vector ﬁeld and α V ∗ is a 1-form, then we deﬁne X α so(V ) ∈ ∈ ! ∈

9 by

(X α)Y , Z α(Y ) X , Z X ,Y α(Z ) (2.2) 〈 ! 〉 = 〈 〉 − 〈 〉 for all Y , Z V . ∈ 2 There is also a natural isomorphism so(V ) Λ V ∗. Given a skew-symmetric endo- =∼ morphism A so(V ), we can ﬁnd a corresponding two-form ω via ∈ A

ω (X ,Y ) X , A(Y ) . (2.3) A = 〈 〉

2 ω Correspondingly, any ω Λ V ∗ deﬁnes a skew-symmetric endomorphism A by ∈

Aω(X ) (ı ω)" , (2.4) = − X

where the sign on the right-hand side turns out to be convenient later on.

Let co(V ) denote those endomorphisms whose symmetric part is proportional to the identity — in other words, co(V ) so(V ) RId . We deﬁne a natural map = ⊕ V

: V V ∗ co(V ) (2.5) • ⊗ →

by X α X α α(X )Id, or more explicitly • = ! +

(X α)(Y ) α(Y )X α(X )Y X ,Y α" (2.6) • = + − 〈 〉

Note that this is manifestly symmetric in X ,Y , so in fact (X α)(Y ) (Y α)(X ). • = •

Similarly, we can exhibit the action of X α on V ∗: •

(X α)(β) (Y ) β((X α)(Y )) α(X )β(Y ) β(X )α(Y ) X ,Y α,β , • = − • = − − + 〈 〉〈 〉 ! " which is also symmetric in α,β and hence (X α)(β) (X β)(α). • = • We collect these observations and other properties of the operator into the follow- • ing useful

Lemma 1. The following identities hold for all X ,Y V, α,β V ∗ and A co(V ): ∈ ∈ ∈

(a) (X α)Y (Y α)X • = •

(b) (X α)β (X β)α • = • 10 (c) [A, X α] A(X ) α X A(α) • = • + •

(d) [X α,Y α]α 0 , • • = where [ , ] denotes the natural commutator of two endomorphisms. − − Proof. We’ve already established (a) and (b).

(c): Both sides of the equation are linear, so it is sufﬁcient to check it for A so(V ) ∈ — obviously, the result holds if A Id , since then both sides vanish identically. = V Therefore, we assume that A is skew-symmetric and compute the left-hand side:

[A, X α](Y ) α(Y )A(X ) X ,Y A(α") α(A(Y )) X X , A(Y ) α" , • = − 〈 〉 − + 〈 〉

On the other hand, the ﬁrst term on the right-hand side gives

(A(X ) α)(Y ) α(Y )A(X ) α(A(X ))Y A(X ),Y α" • = + − 〈 〉 α(Y )A(X ) α(A(X ))Y X , A(Y ) α" , = + + 〈 〉 whereas the second term yields

(X A(α))(Y ) A(α)(Y )X A(α)(X )Y X ,Y A(α)" • = + − 〈 〉 α(A(Y ))X α(A(X ))Y X ,Y A(α") . = − − − 〈 〉

Adding the last two equations gives the result.

(d): We deﬁne the one-form ω(X ,Y ) V ∗ by ∈

ω(X ,Y ) [X α,Y α]α , = • • which is manifestly antisymmetric in X ,Y . We now use (c), (a) and (b), in that order, to ﬁnd

ω(X ,Y ) 1 (ω(X ,Y ) ω(X ,Y )) = 2 − 1 (X α)Y α)α (Y (X α)α ((Y α)X α)α (X (Y α)α)α) = 2 • • + • • − • • − • • 1 ((Y (X α)α)α (X (Y α)α)α) = 2 • • − • • 1 ((Y α)(X α)α (X α)(Y α)α) = 2 • • − • • 1 [Y α, X α]α = 2 • • 1 ω(X ,Y ) = − 2 11 so ω(X ,Y ) 0, which proves the result. =

2.1.2 The Möbius Lie algebra of V

Consider the vector space V co(V ) V ∗. We can make it into a Lie algebra mo(V ), ⊕ ⊕ called the Möbius Lie algebra of V, by introducing the Lie bracket given by

X Y AY B X − A , B [A,B] X γ Y β , (2.7)     =  + • − •  β γ Aγ Bβ      −  which is manifestly antisymmetric. It is a straightforward calculation to show that the Jacobi identity of this bracket vanishes. Using (2.7), the vanishing of

X Y Z Z X Y Y Z X A , B , C C , A , B B , C , A      +      +       α β γ γ α β β γ α                   is equivalent to

0 (X β)Z (Y α)Z (Z α)Y (X γ)Y , = − • + • − • + • 0 [A,Y β] [A, Z β] X Bγ X Cβ B Z α CY α = • − • + • − • − • + • [B, Z α] [B, X γ] Y Cα Y Aγ C X β AZ [C, X β] + • − • + • − • − • + • + • [C,Y α] Z Aβ Z Bα AY γ B X γ , − • + • − • − • + • 0 (X β)γ (Y α)γ (Z α)β (X γ)β (Y γ)α (Z β)α , = • − • − • + • − • + • which can be seen to hold by using the ﬁrst three identities in Lemma 1.

The Möbius Lie algebra mo(V ) is in fact isomorphic to so(Vˆ ), where Vˆ V R1,1, = ⊕ where the inner product on Vˆ (which we also denote by , ) extends the one on 〈− −〉 V .

1,1 1 Let e0,e1 span R and deﬁne e (e0 e1). Then e V and e ,e 1. We ± = -2 ± ± ⊥ 〈 + −〉 = decompose an arbitrary vector Yˆ Vˆ as ∈

Yˆ Y y+e y−e , = + + + −

2 where Y V , and a two-form ωˆ Λ Vˆ ∗ as ∈ ∈

ωˆ ω α e e X ! he e , = + ∧ + + − ∧ + + ∧ − 12 2 where α V ∗, ω Λ V ∗ and X V . Then the action of a skew-symmetric endomor- ∈ ∈ ∈ phism Sˆωˆ on Yˆ is given by (2.4):

ωˆ ω " Sˆ (Yˆ ) S (Y ) y+X y−α (α(Y ) hy+)e (hy− X ,Y )e = + − + − + + − 〈 〉 − Sω X α" Y − α h 0 y+ . =  ! −   X 0 h y− −   This gives us the identiﬁcation

S X α" − (X , A,α) α h 0 so(Vˆ ) , (2.8) 0→  −  ∈ X ! 0 h −  where A S h Id and S so(V ). It is easy to see that using this identiﬁcation, the = + V ∈ matrix commutator on so(Vˆ ) induces the Lie bracket (2.7) on mo(V ), so equation

(2.8) is an explicit isomorphism mo(V ) so(Vˆ ). →

2.1.3 The Clifford algebras of V and Vˆ

Let C&(V ) denote the Clifford algebra of (V, , ), deﬁned by the relation 〈− −〉

X Y Y X 2 X ,Y Id . (2.9) · + · = − 〈 〉

We will frequently need the following formulas for the Clifford product between a

p vector X V and a p-form η Λ V ∗: ∈ ∈

X η X ! η ι η (2.10) · = ∧ − X η X ( 1)p (X ! η ι η) (2.11) · = − ∧ + X

We also introduce the gamma matrices Γi as the generators of the Clifford alge-

bra, corresponding to the image of the pseudo-orthonormal frame ei on Vˆ under

the embedding V % C&(V ). The following Clifford product identities are frequently → useful:

i Γ ιei η pη, (2.12) i · = + Γi θi η (n p)η, (2.13) i · ∧ = − − + 13 The associative algebra C&(V ) can be made into a Lie algebra using the Clifford com-

mutator, and the embedding so(V ) % C&(V ) deﬁned by A 1 ω is a Lie algebra → 0→ − 2 A homomorphism. Furthermore, if X V C&(V ), using (2.10) we see that ∈ ⊂ [ 1 ω , X ] A(X ) . (2.14) − 2 A = This follows because

[ 1 ω , X ] 1 (ω X X ω ) − 2 A = − 2 A · − · A ı ω = − X A A(X ) . = Thus, any C&(V )-module M restricts to a so(V )-module, giving rise to a spinor representation of so(V ). We recall that there is a natural isomorphism between Λ∗V ∗ and C&(V ), which allows us to deﬁne the Clifford action of a p-form on M. When

there is no chance of confusion, we will also denote this action by . · Let S be an irreducible C&(V )-module. It is possible to extend it into a co(V )-module

by introducing a weight: if σ : so(V ) End(S) denotes the representation map, → we deﬁne σw : co(V ) End(S) for all w R by σw (A) σ(A) for A so(V ) and → ∈ = ∈ σw (Id ) w Id and extending linearly. We denote the corresponding co(V )-module V = S by S[w]. Using this representation, we can express the Clifford algebra element

identiﬁed with X α in terms of Clifford products: • w 1 " 1 σ (X α) X α (w )α(X )IdS . • = − 2 · + − 2

2.1.4 Spinor representations of the Möbius algebra

We can also relate the Clifford algebras of V and Vˆ . As associative algebras, C&(Vˆ ) =∼ C&(V ) End(R2). An irreducible C&(Vˆ )-module Sˆ decomposes into a direct sum of ⊗ C&(V )-modules: Sˆ S S , where S KerΓ . (See e.g. [21] for a proof of the = + ⊕ − ± = ± isomorphism.)

In fact, S are isomorphic as C&(V )-modules: the isomorphism ı : S S is given ± − → + by the restriction of Γ to S . Since Γ Γ Γ Γ 2Id, the inverse isomorphism + − + − = − − + − 1 1 ı− is the restriction of Γ to S . − 2 − + 14 The module Sˆ also restricts to a co(V )-module since co(V ) mo(V ) so(Vˆ ). The ⊂ =∼ 1 i j 1 element of C&(Vˆ ) corresponding to A S h IdV co(V ) is S Γi j h(Γ Γ = + ∈ − 4 − 4 − + − 1 1 [ ] [ ] Γ Γ ), and thus Sˆ KerΓ KerΓ S 2 S − 2 as a co(V )-module. + − = + ⊕ − = ⊕ Composing the isomorphism (2.8) with the embedding so(Vˆ ) % C&(Vˆ ), we obtain → the following embedding mo(V ) % C&(Vˆ ): → X 1 i j 1 1 i 1 i A S Γi j h(Γ Γ Γ Γ ) α Γi Γ X Γi Γ (2.15)   0→ −4 − 4 − + − + − + 2 + − 2 − α   The spinorial representation + : mo(V ) End(S S ) induced by the embedding → − ⊕ + is thus given explicitly by

1 i j 1 i 1 S Γ h Id X Γ ı− +(X , A,α) − 4 i j + 2 − i (2.16) = 1 αi Γ ı 1 Si j Γ 1 h Id , 2 i − 4 i j − 2 - Note that this representation automatically gives the right weights to the represen-

tations of A co(V ), so we can rewrite + as ∈ 1 σ 2 (A) X +(X , A,α) −1 · . (2.17) =  1  α σ− 2 (A) 2 ·   We remark that the isomorphism ı could be rescaled by any scalar, although here we

have taken it to be the identity.

2.2 Geometric preliminaries

We now ﬁx our geometric conventions and introduce a number of useful tensors.

Our conventions are consistent with [22].

Let (Mn,g ) be an n-dimensional pseudo-riemannian manifold and let denote the ∇

Levi-Cività connection. Apart from T M and T ∗M (whose sections will be denoted

X (M) and Ω1(M), respectively), we will be somewhat cavalier about the distinction

between bundles and their sections, often taking A End(T M) to mean that A is a ∈ smooth section of End(T M).

The algebraic machinery introduced in Section 2.1 naturally carries over to this global

setting if we now take the vector space V to be T M, where p M, and operations p ∈

15 such as naturally extend to X (M) and Ω1(M) as well if we replace the inner prod- • uct , with the metric g . Let so(T M) End(T M) denote those endomorphisms 〈− −〉 ⊂ of T M which are skew-symmetric relative to g , and co(T M) so(T M) Id denote = ⊕ 〈 〉 those endomorphisms whose symmetric part is proportional to the identity. We can

redeﬁne X α so(T M) as ∧ ∈

g ((X α)Y , Z ) α(Y )g (X , Z ) g (X ,Y )α(Z ) (2.18) ∧ = −

Similarly we deﬁne X α : X α α(X )Id co(T M). • = ∧ + ∈ We note that since equation (2.18) is homogenenous in g , it is apparent that X α ∧ and X α depend only on the conformal class of the metric. Indeed, introducing the •

musical isomorphisms " : T ∗M T M and ! : T M T ∗M deﬁned by the metric g , → → we can rewrite X α as ∧

" ! X α X α α X T M T ∗M End(T M) , ∧ = ⊗ − ⊗ ∈ ⊗ =∼

which is manifestly invariant under conformal rescalings of the metric.

It is also clear that any section (X , A,α) of T M co(T M) T ∗M deﬁnes an endomor- ⊕ ⊕ phism of the bundle S S (where S is the spinor bundle on M, which we take to be ⊕ a bundle of irreducible C&(T M)-modules over M) — sometimes known as the “local

twistor bundle” — via the global version of (2.17).

2.2.1 The Riemann curvature tensor and its relatives

The Riemann curvature operator of the Levi-Cività connection is deﬁned by ∇

R(X ,Y )Z Z Z Z . (2.19) = ∇[X ,Y ] − ∇X ∇Y + ∇Y ∇X

and the corresponding curvature tensor is

R(X ,Y , Z ,U) g (R(X ,Y )Z ,U) . (2.20) =

The curvature operator satisﬁes the algebraic Bianchi identity

R(X ,Y )Z R(Y , Z )X R(Z , X )Y 0 . (2.21) + + = 16 The Ricci tensor is deﬁned as the trace with respect to the the second argument of the curvature operator:

r (X ,Y ) tr(Z R(X , Z )Y ) . (2.22) = 0→

It turns out to be convenient to introduce the Ricci operator Ric : T M T M via →

g (Ric(X ),Y ) r (X ,Y ) , (2.23) = whose trace gives the scalar curvature s. The Riemann curvature tensor has a natural

decomposition

R(X ,Y , Z ,U) W (X ,Y , Z ,U) (& g )(X ,Y , Z ,U) (2.24) = + 2

where W is the (conformally invariant) Weyl curvature tensor and stands for the 2 Kulkarni–Nomizu product of two symmetric tensors which guarantees that a b has 2 the symmetries of a curvature tensor:

(a b)(X ,Y , Z ,U) a(X , Z )b(Y ,U) a(Y ,U)b(X , Z ) a(Y , Z )b(X ,U) a(X ,U)b(Y , Z ) , 2 = + − − (2.25) and & is the Schouten tensor

1 s &(X ,Y ) r (X ,Y ) g (X ,Y ) , (2.26) = n 2 − 2(n 1) − , − - which can be thought as the ﬁrst quotient of the division of the Riemann tensor R

by the metric.

Alternatively, one may deﬁne the Schouten tensor via its associated map L : T M →

T ∗M, by

R(X ,Y ) W (X ,Y ) X L(Y ) Y L(X ) End(T M) , (2.27) = − • + • ∈ 2 Another useful tensor will be the Cotton-York tensor C : Λ T M T ∗M, deﬁned as →

C(X ,Y ) : ( L)(Y ) ( L)(X ) . (2.28) = ∇X − ∇Y

Taking the appropriate traces in the differential Bianchi identity we can show that

the Cotton-York tensor is nothing more than the divergence of the Weyl tensor:

(divW )(X ,Y ) (n 3)C(X ,Y ) , (2.29) = − 17 or in local coordinates,

dW (n 3)( L L ) . ∇ dcab = − ∇a bc − ∇b ac

18 Chapter 3

The Lie algebra of conformal Killing vectors

In this chapter we introduce the notion of conformal Killing transport and study the

Lie algebra of conformal Killing vector ﬁelds. We also deﬁne the Weyl structure on a

semi-Riemannian manifold and present manifestly Weyl-invariant versions of both

the conformal Killing transport equations and the conformal Lie algebra.

3.1 Conformal Killing transport

Deﬁnition 2. A vector ﬁeld X on (M,g ) is a conformal Killing vector if L g 2h g X = − X

for some smooth function h C ∞(M). X ∈

The equation L g 2h is equivalent to X = − X

g ( X , Z ) g (Y , X ) 2h g (Y , Z ) (3.1) ∇Y + ∇Z = − X for all vector ﬁelds Y , Z . In local coordinates this equation is often called the conformal Killing equation

X X 2h g , (3.2) ∇a b + ∇b a = − X ab from which we see that h 1 div X , where div X X a. Equivalently, confor- X = n = −∇a mal Killing vectors are characterized in terms of the endomorphism A End(T M), X ∈ deﬁned by

A Y X for all Y C ∞(M,T M). (3.3) X = −∇Y ∈ Indeed, we have 19 Lemma 3. A vector ﬁeld X is a conformal Killing vector if and only if A co(T M). X ∈

Proof. Let S so(T M) denote the skew-symmetric part of A . Then if A co(T M), X ∈ X X ∈ we have

A S h Id , (3.4) X = X + X or in local coordinates

X S h g . (3.5) ∇a b = ab − X ab

Then

g (A Y , Z ) g (Y , A Z ) g ( X , Z ) g (Y , X ) 2h g (Y , Z ) , (3.6) X + X = − ∇Y − ∇Z = X so obviously X satisﬁes the conformal Killing equation (3.1). The converse is obvious, since if X satisﬁes (3.1), then clearly AX can be written in the form (3.4) and is

thus an element of co(T M).

Differentiating A further we obtain A Ω1(M,co(T M)). X ∇ X ∈

Lemma 4.

A R(Y , X ) Y α co(T M) , ∇Y X = + • X ∈

where α dh . X = X

Proof. We can write the covariant derivative of AX as

( A )Y (A Y ) A Y ∇Z X = ∇Z X − X ∇Z

Z Y X Y X = −∇ ∇ + ∇∇Z

( A )Y ( A )Z R(Z ,Y )X ∇Z X − ∇Y X = R(Z , X )Y R(Y , X )Z , = =

where the last line follows from the algebraic Bianchi identity (2.21). This means

that if we deﬁne

F (Z ,Y ,U) : g (( A R(Z , X ))Y ,U) , = ∇Z X − 20 then F is symmetric in the ﬁrst two arguments:

F (Z ,Y ,U) F (Y , Z ,U) (3.7) =

We can also take of the conformal Killing equation (3.6), obtaining ∇

g (( A )Y ,U) g (( A )U,Y ) 2α (Z )g (Y ,U) , ∇Z X + ∇Z X = X where α dh . It follows that X = X

F (Z ,Y ,U) F (Z ,U,Y ) α(Z )g (Y ,U) , (3.8) = − +

and combining this with (3.7), we have

F (Z ,Y ,U) α(Z )g (Y ,U) α(U)g (Z ,Y ) α(Y )g (U, Z ) . (3.9) = − +

Comparing this with (2.18), we conclude that

A R(Y , X ) Y α . (3.10) ∇Y X = + • X

In local coordinates,

S X d R g α g α and h α . (3.11) ∇a bc = dabc + ab c − ac b ∇a X = a

In other words,

X X d R g α g α α g , (3.12) ∇a∇b c = dabc + ab c − ac b − a bc

whence tracing with g ab, we obtain

2X (n 2)α R X d , (3.13) ∇ c = − c − cd

1 2 b αa Xa Rab X . (3.14) = n 2 ∇ + − . / Differentiating further we ﬁnd

α ( L )X c L c S L c S 2L h , (3.15) ∇a b = ∇c ab + a bc + b ac − ab X 21 where Lab are the components of the Schouten tensor. Now note that the Lie derivative of a vector ﬁeld can be written as L Y [X ,Y ] X = = Y X Y A Y , where A is the endomorphism associated to the co- ∇X − ∇Y = ∇X + X X variant derivative of X as deﬁned in Equation 3.3. (In fact, the Lie derivative of any

tensor can be written as L +(A ), where + is the representation of A acting X = ∇X + X X on the appropriate bundle.)

Rewriting Equation 3.15 using this observation gives

Lemma 5.

α ( L)(Y ) L(Y ) A L(A Y ) ( L)(Y ) (A L)(Y ) (L L)(Y ) . ∇Y X = ∇X − ◦ X − X = ∇X + X = X

Putting Lemmas 4 and 5 together, we arrive at the characterization of conformal

Killing vectors in terms of conformal Killing transport.

Proposition 6. Conformal Killing vectors are in bijective correspondence with sec-

tions of the bundle T M co(T M) T ∗M which are parallel relative to the following ⊕ ⊕ connection which we call the Geroch connection:

X X AY ∇Y + D A : Y A R(X ,Y ) Y α . (3.16) Y   =  ∇ + − •  α Y α ( X L)(Y ) L(Y ) A L(AY )   ∇ − ∇ + ◦ +  Indeed a Killing vector X determines and is determined uniquely by a parallel section

(X , AX ,αX ).

We remark that in terms of β : α L(X ), the Killing transport equations can be = − rewritten in terms of the Weyl, Schouten and the (normalized) Cotton–York tensor:

X X AY ∇Y + D A : Y A W (X ,Y ) Y β X L(Y ) . (3.17) Y   = ∇ + − • − •  β Y β C(X ,Y ) L(Y ) A    ∇ + + ◦  3.2 The conformal Lie algebra

If X ,Y are conformal Killing vectors, so is their Lie bracket; indeed,

L g 2h g , [X ,Y ] = − [X ,Y ] 22 where

h α (X ) α (Y ) β (X ) β (Y ) . (3.18) [X ,Y ] = Y − X = Y − X This means that conformal Killing vectors span a Lie subalgebra of the Lie algebra

of vector ﬁelds, which we call the conformal Lie algebra of (M,g ).

Proposition 6 implies that a conformal Killing vector is uniquely determined by

its conformal Killing transport data (X , A ,α ) or (X , A ,β ) at a point p M. X X X X ∈ This exhibits the conformal Lie algebra of (M,g ) at a point as a vector subspace of mo(Tp M). We will now determine the Lie bracket for this algebra.

The conformal Killing transport data for [X ,Y ] is ([X ,Y ], A[X ,Y ],α[X ,Y ]), or with β[X ,Y ] replcing α[X ,Y ]. Since the Levi-Cività connection is torsion-free,

[X ,Y ] Y X A Y A X . = ∇X − ∇Y = X − Y

Differentiating [X ,Y ] we obtain

A [A , A ] X α Y α R(X ,Y ) , (3.19) [X ,Y ] = X Y + • Y − • X +

or, in terms of β,

A [A , A ] X β Y β W (X ,Y ) . (3.20) [X ,Y ] = X Y + • Y − • X +

Differentiating h[X ,Y ] we obtain

α α A α A C(X ,Y ) L(A Y A X ) L(Y ) A L(X ) A (3.21) [X ,Y ] = X ◦ Y − Y ◦ X − + X − Y + ◦ X − ◦ Y

or, in terms of β,

β β A β A C(X ,Y ) . (3.22) [X ,Y ] = X ◦ Y − Y ◦ X − In summary, we have the following Lie brackets for the conformal Killing transport

data: X Y A Y A X X − Y A , A [A , A ] X β Y β W (X ,Y ) , (3.23)  X   Y  =  X Y + • Y − • X +  βX βY βY AX βX AY C(X ,Y )      − ◦ + ◦ −  which shows that the Weyl curvature measures the failure of this Lie bracket to agree

with the algebraic Lie bracket on sections of T M co(T M) T ∗M, given by equation ⊕ ⊕ (2.7). 23 3.3 Conformal changes of the metric

2f Let g e g , where f C ∞(M) is a smooth function, be a conformal rescaling of the = ∈ metric. If X is a conformal Killing vector for g , it is also a conformal Killing vector

for g . Indeed,

L g 2h g , X = − X where h h d f (X ). As this calculation already shows, the conformal Killing X = X − transport data for X does depend on the metric and not just on its conformal class.

Let (X , AX ,αX ) denote the conformal Killing transport data associated to X relative

to the conformally rescaled metric g . To relate (X , AX ,αX ) to the conformal Killing

transport data (X , AX ,αX ) relative to the original metric, we need to see how certain

geometric objects behave under conformal rescalings of the metric. The Levi-Cività

connection changes by [22]

X d f , (3.24) ∇X = ∇X + • whence

A A X d f . (3.25) X = X − • Finally,

α dh α dι d f α L d f . (3.26) X = X = X − X = X − X In summary, under a conformal rescaling of the metric, the conformal Killing trans-

port data associated to a conformal Killing vector X changes by X X 0 A AX X d f . (3.27)  X  =  −  •  αX αX LX d f       Things are a little more complicated in terms of β, since the Schouten tensor has

more complicated transformation laws under conformal rescalings of the metric.

The (4,0) Riemann curvature tensor transforms as [22]

R e2f R g d f (d f )2 1 d f 2g , (3.28) = − 2 ∇ − + 2 | | ! " with d f the Hessian of f . On the other hand, from (2.24), we have that ∇

R W g L e2f W g L , = + 2 = + 2 24 whereas on the other hand, inserting (2.24) into (3.28), we have that

R e2f W g L d f (d f )2 1 d f 2g . = + 2 − ∇ + − 2 | | ! " Comparing the two expressions, we can read off how the Schouten tensor trans-

forms:

L L d f (d f )2 1 d f 2g . (3.29) = − ∇ + − 2 | |

In local coordinates,

L L f f f 1 g cd f f g . (3.30) ab = ab − ∇a∇b + ∇a ∇b − 2 ∇c ∇d ab

The Cotton-York tensor transforms in a particularly simple way under Weyl trans-

formations:

C(X ,Y ) C(X ,Y ) W (X ,Y )d f . (3.31) = + We note that d f dι d f d f L d f A d f , whence ∇X − X = ∇X − X = − x

β β A d f d f (X )d f 1 d f 2X ! . (3.32) X = X − X − − 2 | |

Since the conformal Lie algebra is an invariant of the conformal structure, we would

like to ﬁnd a version of the Geroch connection (3.16) which is manifestly invariant

under conformal rescalings of the metric. This requires introducing a Weyl connec-

tion.

3.4 Weyl connections

By a Weyl connection we mean a torsion-free connection D on T M preserving the conformal class of the metric; that is, a connection which obeys, for any vector ﬁeld

X ,

D g 2θ(X )g , (3.33) X =

where θ is a 1-form. This connection is invariant under conformal rescaling of the

metrics g e2f g , provided that the one-form θ transforms as θ θ d f . We call = = + such a transformation a Weyl transformation .

25 A manifold M equipped with a conformal class of metrics and a Weyl connection is often said to have a Weyl structure: in fact, it can be realised as a reduction of the frame bundle of M to CO(T M). We will work with a ﬁxed representative metric g , but one frequently encounters the more general viewpoint in conformal geometry literature [23].

One can derive an explicit formula for D in terms of the Levi-Cività connection of g :

D X θ . (3.34) X = ∇X − •

The curvature RD of the Weyl connection is deﬁned by

RD (X ,Y ) D D D D D , = [X ,Y ] − X Y + Y X

and using the above expression for D X , can be related to the Riemann curvature R

of g by

RD (X ,Y ) R(X ,Y ) X θ Y θ [X θ,Y θ] . (3.35) = − • ∇Y + • ∇X − • •

Inserting (2.27) into this equation and decomposing the result in a way similar to

(2.27) itself, we ﬁnd

RD (X ,Y ) W (X ,Y ) X LD (Y ) Y LD (X ) , (3.36) = − • + •

where

LD (X ) L(X ) θ θ(X )θ 1 θ 2X ! , (3.37) = + ∇X + − 2 | |

or, equivalently, using that D θ θ 2θ(X )θ θ 2 X !, X = ∇X − + | |

LD (X ) L(X ) D θ θ(X )θ 1 θ 2X ! . (3.38) = + X − + 2 | |

Unlike the Schouten tensor L, the map LD is not symmetric, and we can make this

manifest by rewriting (3.36) as follows

RD (X ,Y ) W (X ,Y ) F D (X ,Y )Id X LD (Y ) Y LD (X ) , (3.39) = + − ! + !

where we have introduced the Faraday 2-form F D dθ, which is invariant under = Weyl transformations. It follows from Equation 3.39 that LD is also Weyl-invariant, 26 a fact which can also be checked directly from equations (3.29) and (3.34) and using equation (3.24).

Similarly, we can construct the Weyl-invariant analogue of the Cotton-York tensor.

From the Weyl transformation law (3.31), we immediately see that

C D (X ,Y ) : C(X ,Y ) W (X ,Y )θ (3.40) = −

is Weyl-invariant. Note that this is in fact equal to the “naive” Weyl-covariantisation

of the Cotton-York tensor, i.e. obtained simply by replacing the Levi-Cività connec-

tion with the Weyl connection in equation (2.28) and the Schouten tensor with its

Weyl-invariant analogue:

C D (X ,Y ) (D LD )(Y ) (D LD )(X ) . = X − Y

Naturally, one can also show that C D can be obtained as the D-divergence of the

Weyl tensor.

3.5 Weyl-invariant conformal Killing transport

Defining manifestly Weyl-invariant conformal Killing transport requires redefining the conformal Killing transport data itself — although the vector field X itself is con-

formally invariant, because of (3.24) AX and αX are not. We will remedy the situa-

tion by adding θ-dependent terms to them in such a way that the resulting data

D D (X , AX ,αX ) is Weyl-invariant. Taking into account the transformation property of θ and equation (3.25), it is easy to see that

AD : A X θ (3.41) X = X + •

is Weyl-invariant. However, the one-form αX actually has a one-parameter family

of Weyl-invariant extensions, since we can always add the manifestly Weyl-invariant

D term ιX F to it. From equation (3.26) it is apparent that

αD : α L θ tι F D (3.42) X = X + X + X 27 is Weyl-invariant for all t. Similarly, for any choice of t,

βD : αD LD (X ) (3.43) X = X −

is manifestly Weyl-invariant.

D D It remains to be shown that (X , AX ,αX ) is also parallel with respect to a suitable

connection on the Möbius bundle T M co(T M) T ∗M. Using equation (3.34) and ⊕ ⊕ (3.41), we observe that in fact

AD Y A Y (X θ)Y X (Y θ)X D X , (3.44) X = X + • = −∇Y + • = − Y

which would suggest attempting a naive covariantisation of the conformal Killing

transport equations (3.16) simply by replacing by D. Indeed, applying D to (3.41) ∇ and using (3.34) and (3.35) yields

D AD RD (Y , X ) Y (α L θ) , (3.45) Y X = + • X + X

which in comparison with (3.42) and (3.16) suggests setting the parameter t 0 in = D the deﬁnition of αX . This allows us to rewrite equation (3.45) in a more familiar form:

D AD RD (Y , X ) Y αD , (3.46) Y X = + • X D which can also be rewritten in terms of βX with the help of equation (3.35) as

D AD W (Y , X ) Y βD X LD (Y ) . (3.47) Y X = + • X + •

D Similarly, we can calculate DY αX and ﬁnd

D αD (D LD )(Y ) (AD LD )(Y ) (L LD )(Y ) , (3.48) Y X = X + X = X

D or using βX , D βD C D(X ,Y ) AD LD (Y ) . (3.49) Y X = − + X Combining these results allows us to deﬁne the manifestly Weyl-invariant version of

the Geroch connection, whose parallel sections are in one-to-one correspondence

with conformal Killing vectors on M:

X DY X AY D + D DY A DY A W (X ,Y ) Y β X L (Y ) . (3.50)   =  + D − • − D •  β DY β C (X ,Y ) AL (Y )    + −  28 3.6 The conformal Lie algebra of a Weyl structure

The conformal Lie algebra of a Weyl structure (M,g ,θ) can now be determined with

the help of the Weyl-invariant Geroch connection (3.50). Since D is torsion-free, the

T M-component of the bracket does not change:

[X ,Y ] D Y D X AD Y AD X . = X − Y = X − Y

Proceeding in a similar fashion as before, we ﬁnd

AD [AD , AD ] X βD Y βD W (X ,Y ) (3.51) [X ,Y ] = X Y + • Y − • X +

and

βD AD βD AD βD C D (X ,Y ) . (3.52) [X ,Y ] = X Y − Y X −

Combining these, we obtain the Weyl-invariant version of equation (3.23):

X Y AD Y AD X X − Y AD , AD [AD , AD ] X βD Y βD W (X ,Y ) , (3.53)  X   Y  =  X Y + • Y − • X +  βD βD βD AD βD AD βD C D(X ,Y ) X Y [X ,Y ] = X Y − Y X −       where the Weyl curvature again measures the failure of this bracket to agree with the

natural Lie bracket of the Möbius algebra.

3.7 Normal conformal Killing vectors

Naturally, when (M,g ) is conformally ﬂat and thus W 0, the bracket (3.53) agrees = with the algebraic one. However, even in general, (M,g ) may possess a Lie subalgebra of conformal Killing vectors whose Lie bracket does agree with the Möbius algebra bracket.

Deﬁnition 7. If X is a conformal Killing vector ﬁeld of (M,g ,θ) and in addition

W (X ,Y ) C(X ,Y ) 0 (3.54) = =

for any vector ﬁeld Y X (M), we call X a normal conformal Killing vector ﬁeld. ∈

29 While we will content ourselves with the above deﬁnition for the purposes of this

thesis, the term “normal” here is motivated by the fact that normal conformal Killing

vector ﬁelds arise as parallel sections of mo(T M) with respect to not just the Geroch

connection but a connection induced from the so-called normal conformal Cartan

connection of M [24, 25, 26], an important tool in conformal geometry. In our nota-

tion, this connection can be written as [24]

D NC adY adL(Y ) , (3.55) Y = ∇Y + + where the connection acts on the Möbius bundle and the adjoint action is with respect to the bracket (2.7). More explicitly, we can write

X Y X AY NC ∇ + D A Y A Y β X L(Y ) . (3.56) Y   = ∇ − • − •  β Y β AL(X )    ∇ −  Comparing equation (3.56) with (3.17), we see that a conformal Killing vector which

is also parallel with respect to the normal conformal Cartan connection satisﬁes

equation (3.54).

30 Chapter 4

Conformal Killing spinors

Let (M,g ) be a n-dimensional Lorentzian spin manifold1, S its spinor bundle and

the spin connection acting on S induced from the Levi-Cività connection. We ∇ denote the Clifford multiplication of a spinor by a p-form — the Clifford action of

the Clifford bundle C&(T M) on S — by . The spin connection respects the Clifford · product in the following way: if Y is any vector ﬁeld and ψ is a spinor, then

(Y ψ) Y ψ Y ψ . (4.1) ∇Z · = ∇Z · + ·∇Z

Note that the following relationship holds between the Riemann curvature tensor

and the curvature of the spin connection:

R(X ,Y )ψ ψ [ , ]ψ 1 R(X ,Y ) ψ , (4.2) = ∇[X ,Y ] − ∇X ∇Y = − 2 ·

where the right-hand side means the Clifford action of the Riemann curvature ten-

sor R(X ,Y ) considered as a two-form.

There are two natural ﬁrst-order operators acting on S derived from the connection

and the Clifford product. The Dirac operator / is the connection composed with ∇ ∇ the Clifford product. Given a spinor ψ S, if e is a local pseudo-orthonormal frame ∈ a and ea is the coframe deﬁned by g (e ,eb) δ b, then the Dirac operator acting on ψ a = a can be expressed as

a / ψ e ea ψ . (4.3) ∇ = a ·∇ + 1Henceforth we will always assume that g has Lorentzian signature unless explicitly stated otherwise.

31 The Penrose operator P is complementary to the Dirac operator in the following

sense. Let µ : T M S S denote the Clifford multiplication by a vector. Then Kerµ ⊗ → is a subbundle of T M S. Let p : T M S Kerµ be the projection on the kernel. ⊗ ⊗ → Then P is deﬁned to be the composition of the spin connection and the projection

p. p P : Γ(S) ∇ Γ(T ∗M S) Kerµ. (4.4) −→ ⊗ −→ Much of our treatment is concerned with spinors that lie in the kernel of the Penrose

operator.

Deﬁnition 8. A spinor ψ S is called a conformal Killing spinor iff P ψ 0 for all ∈ X = X X (M). Equivalently, ψ satisﬁes the differential equation ∈

ψ 1 X / ψ 0 . (4.5) ∇X + n ·∇ =

S admits a Spin-invariant inner product ( , ) whose properties depend on dimen- − − sion and signature of g . Details can be found in standard texts, for example [21], but the only properties we will need are the following:

(ψ,χ) .(χ,ψ) (4.6) = (X ψ,χ) .(ψ, X χ) (4.7) · = · X (ψ,χ) ( ψ,χ) (ψ, χ) , (4.8) = ∇X + ∇X ψ where . is a sign. Now given any two spinors , one can deﬁne a vector ﬁeld V χ ψ,χ , - using the spinor inner product, sometimes known as the Dirac current.

It is deﬁned by the following equation:

g (Y ,V ) (ψ,Y χ) for all vector ﬁelds Y . (4.9) ψ,χ = ·

The reason why spinors in the kernel of P are called conformal Killing spinors2 is

explained by the following proposition.

Proposition 9. If ψ,χ are conformal Killing spinors, Vψ,χ is a conformal Killing vector.

2Conformal Killing spinors are also often known as twistor spinors in the literature due to their relationship with the twistor bundle in four dimensions[15]

32 Proof. Taking of equation (4.9), we ﬁnd: ∇Z

g ( V ,Y ) g (V , Y ) ( ψ,Y χ) (ψ, Y χ) (ψ,Y χ) , (4.10) ∇Z ψ,χ + ψ,χ ∇Z = ∇Z · + ∇Z · + ·∇Z

g ( V ,Y ) ( ψ,χ) (ψ,Y χ) ∇Z ψ,χ = ∇Z + ·∇Z 1 (Z / ψ,Y χ) 1 (ψ,Y Z / χ) = − n ·∇ · − n · ·∇ . 1 ( 1) + − (Y Z / ψ,χ) 1 (ψ,Y Z / χ) , = n · ·∇ − n · ·∇

whence

2 g ( Z Vψ,χ,Y ) g ( Y Vψ,χ, Z ) (ψ, / χ) .(/ ψ,χ) g (Y , Z ) , (4.11) ∇ + ∇ = n ∇ + ∇ ! " where we have used the deﬁning relation of the Clifford algebra (2.9). Equation

(4.11) shows that in fact Vψ,χ satisﬁes the conformal Killing equation (3.1), with

h 1 (ψ, / χ) .(/ ψ,χ) . Vψ,χ = n ∇ + ∇ ! " 4.1 Spinorial conformal Killing transport

We saw in the previous chapter that conformal Killing vectors deﬁne parallel sections of the Möbius bundle T M co(T M) T ∗M. Similarly, there exists a charac- ⊕ ⊕ terisation of conformal Killing spinors as parallel sections of S S with respect to a ⊕ certain connection. We will determine this connection by rewriting the conformal

Killing spinor equation (4.5) as a ﬁrst-order system.

Lemma 10. If ψ is a conformal Killing spinor, then the following identities are satis-

ﬁed for every vector ﬁeld X :

2 1 n (a) / ψ 4 n 1 sψ ∇ = −

(b) / ψ n L(X ) ψ, ∇X ∇ = − 2 ·

where L(X ) is the Schouten map deﬁned in equation (2.27).

33 Proof. For this calculation, it is convenient to assume that the pseudo-orthonormal

frame e arises from a basis of T M for some p M via parallel transport along a p ∈ geodesics, so that

e (p) 0 ∇ a = [e ,e ](p) 0 . a b =

(a): Differentiating the conformal Killing spinor equation (4.5), using the property

(4.1) and taking the trace, we obtain (at the point p M): ∈

1 0 a aψ n a ea / ψ = a ∇ ∇ + ∇ ·∇ + ! " ∆ψ 1 / 2 ψ , = − + n ∇

where ∆ is the spin connection Laplacian. We now apply the Weitzenböck formula

[22] / 2 ∆ 1 s to obtain ∇ = + 4 2 1 n / ψ n 1 sψ . ∇ = 4 − (b): The conformal Killing spinor equation implies

R(X ,e )ψ 1 e / ψ 1 X / ψ , i = − n i ·∇X ∇ + n ·∇i ∇

and taking the Clifford trace, we obtain

2 Ric(X ) ψ 2 X / ψ n ei X i / ψ · = − ∇ ∇ − i · ·∇ ∇ + 4 2 / ψ 2 X / 2 ψ / ψ , = − ∇X ∇ + n ·∇ + ∇X ∇

and substituting the result of (a) and using the deﬁnition of L in equation (2.26), we

arrive at the result.

Using Lemma 10, we can immediately see what the spinorial analogue of Proposi- tion 6 is.

Proposition 11. Conformal Killing spinors are in one-to-one correspondence with sections of the bundle S S which are parallel with respect to the following connec- ⊕ tion: ψ X ψ P ∇X · X χ = 1 L(X ) χ , - , 2 ∇X -, - 34 A conformal Killing spinor determines and is determined uniquely by a parallel section (ψ, 1 / ψ). Furthermore, the connection P has curvature n ∇ W (X ,Y ) 0 RP(X ,Y ) P [P ,P ] 1 . (4.12) = [X ,Y ] − X Y = − 2 C(X ,Y ) W (X ,Y ) , - Proof. The ﬁrst part follows immediately from Lemma 10. As for the curvature, we

simply compute

[X ,Y ] P [P ,P ] ∇[X ,Y ] · [X ,Y ] − X Y = 1 L([X ,Y ]) , 2 · ∇[X ,Y ] - [ , ] 1 X L(Y ) 1 Y L(X ) Y X ∇X ∇Y + 2 · − 2 · ∇X − ∇Y − 1 ( L)(Y ) 1 ( L)(X ) 1 L([X ,Y ]) [ , ] 1 X L(Y ) 1 Y L(X ) , 2 ∇X − 2 ∇Y + 2 ∇X ∇Y + 2 · − 2 · - W (X ,Y ) 0 1 , = − 2 C(X ,Y ) W (X ,Y ) , - where we have used equation (2.27).

4.2 Conformal covariance of the Penrose operator

An important property of the Penrose operator P is conformal covariance under

Weyl transformations. This implies that KerP — the space of conformal Killing

spinors — is an invariant of the conformal structure on M. In this section we show

how spinors transform under Weyl transformations.

Let (M,g ) and (M,g e2f g ) be conformally related pseudo-riemannian spin man- =

ifolds. Given a a pseudo-orthonormal frame (ea) for g , we can readily construct

f a frame (e ) for g by setting e e− e . This deﬁnes a bundle isomorphism be- a a = a tween the two frame bundles ξ : P (M,g ) P (M,g ), which in turn lifts to a bun- SO → SO dle isomorphism of the spin bundles ξ˜ : P (M,g ) P (M,g ), provided that Spin → Spin we choose the same topological spin structure for both manifolds. Now let S and

S be the corresponding spinor bundles constructed as associated bundles of the spinor representation σ : Spin GL(S) — in other words, S P (M,g ) S and → = Spin ×σ S P (M,g ) S. The bundle isomorphism ξ˜ is Spin-equivariant, and thus in- = Spin ×σ duces a further bundle isomorphism Ξ : S S, which obeys →

Ξ(e ψ) ξ(e ) Ξ(ψ) . (4.13) a · = a · 35 To keep the notation from getting out of hand, we often simply denote Ξ(ψ) ψ, = rewriting the previous equation as

e ψ e ψ . a · = a ·

f This means that for all vector ﬁelds X X (M), X ψ e− X ψ. ∈ · = · It is also convenient to introduce bundle isomorphisms Ξ : S S for every w R, w → ∈ deﬁned by

Ξ (ψ) ew f Ξ(ψ) ew f ψ . (4.14) w = =

Using these isomorphisms, we can now compute what happens to the spin connec-

tion, the Dirac operator and the Penrose operator under a Weyl transformation.

Proposition 12. The spin connection, Dirac and Penrose operators of (M,g ) and

(M,g ) are related as follows:

Ξ Ξ ( 1 X grad f 1 d f (X )Id) ∇X ◦ = ◦ ∇X − 2 · − 2 1 / Ξ Ξ 1 (/ (n 1)grad f ) ∇ ◦ = − ◦ ∇+ 2 − 1 1 P X Ξ Ξ (PX d f (X )Id X grad f ) . ◦ = ◦ − 2 −2n ·

Proof. Beginning with the spin connection, it is easy to see that the transformation

law for follows using the structure equations: ∇

de a ω a e b 0 + b ∧ = d(e f ea) ωa eb = + b ∧ e f d f ea e f dea ωa eb = ∧ + + b ∧ d f e a e f ωa eb ω a e b = ∧ − b ∧ + b ∧ ∂ f e a e b ωa e b ω a e b , = − b ∧ − b ∧ + b ∧

which implies that

ω ab ωab ∂a f e b . (4.15) = +

36 Substituting this identity into the expression for the spin connection in local coordinates, we have

a 1 ab X X ∂a ω Σab ∇ = + 4 1 ∂b f X aΓ = ∇X − 2 ab 1 X grad f 1 d f (X )Id , = ∇X − 2 · ·− 2

where we have used Σ 1 Γ and Γ Γ Γ η . The expression for the ab = − 2 ab ab = a b + ab Dirac operator now follows simply by taking the Clifford trace, and combining the

two gives the expression for the transformed Penrose operator.

It follows that the Dirac and Penrose operators are covariant, provided that they act on spinors with the correct weight.

Corollary 13.

/ Ξ n 1 Ξ n 1 / − + ∇ ◦ − 2 = 2 ◦ ∇

P X Ξ 1 Ξ 1 PX , ◦ 2 = − 2 ◦

f where X e− X . = 1 In particular, it follows that if ψ is a conformal Killing spinor on (M,g ), then e 2 f ψ is

a conformal Killing spinor on (M,g ).

4.3 Weyl-invariant spinorial conformal Killing transport

As in the case of conformal Killing vectors, we would like to modify the spinorial conformal Killing transport equation and make it transform covariantly under Weyl

transformations. This requires determining how the Weyl connection D acts on

spinors: that is, we must specify how spinors transform under the action of co(T M).

A spinor is said to have a weight w R if for any A S h Id co(T M), Aψ ∈ = + ∈ = 1 S ψ whψ, where S so(T M) has been identiﬁed with the corresponding two- − 2 · + ∈ 2 form in Λ T ∗M and we use the identiﬁcation of the spin representation of S with

37 1 times the Clifford multiplication by S. As before, we will denote the bundle of − 2 spinors with weight w by S[w] and the Weyl connection acting on it by Dw . Using

these deﬁnitions and equation (3.34), it is easy to see how Dw acts on S[w]:

Dw ψ ψ 1 X θ ψ wθ(X )ψ X = ∇X + 2 ∧ · − ψ 1 X θ ψ ( 1 w)θ(X )ψ . = ∇X + 2 · · + 2 −

The Clifford trace of the last equation gives an expression for the corresponding

Dirac operator:

D/ w ψ / ψ ( 1 n w)θ ψ . (4.16) = ∇ + 2 − 2 − ·

Finally, combining these two results, we obtain the Penrose operator P D,w associ-

ated to the Weyl connection D.

P D,w ψ P ψ ( 1 w)( 1 X θ ψ θ(X )ψ) . (4.17) X = X + 2 − n · · + 1 It is apparent from this expression that in fact, P D 2 P . This and Corollary 13 then X = X give rise to the following result:

Proposition 14. Conformal Killing spinors are in one-to-one correspondence with 1 1 [ ] [ ] sections of the bundle S 2 S − 2 which are parallel with respect to the following ⊕ connection: 1 ψ D 2 X ψ PD X · (4.18) X χ =  1  χ , , - 1 LD (X ) D− 2 , -  2 · X    1 1 2 with a conformal Killing spinor ψ determining a unique parallel section (ψ, n D/ ψ). In addition, PD has curvature

D W (X ,Y ) 0 RP (X ,Y ) PD [PD ,PD ] 1 . (4.19) = [X ,Y ] − X Y = − 2 C D(X ,Y ) W (X ,Y ) , -

38 Chapter 5

Conformal Killing superalgebras

In this section, our aim is to construct a superalgebra h h h associated to (M,g ,θ) = 0⊕ 1 which is also a conformal invariant. The natural object turns out to be the algebra

for which h0 consists of the normal conformal Killing vectors of (M,g ) and h1 is the

space of conformal Killing spinors. In this section we deﬁne the natural product

structure of this algebra. Unlike in the analogous Killing superalgebra case, we will

see that h is not in general a Lie superalgebra: one of the Jacobi identities of the algebra can fail. Nevertheless, we find that there are well-defined maps h h h , 0 × 0 → 0 h h h and h h h . In Chapter 3, we have already defined the natural Lie 0 × 1 → 1 1 × 1 → 0 bracket of conformal Killing vectors. In this chapter we define the two remaining

maps and study the structure of the superalgebra we thus obtain.

5.1 From conformal Killing spinors to conformal Killing vectors

We begin by determining the odd-odd bracket [ , ] : h h h . In other words, − − 1 × 1 → 0 1 1 2 [ ] [ ] we want to deﬁne a map [ , ] : S S 2 S − 2 mo(T M) which preserves paral- − − ⊕ → , - lel sections with respect to P and the Geroch connection D.

ψ Proposition 15. The map [ , ] : (X , A,β) is deﬁned by − − χ 0→ , - g (X ,Y ) (ψ,Y ψ) (5.1) = · g (Y , AZ ) 2(ψ,Y Z χ) (5.2) = · · β(Y ) 2(Y χ,χ) , (5.3) = · 39 when (ψ,χ) KerP. ∈ Proof. A natural way to deﬁne X is to take it to be the Dirac current X V . To = ψ,ψ obtain A, we differentiate it, obtaining

g (X ,Y ) g ( X ,Y ) g (X , Y ) ∇Z = ∇Z + ∇Z ( ψ,ψ) (ψ, Y ψ) (ψ,Y ψ) , = ∇Z + ∇Z · + ·∇Z and since X A Z and ψ Z χ, ∇Z = − X ∇Z = − · g (A Z ,Y ) (Z χ,Y ψ) (ψ,Y Z χ) X = · · + · · 2(ψ,Y Z χ) = ·· 2(ψ,Y Z χ) 2g (Y , Z )(ψ,χ) , = ∧ · − which also implies that if A S h Id, h 2(ψ,χ) and ω (Y , Z ) 2(ψ,Y Z X = X + X X = − SX = ∧ · χ).

Now

α (Y ) h X = ∇Y X 2( ψ,χ) 2(Ψ, χ) = − ∇Y − ∇Y 2(Y χ,χ) (ψ,L(Y ) ψ) , = · + · and since β α L(X ), we have X = X − β (Y ) 2(Y χ,χ) . (5.4) X = · Proposition 9 guarantees that (X , A,β) is parallel with respect to the Geroch connec-

tion D.

1 1 2 [ ] [ ] We extend the map deﬁned in the previous Proposition to S S 2 S − 2 using a ⊕ , - standard polarisation argument.

5.2 Spinorial Lie derivatives

The general question of constructing a Lie derivative for spinors was ﬁrst studied by Kosmann-Schwarzbach [27] and by Bourguignon and Gauduchon [28]. Bour- guignon and Gauduchon construct the so-called metric Lie derivative for spinor 40 ﬁelds. Computing the Lie derivative of a spinor with respect to a conformal Killing vector, however, requires comparing spinors on manifolds with different metrics.

Using the isomorphism for identifying the spinor bundles on manifolds with conformally related metrics which we introduced in Chapter 4, Section 4.2 makes it possible to deﬁne a Lie derivative in a classical manner — that is, along a parametrized curve generated by the vector ﬁeld, and this is the approach taken in [27, 20].

For us, however, this approach is not completely natural as we want to empha- sise the underlying algebraic structure of conformal Killing spinors and conformal

Killing vectors. Furthermore, there seems to be some confusion about the correct definition of the spinorial Lie derivative and its properties. Therefore, we choose to work from first principles and define our Lie derivative in terms of the connection

PD and the natural spinorial representation of the conformal Killing data +. We will

show that the action of this Lie derivative agrees with the Kosmann-Schwarzbach

Lie derivative.

Deﬁnition 16. By a spinorial Lie derivative we mean an endomorphism LX of sec- 1 1 [ ] [ ] tions of the local twistor bundle S 2 S − 2 associated to any conformal Killing ⊕ vector X , satisfying the following properties when X and Y are conformal Killing

vectors and Z is any vector ﬁeld:

(a) L (f ψ) X (f )ψ f L ψ, i.e. L is a derivation X = + X X

(b) [L ,L ] L , so the map X L is a homomorphism from the Lie X Y = [X ,Y ] 0→ X 1 [ ] algebra of conformal isometries to the Lie algebra of endomorphisms of S 2 ⊕ 1 [ ] S − 2 ;

A Lie derivative of a section of any vector bundle with respect to a vector ﬁeld can be

written as a sum of a connection and a suitable representation acting on the section. 1 1 [ ] [ ] Acting on the bundle S 2 S − 2 , a natural candidate is thus ⊕

L PD +(X , AD ,βD ) , (5.5) X = X + X X 41 where +(X ) is the spinor representation of mo(T M) deﬁned in (2.16). Clearly, this

Lie derivative satisﬁes property (a) in Deﬁnition 16. The other two properties can be

rewritten in the following way.

Proposition 17. The operator LX deﬁned above is a spinorial Lie derivative if for

every X ,Y conformal Killing vectors, the following equalities hold:

D [PD ,+(Y )] RP (X ,Y ) R+(X ,Y ) , (5.6) X = =

where

R+(X ,Y ) : +([X ,Y ]) [+(X ),+(Y )] . (5.7) = −

Proof. The properties that need to be checked are (b) and (c). We begin with (c) and

compute

L ,PD PD PD ,PD PD +(X ),PD X Y − [X ,Y ] = X Y − [X ,Y ] + Y 2 3 2 D 3 2 3 RP (Y , X ) PD ,+(X ) , = − Y 2 3 which clearly vanishes if the ﬁrst equality in (5.6) is satisﬁed.

Similarly, for the property (b) we can compute

L [L ,L ] PD +([X ,Y ]) PD ,PD PD ,+(Y ) +(X ),PD +(X ),+(Y ) , [X ,Y ] − X Y = [X ,Y ] + − X Y − X − Y − 2 3 2 3 2 3 2 3 which also vanishes provided equation (5.6) holds.

Using the deﬁnition of + and the Weyl-invariant bracket of conformal vector ﬁelds

D (3.53), one ﬁnds that R+(X ,Y ) RP (X ,Y ) as required. However, the ﬁrst equality = fails to hold. Instead, we have:

W (X ,Y ) 0 [PD ,+(Y )] 1 , X = − 2 C D(X ,Y ) W (X ,Y ) ,− - The offending term is the lower left-hand corner, which has the wrong sign. This

means that neither property (b) or (c) are satisﬁed unless C D (X ,Y ) 0. Since we = are primarily interested in conformal Killing spinors in supergravity, we might be

content with this, since in the absence of fluxes and cosmological constant terms, 42 the supergravity field equations require (M,g ) to be Ricci-flat, in which case this

condition is automatically satisﬁed. It seems counterintuitive that such a natural

candidate for the spinorial Lie derivative should fail in the general case.

However, equation (5.6) is satisﬁed for normal conformal Killing vectors: this is obvious since (recalling equation (3.40)) C D(X ,Y ) C(X ,Y ) W (X ,Y )θ. Thus, it is = −

natural to deﬁne h0 as the Lie algebra of normal conformal Killing vectors on (M,g ).

This is admittedly a strong restriction, but without it we have little hope of ﬁnding a

well-deﬁned conformal Killing superalgebra.

We now show that conformal Killing vectors arising as Dirac currents of conformal

Killing spinors are actually normal, so that the bracket [ , ] : S2h h is well- − − 1 → 0 deﬁned.

Proposition 18. Let X V be a conformal Killing vector obtained as the Dirac = ψ,ψ current of a conformal Killing spinor ψ. Then X is a normal conformal Killing vector:

that is, W (X ,Y ) 0, where Y is any vector ﬁeld (not necessarily conformal). =

Proof. We begin by showing that W (X ,Y ) 0. From the conformal Killing transport = equations (3.17) we know that for any conformal Killing vector X ,

W (X ,Y ) A Y β X L(Y ) . (5.8) = −∇Y X + • X + •

We will show that if X is the Dirac current of a conformal Killing spinor ψ, the right-

hand side of this equation vanishes.

Recall that if X V , g (A Z ,U) 2(ψ,U Z χ). Differentiating this, we ﬁnd = ψ,ψ X = · ·

g (( A )Z ,U) 2(Y χ,U Z χ) (ψ,U Z L(Y ) ψ) . (5.9) ∇Y X = · · · + · · ·

Similarly — again using Proposition 15 — for the second term in (5.8) we have

g (U,(Y β )) g (U,β (Z )Y β (Y )Z g (Y , Z )β" ) • X = X + X − X g (U,Y )β (Z ) g (U, Z )b (Y ) g (Y , Z )β (U) = X + X − X 2g (U,Y )(Z χ,χ) 2g (U, Z )(Y χ,χ) 2g (Y , Z )(Y χ,χ) . = · + · − ·

43 Finally, the last term in equation (5.8) can be rewritten as

g (U,(X L(Y ))Z ) L(Y , Z )g (U, X ) L(X ,Y )g (U, Z ) L(U,Y )g (X , Z ) • = + − L(Y , Z )(ψ,U ψ) L(U,Y )(ψ, Z ψ) g (U, Z )(ψ,L(Y ) ψ) . = · − · + ·

The right-hand side of equation (5.9) can be rewritten using the fact that

U Z L(Y ) U Z L(Y ) g (U, Z )L(Y ) L(Y ,U)Z L(Y , Z )U , · · = ∧ ∧ − + − and

Z U Y Z U Y g (U, Z )Y g (Y , Z )U g (U,Y )Z , · · = ∧ ∧ − + − where we have repeatedly used the formula (2.10). Substituting these results into equation (5.9) and into (5.8), we ﬁnd that

g (W (X ,Y )Z ,U) (ψ,U Z L(Y ) ψ) 2(Z U Y χ,χ) . (5.10) = ∧ ∧ · + ∧ ∧ ·

It is possible to show that the right-hand side of this equation vanishes by symmetry.

For any spinor ψ, it holds that

(ψ,Γ ψ) ( 1)3.(Γ ψ,ψ) abc = − cba ( 1)3.(Γ ψ,ψ) = − − abc ( 1)4.(ψ,Γ ψ) = − − abc (ψ,Γ ψ) , = − abc

so in fact — given the assumptions we made about the spinor inner product —

any three-form constructed out of a spinor must vanish. Applying this to equation

(5.10), we see that

W (X ,Y ) 0 , =

as required.

Similarly, from the conformal Killing transport equations (3.17) we know that when

X is a conformal Killing vector,

C(X ,Y ) β L(Y ) A . (5.11) = −∇Y X + ◦ X 44 When X is the Dirac current of a conformal Killing spinor ψ, we compute (using

Proposition 15 as before):

( β )(Z ) 2(Z χ,χ) 2(Z χ, χ) ∇Y X = ·∇Y + · ∇Y 2(ψ,L(Y ) Z χ) = − · · 2&(Y ,e )(ψ,e Z χ) = − a a · · &(Y , A Z ) . = X

But the second term in (5.11) gives

(L(Y ) A )(Z ) &(Y , A Z ) , ◦ X = X

so the RHS of equation (5.11) vanishes and C(X ,Y ) 0 as required. =

We remark that it can be shown [13] that the normal conformal Cartan connection 1 1 [ [ ] induces the connection P on the bundle S 2 S − 2 as well as the connection deﬁn- ⊕ ing normal conformal Killing vectors we mentioned in section 3.6. In this sense,

both h0 and h1 originate as parallel sections of the same connection acting on dif-

ferent vector bundles on M, and thus it is not surprising that there exists a natural

algebraic structure involving them both. Since there is no analogue of the normal

conformal Cartan connection in the supergravity case which we will discuss later,

we forego presenting this uniﬁed viewpoint in more detail and refer the interested

reader to the literature [29, 30].

Note that because of the Weyl-invariant way we have deﬁned both conformal Killing

transport, its spinorial counterpart and the Lie derivative L , h is manifestly a con-

formal invariant of (M,g ).

5.3 The Kosmann-Schwarzbach Lie derivative

We now introduce the deﬁnition of the spinorial Lie derivative that has become standard in the literature [20, 27, 28].

45 ˆ Deﬁnition 19. The Kosmann-Schwarzbach Lie derivative LX of a spinor ψ with re-

spect to a conformal Killing vector ﬁeld X is deﬁned as follows [27]:

1 Lˆ ψ ψ 1 S ψ 1 h ψ σ 2 (A ) . (5.12) X = ∇X − 2 X · + 2 X = ∇X + X

The Kosmann-Schwarzbach Lie derivative fails to respect the Clifford product, as

the following Lemma shows.

Lemma 20. The Kosmann-Schwarzbach Lie derivative has the following properties

with respect to Clifford multiplication when X is a conformal Killing vector, ψ is a

conformal Killing spinor, Y is an arbitrary vector ﬁeld and η is a p-form.

(a) Lˆ (Y ψ) L Y ψ Y Lˆ ψ h Y ψ X · = X · + · X − X ·

(b) Lˆ (η ψ) L η ψ η Lˆ ψ ph η ψ X · = X · + · X + X ·

Proof. (a): Recall that we can write L Y [X ,Y ] Y X Y A Y , X = = ∇X − ∇Y = ∇X + X where A S h Id. Using Deﬁnition 19, the properties of the spin connection X = X + X and equation (2.14), we compute

Lˆ (Y ψ) Y ψ Y ψ 1 S Y ψ 1 h Y ψ X · = ∇X · + ·∇X − 2 X · · + 2 X · Y ψ [ 1 S ,Y ] ψ Y Lˆ ψ = ∇X · + − 2 X · + · X Y ψ S Y ψ Y Lˆ ψ = ∇X · + X · + · X Y ψ A Y ψ h Y ψ Y Lˆ ψ = ∇X · + X · − X · + · X L Y ψ Y Lˆ ψ h Y ψ . = X · + + · X − X ·

(b): Now equation (2.1) implies that if β is a one-form, L β β β A . This X = ∇X − ◦ X

leads to an almost identical calculation as above, except that now the remaining hX term has a different sign. It is straightforward to extend the result to p-forms by linearity.

Given the suggestive name of this operator, it is not surprising that we can prove the following proposition.

46 Proposition 21. The Kosmann-Schwarzbach Lie derivative Lˆ is a spinorial Lie deriva-

tive in the sense of Deﬁnition 16.

Proof. Property (a) in Deﬁnition 16 is again obvious.

(b): We simply compute

1 1 1 [L ,L ] [ , ] [ ,σ 2 (A )] [ ,σ 2 (A )] σ 2 ([A , A ]) X Y = ∇X ∇Y + ∇X Y − ∇Y X + X Y 1 1 1 4 4 R(X ,Y ) σ 2 ( A ) σ 2 ( A ) σ 2 ([A , A ]) = − + ∇[X ,Y ] + ∇X Y − ∇Y X + X Y R(X ,Y ) = − + ∇[X ,Y ] 1 σ 2 R(X ,Y ) X α R(Y , X ) Y α A R(X ,Y ) X α Y α + + • Y − − • X + [X ,Y ] − − • Y + • X ! 1 " σ 2 (A ) = ∇[X ,Y ] + [X ,Y ] L , = [X ,Y ] 1 where we have4used the identity R∇(X ,Y ) R(X ,Y ), the conformal Killing trans- − 2 · = port equations and the Lie bracket for conformal Killing data deﬁned in equation

(3.23).

(c): We begin by computing the commutator of L and the spin connection . Now ∇ 1 4 [L , ] [ , ] [σ 2 (A ), ] X ∇Y = ∇X ∇Y + X ∇Y 1 4 R(X ,Y ) σ 2 ( A ) = − + ∇[X ,Y ] − ∇Y X 1 R(X ,Y ) σ 2 (R(X ,Y ) Y α ) = − + ∇[X ,Y ] − + • X 1 Y α . = ∇[X ,Y ] + 2 · X ·

Using Lemma 20, we can now show that

[L , Z ! ] L Z ! Z ! [L , ] h Z , (5.13) X ·∇Y = X ·∇Y + · X ∇Y + X ·∇Y 4 4 and taking the trace of this equation over Y and Z , we obtain the following result for

the Dirac operator / : ∇

[L , / ] L ea ea n α h / . (5.14) X ∇ = X ·∇a + ·∇[X ,ea ] − 2 X ·+ X ∇ 4 Combining these results and using Lemma 20 again, it follows that

[L ,P ] P . (5.15) X Y = [X ,Y ] 4 47 Thus, L preserves the space of conformal Killing spinors on M.

Becaus4e of Proposition 21 , the Kosmann-Schwarzbach Lie derivative would also appear to be a natural candidate for a spinorial Lie derivative. However, it only contains the T M- and the co(T M)-components of the conformal Killing data deﬁning

X .

Given that we have shown that there is a well-deﬁned action of (X , A,β) on the local 1 1 [ ] [ ] twistor bundle S 2 S − 2 — including the one-form part — our deﬁnition appears ⊕ more natural, even if there is an obstruction (proportional to the Cotton-York tensor

C D) for it to be a spinorial Lie derivative. 1 [ ] We now show that in fact the actions of LX and LX on S 2 agree when X is nor-

mal, so that there is no ambiguity in the w4ay we deﬁne the even-odd bracket of the conformal Killing superalgebra h. 1 1 [ ] [ ] Suppose that X is normal and (ψ,χ) is any section of S 2 S − 2 . Then ⊕ 1 1 ψ D 2 χ X χ σ 2 (AD )ψ X χ X + · + X − · LX  1 1  χ = 1 D 2 1 D D , - L (X ) D− χ β ψ σ− 2 (A )χ  2 + X + 2 X · + X   1  X ψ σ 2 (AX )ψ ∇ 1+ , =  1  χ σ− 2 (A )χ α ψ ∇X + X + 2 X · where we have used the deﬁnitions of the Weyl-invariant conformal Killing data and 1 the Weyl connection from Chapter 3, Section 3.5. The action on the S 2 -component

clearly agrees with the Kosmann-Schwarzbach Lie derivative. Since L and L both 1 [ ] preserve the kernel of P, it is also clear that the induced action on the S4− 2 -component ψ agrees with that of L when deﬁnes a conformal Killing spinor. χ , - When we have explicit expressions for conformal Killing spinors,the Kosmann-Schwarzbach

Lie derivative L is often more convenient, whereas the “natural” spinorial Lie deriva-

tive L makes t4he uniﬁed origin of conformal Killing vectors and conformal Killing spinors as parallel sections of bundles with a natural algebraic Lie bracket more

manifest. Since their actions agree on conformal Killing spinors, for the remain-

der of this thesis we will denote both Lie derivatives simply by L , trusting that it

will be apparent from context which one we are using. 48 5.4 The Jacobi identities

The Jacobi identity for a Lie superalgebra g g g can be written as = 0 ⊕ 1

X Y [X ,[Y , Z ]] [[X ,Y ], Z ] ( 1)| || | [Y ,[X , Z ]] , (5.16) = + − where X ,Y , Z g are homogeneous elements and X denotes the degree of X . In ∈ | | this section we show that the conformal Killing superalgebra h is not a Lie superal-

gebra in general. We will examine each of the four possible Jacobi identities in turn.

We already know that the natural Lie bracket on h0 satisﬁes the Jacobi identity, since

for normal conformal Killing vectors the Lie bracket (3.53) reduces to the natural al-

gebraic Lie bracket on mo(T M). We have also shown that the even-even-odd Jacobi

identity holds: it is easy to see that this is equivalent to property (b) in Deﬁnition 16.

Assume that Ψ : (ψ,χ) deﬁnes a conformal Killing spinor. Checking the even-odd- = odd Jacobi identity amounts to showing that

[X ,[Ψ,Ψ]] [L Ψ,Ψ] [Ψ,L Ψ] . (5.17) = X + X

It is enough to check this identity for the T M-component, since we are dealing with

D-parallel sections of mo(T M). The other components are then ﬁxed by the confor-

mal Killing transport equations. We denote the Dirac current of ψ by V : V and = ψ,ψ

the T M-component of [LX Ψ,Ψ] by VLX ψ,ψ . Let us begin by computing the T M-component of the left-hand side of equation

(5.17). We recall that [X ,V ] A V B X (where B V ) and obtain = X − V V = −∇

g (A V, Z ) g (B X , Z ) g (A V, Z ) 2(ψ, Z X χ) X − V = X − · · g (V,S Z ) h g (V, Z ) 2(ψ, Z X χ) = − X + X − · · (ψ,S Z ψ) h (ψ, Z ψ) 2(ψ, Z X χ) . = − X · + X · − · ·

On the other hand, computing the right-hand side of (5.17) yields

1 1 g (V , Z ) g (V ) (σ 2 (A )ψ, Z ψ) (X χ, Z ψ) (ψ, Z σ 2 (A )ψ) (ψ, Z X χ) LX ψ + ψ,LX ψ = X · − · · + · X − · · (ψ,[ 1 S , Z ] ψ) h (ψ, Z ψ) 2(ψ, Z X χ) , = − − 2 X · + X · − · · 49 where we have used the fact that the adjoint of a two-form η with respect to the

spinor inner product is η and equation (2.14). The even-odd-odd Jacobi identity is − thus satisﬁed.

By a standard polarisation argument, the vanishing of the odd-odd-odd Jacobi iden-

tity would be equivalent to ψ L 0. (5.18) Vψ,ψ χ = , - Unfortunately, we ﬁnd that

1 ψ σ 2 (AV )ψ V χ LVψ,ψ −1 · . χ =  1  , - βV ψ σ− 2 (AV )χ − 2 · +   The vanishing of this expression is equivalent to

(ψ,Γ ψ)Γ χ (ψ,Γ χ)ψ (ψ,χ)ψ 0 a a + ab + = (Γ χ,χ)Γ ψ (ψ,Γ χ)Γ χ (ψ,χ)χ 0 . a a + ab ab − =

However, the expressions on the left-hand side do not vanish in general for arbitrary spinors ψ,χ, and thus we are forced to conclude that h is not necessarily a Lie

superalgebra. This result is not new: Habermann [20] studies the algebra of con-

formal Killing vectors and conformal Killing spinors and presents an explicit (non-

complete) example where the fourth Jacobi identity is not satisﬁed. However, from

the above considerations it is evident that the fourth Jacobi identity may fail purely

for algebraic reasons.

5.5 The Minkowski conformal Killing superalgebra

In this section, we exhibit the simplest possible example of a conformal Killing su-

1,n 1 peralgebra: that of the ﬂat Minkowski space (R − ,ηab). For convenience, we com-

pute the brackets of the algebra using the Kosmann-Schwarzbach Lie derivative and

the Lie bracket of vector ﬁelds, although of course we could have used the natural

1,n 1 bracket of mo(R − ) and the spinorial Lie derivative L as well.

1,n 1 a The conformal algebra of the Minkowski space R − with coordinates x and the

50 standard ﬂat metric ηab is generated by Pa ,Mab,D,Ka — corresponding to transla-

tions, rotations, the dilatation and the special conformal transformations.

P ∂ , (5.19) a = a M x ∂ x ∂ , (5.20) ab = a b − b a D xa∂ , (5.21) = a K 2x xb∂ (x,x)∂ . (5.22) a = a b − a

The even part of the algebra is given by

[P ,P ] 0, (5.23) a b = [M ,P ] η P η P , (5.24) ab c = bc a − ac b [M ,M ] η M η M η M η M , (5.25) ab cd = bc ad − ac bd − bd ac + ad bc [P ,D] P , (5.26) a = a [K ,D] K , (5.27) a = − a [P ,K ] 2η D 2M , (5.28) a b = ab − ab [M ,K ] η K η K . (5.29) ab c = bc a − ac b

The conformal Killing spinor equation is readily solved using Lemma 10: / ψ 0, ∇ ∇ = so / ψ χ, a constant spinor. Substituting this into the equation P ψ 0,!we h"ave ∇ = X =

ψ ψ x χ, (5.30) = 0 + ·

where x χ : xa Γ χ and ψ ,χ are arbitrary constant spinors. It is straightforward · = a 0 to compute the action of the conformal algebra generators on ψ via the Kosmann-

Schwarzbach spinorial Lie derivative. This yields

L ψ Γ ψ , (5.31) Pa = a 2 L ψ 1 M ψ 1 x M ψ , (5.32) Mab = − 2 ab · 1 − 2 · ab · 2 L ψ 1 ψ 1 x ψ , (5.33) D = − 2 1 + 2 · 2 L ψ x Γ ψ . (5.34) Ka = · a 1 51 Finally, we compute the Dirac current V : V associated to a conformal Killing = ψ,ψ spinor ψ. In local coordinates, we have

V a (ψ,Γa ψ) = (ψ ,Γaψ ) xb (ψ ,ΓaΓ χ) xb (Γ χ,Γa ψ ) xb xc (Γ χ,Γa Γ χ) = 0 0 + 0 b + b 0 + b c (ψ ,Γaψ ) 2xb(ψ ,Γa χ) 2xa (ψ ,χ) xb xc (Γ bχ,ΓaΓ χ) = 0 0 + 0 b − 0 + b c (ψ ,Γaψ ) 2xb(ψ ,Γa χ) 2xa (ψ ,χ) 2xa xc (Γ χ,χ) x 2(Γaχ,χ) , = 0 0 + 0 b − 0 − c + | |

where we have used the Clifford algebra identity Γ Γ Γ Γ η Γ η Γ abc = a b c − ac b + ab c +

ηcbΓa and the fact that the three-form constructed out of the spinor χ vanishes due to the symmetry properties of the spinor inner product. We can rewrite this in terms of the conformal Lie algebra generators as

V (ψ ,Γaψ )∂ 2(ψ ,Γabχ)M 2(ψ ,χ)D (Γa χ,χ)K . (5.35) ψ,ψ = 0 0 a + 0 ab − 0 − a

In particular, this shows that all the conformal Killing vectors of Minkowski space

are normal, since they arise as Dirac currents of conformal Killing spinors.

52 Chapter 6

Conformal Killing spinors in M-theory

In this chapter we generalise the concept of conformal Killing spinors to eleven- dimensional supergravity. We show that M-theory backgrounds that admit a su-

pergravity conformal Killing spinor distinct from supergravity Killing spinors and geometric conformal Killing spinors must be of a very particular type: the metric must be one of the so-called Bryant metrics and the four-form must satisfy a strong integrability condition.

6.1 M-theory backgrounds

A (bosonic) background of eleven-dimensional supergravity is a triple (M,g ,F),where

M is an 11-dimensional Lorentzian spin manifold with metric g , and F is a closed

four-form subject to the following equation:

d + F 1 F F (6.1) = 2 ∧

There is also an Einstein-type equation relating the Ricci curvature of g to the stress-

energy tensor of F.

r (X ,Y ) 1 ι F,ι F 1 g (X ,Y ) F 2 , (6.2) = 2 〈 X Y 〉 − 6 | |

where , is the inner product on p-forms induced by the metric g . 〈− −〉 We remark that equations (6.1) and (6.2) actually arise as the Euler-Lagrange equa-

tions of the eleven-dimensional supergravity action: we will not need the explicit

form of the action here, and refer the interested readed to e.g. [5] for details.

53 For future reference we note that equation (6.2) also implies that the Schouten tensor L can also be expressed in terms of F. A straightforward computation using the deﬁnition of L and equation (6.2) gives

L(X ,Y ) 1 ι F,ι F 7 g (X ,Y ) F 2 . (6.3) = 18 〈 X Y 〉 − 360 | |

We remark that since the geometric conformal Killing spinors are conformally co-

variant objects, one might worry that the lack of conformal invariance in M-thery

might prevent one from deﬁning their supergravity analogue. It is known, however,

that the M-theory equations of motion (6.2), (6.1) admit a scaling symmetry

2 g λ− g 0→ (6.4) 3 F λ− F , 0→ where λ is a constant. It is easy to see that this transformation maps M-theory back-

grounds to other M-theory backgrounds since the equations of motion transform

homogeneously under (6.4). One might therefore expect that a M-theory back-

ground admits some sort of scale-invariant structure,characterised by scale-invariant

spinorial objects.

The spinors in M-theory are real and 32-dimensional. There are two possible Clif-

ford modules, both isomorphic to R32; we choose the one for which the action of the

eleven-dimensional volume element is nontrivial. The spinor inner product ( , ) is − − now symplectic and obeys (ψ,Y χ) (Y ψ,χ) for any vector ﬁeld Y . Moreover, if η · = − · is a p-form, its Clifford adjoint (considered as a spinor endomorphism) with respect

p(p 1) + + to the spinor inner product is η ( 1) 2 η. = − With these conventions, the supercovariant connection acting on S is given by

D 1 ι F 1 X ! F : Ω . (6.5) X = ∇X + 6 X + 12 ∧ = ∇X + X

The curvature of D is deﬁned in the usual way:

RD(X ,Y ) D [D ,D ] . (6.6) = [X ,Y ] − X Y

It is an important fact that the vanishing of the Clifford trace of RD considered as a

Clifford endomorphism is equivalent to the equations of motion. [31, 32]. In other 54 words,the identity

a D e R (X ,ea) 0 . (6.7) a · = + is equivalent to (??).

Unlike the Levi-Cività spin connection , D does not respect the Clifford product. ∇ Instead, we have the identity

D Z ψ Z ψ Z D ψ 1 Z ! ι F ψ 1 Z ! Y ! F ψ . (6.8) Y · = ∇Y · + · Y − 3 ∧ Y · − 6 ∧ ∧ · ! " 6.2 The M-theory Penrose operator

We wish to consider spinors in the kernel of a Penrose-type operator deﬁned using the supercovariant connection. Note that this can be thought of as the composition of the projection to the kernel of Clifford multiplication with the supercovariant connection. We therefore deﬁne

P D 1 Y D/ , (6.9) Y = Y + 11 ·

where D/ ea D / 1 F is the Dirac operator associated to the supercovari- = a · a = ∇+ 12 · ant connec5tion. We note that P can be written as a sum of the geometric Penrose

operator and the F-dependent terms as

P ψ P ψ Ω ψ 1 X F ψ . X = X + X + 121 · ·

We call the spinors in the kernel of P supergravity conformal Killing spinors (SCKS)1.

Since this is a somewhat unwieldy term, for the rest of the chapter we refer to them

simply as conformal Killing spinors, making the distinction between them and geo-

metric conformal Killing spinors when necessary.

The reason for this nomenclature is that spinors in the kernel of P have the proper-

ties that we would expect supergravity generalisations of conformal Killing spinors

to have. The D-parallel spinors – the supergravity Killing spinors – are obviously in

KerP, for instance. In addition, when F 0, supergravity conformal Killing spinors = 1With some reluctance, we refrain from introducing the acronym SUCKS, suggested by José Figueroa-O’Farrill. Nevertheless, look out for a forthcoming paper [33].

55 reduce to geometric conformal Killing spinors. We now show that the Dirac currents of supergravity twistor spinors are conformal Killing vectors, obtaining a M-theory analogue of Proposition 9.

Proposition 22. Suppose that ψ,χ KerP are supergravity conformal Killing spinors. ∈

Then the Dirac current Vψ,χ is a conformal Killing vector.

Proof. We proceed in a similar fashion as in the geometric case, taking the covariant

derivative of the Dirac current.

g ( V , Z ) ( ψ,Y χ) (ψ, Z χ) ∇Y ψ,χ = ∇Y · + ·∇Y 1 (ι F ψ, Z χ) 1 (ψ, Z ι F χ) 1 (Y ! F ψ, Z χ) = − 6 Y · · − 6 · Y · − 11 ∧ · · 1 (ψ, Z Y ! F χ) 1 (Y D/ ψ, Z χ) 1 (ψ, Z Y D/ χ) = − 12 · ∧ · − 11 · · − 11 · · 1 (ι ι F ϕ ,χ) 1 (Z ! Y ! F ψ,χ) = − 3 Z Y · 1 + 6 ∧ ∧ · 1 (Z Y D/ ψ,χ) (Y Z ψ,D/ χ) = + 11 · · − · · ! " where we have made use of the fact that the Clifford adjoints of the terms appearing

in Ω are (ι F)+ ι F and (Y ! F)+ Y ! F. The ﬁrst two terms in the ﬁnal Y Y = Y ∧ = − ∧ expression are manifestly antisymmetric, so antisymmetrising it gives

g ( V , Z ) g (Y , V ) 2h g (Y , Z ) , ∇Y ψ,χ + ∇Z ψ,χ = − Vψ,χ where h 1 (ψ,D/ χ) (D/ ψ,χ) . In other words, the conformal Killing equa- Vψ,χ = − 11 − tion is satisﬁed. 2 3

6.3 Supercovariant conformal Killing transport

We will now ﬁnd a connection that allows us to identify KerP with parallel sections

of S S. ⊕

Lemma 23. If ψ is a conformal Killing spinor, D/ ψ satisﬁes the following identity:

D D/ ψ 1 D/ 2 ψ 7 ι F D/ ψ 2 X ! F D/ ψ. (6.10) X = 9 + 27 X · + 27 ∧ ·

56 Proof. Suppose ea is again a geodesic frame as in the proof of Lemma 10. We differ-

entiate the conformal Killing spinor equation at a point p M, obtaining ∈

D D ψ 1 D e D/ ψ 0, X a + 11 X a · = ! " ! " D D ψ 1 D X D/ ψ 0 . a X + 11 i · = ! " ! " Subtracting the second equation from the ﬁrst and applying (6.8),we obtain

0 RD(X ,e )ψ 1 (e D D/ ψ 1 ea ι F D/ ψ = a + 11 a · X − 3 ∧ X · 1 ea X ! F D/ ψ X D D/ ψ 1 X ! ι F D/ ψ 1 X ! ea F D/ ψ) . − 6 ∧ ∧ · − · a − 3 ∧ a · − 6 ∧ ∧ ·

Taking the Clifford trace of this equation and using equations (2.12) and (6.7), this

yields

D D/ ψ 1 D/ 2 ψ 7 ι F D/ ψ 2 X ! F D/ ψ. (6.11) X = 9 + 27 X · + 27 ∧ ·

We can compute D/ 2 ψ by taking the Clifford trace of this equation:

D/ 2 ψ 14 F D/ ψ (6.12) = 60 ·

and substituting this expression back to (6.11) gives

D/ ψ 1 ι F D/ ψ 1 X ! F D/ ψ. ∇X = 15 X · + 60 ∧ ·

We can also write this as

D˜ D/ ψ 0 , X =

where D˜ 1 ι F 1 X ! F – a connection similar to the supercovariant con- X = ∇X − 15 X − 60 ∧ nection, but with different numerical coefﬁcients.

This immediately gives us the supercovariant version of Proposition 11.

Proposition 24. M-theory conformal Killing spinors are in one-to-one with with parallel sections of S S with respect to the connection ⊕ ψ D X ψ ℘ X · , (6.13) X χ = 0 D˜ χ , - , X -, -

57 1 with a conformal Killing spinor ψ uniquely determining a parallel section (ψ, 11 D/ ψ). Furthermore, ℘ has curvature

D 1 ! 1 ! 14 2 ! ! ℘ R (X ,Y ) 5 X ιY F 5 Y ιX F 30 ιX ιY F 15 X Y F R (X ,Y ) − ∧ + ∧ D˜ − − ∧ ∧ . = 6 0 R (X ,Y ) 7 (6.14)

Proof. The ﬁrst part follows from Lemma 23. Obtaining the curvature of ℘ is a

straightforward calculation using equation (6.8)and the usual Clifford product iden-

tities (2.10).

We observe that knowing the expression for the curvature (6.14) immediately allows us to determine when a (simply connected) M-theory background admits a maxi-

mal number of conformal Killing spinors. This happens when R℘ 0, which in turn = ˜ implies that RD and RD must vanish as well. The vanishing of RD means that the

background must be maximally supersymmetric. Such eleven-dimensional back-

grounds have been classiﬁed [34], up to local isometry, and in fact the only possibil-

ities are the AdS S7 and AdS S4 Freund-Rubin solutions, the maximally super- 4 × 7 × symmetric Hpp-wave and ﬂat Minkowski space.

The vanishing of the remaining component of the curvature gives

1 X ! ι F 1 Y ! ι F 14 ι ι F 2 X ! Y ! F 0. 5 ∧ Y − 5 ∧ X + 30 X Y + 15 ∧ ∧ =

But this implies that F 0 and hence the only remaining possibility is Minkowski = space.

What about the non-maximal case? We now show that an important corollary of

Proposition 24 is a very strong integrability condition.

Corollary 25. Let (ψ,χ) Ker℘ deﬁne a conformal Killing spinor on (M,g ,F). Then ∈ the spinor χ must in fact be parallel with respect to the Levi-Cività spin connection . ∇ In addition,

X ! F χ ι F χ F χ 0 . (6.15) ∧ · = X · = · =

58 Proof. Since (ψ,χ) is parallel with respect to ℘, we have that

ψ R℘(X ,Y ) 0 χ = , - for any X ,Y X (M). This is equivalent to ∈

RD(X ,Y )ψ 1 X ! ι F 1 Y ! ι F 14 ι ι F 2 X ! Y ! F χ , = 5 ∧ Y − 5 ∧ X + 30 X Y + 15 ∧ ∧ · ˜ . / RD(X ,Y )χ 0 . =

Taking the Clifford trace of the ﬁrst equation with respect to Y and using (6.7) again

gives (after some manipulation)

X ! F χ 0 . ∧ · =

Taking a Clifford trace of this equation and using (2.12) then gives F χ 0. This tells · = us that for any vector ﬁeld X , X F χ (X ! F ι F) χ ι F χ 0. Using equation · · = ∧ − X · = X · = (6.10), we obtain

D˜ χ D χ χ 0 . Z = Z = ∇Z =

M-theory backgrounds that admit conformal Killing spinors must thus be rather special. Not only must they admit solutions of the conformal Killing spinor equation, they must also possess (supergravity) Killing spinors χ which are parallel.

We thus have a set of necessary (but not sufﬁcient) conditions for a M-theory background M to admit supergravity conformal Killing spinors. If χ 1 D/ ψ vanishes, = 11 the supergravity conformal Killing equation reduces to the usual supergravity Killing

equation. We already know that if F 0, the supergravity Penrose operator agrees = with the geometric Penrose operator. The remaining possibility (that allows for the

existence of supergravity conformal Killing spinors distinct from geometric confor-

mal Killing spinors or supergravity Killing spinors) is that χ is nonzero and parallel

(so that M has constrained holonomy), F is nonzero and moreover satisﬁes the in-

tegrability condition (6.15).

We summarise these ﬁndings in the following proposition. 59 Corollary 26. If (ψ,χ) Ker℘ is a conformal Killing spinor on a M-theory back- ∈ ground (M,g ,F), then one of the following holds.

(a) M is Ricci-ﬂat and ψ is a geometric conformal Killing spinor. This occurs when

X ! F 0 for all X , which implies F 0. ∧ = =

(b) χ 1 D/ ψ 0 and thus P ψ D ψ 0; in other words, ψ is a supergravity = 11 = X = X = Killing spinor.

In the sequel we will mostly be interested in case (c) since the two other cases have

been studied extensively in the literature.

6.4 M-theory backgrounds admitting conformal Killing spinors

Using Corollary 25, we will now attempt to characterise M-theory backgrounds which admit non-trivial (i.e. distinct from geometric conformal Killing and supergravity

Killing) solutions to the SCKS equation.

Suppose that (M,g ,F) is a M-theory background and that in addition, (M,g ) admits a parallel spinor χ. Metrics of this type have been studied extensively for example in

[35, 36] and in supergravity context in [37].

It is well known that the existence of a parallel spinor constraints the holonomy of a manifold. In eleven dimensions, the subgroups H Spin(1,10) that leave a spinor ⊂ invariant have been classiﬁed by Bryant [35, 36]. There are two possibilities, distinguished by the type of the Dirac current Vχ of χ. Note that since χ is parallel, Vχ is

parallel as well.

As Bryant shows, if V is time-like, Hol(M) must be contained in SU(5) Spin(10). χ ⊂ This means that (M,g ) is locally isometric to a product R N with metric ×

g dt 2 h , (6.16) = − + 60 where (N,h) is any Calabi-Yau 5-fold. Such spacetimes are automatically Ricci-ﬂat

(as the product of a ﬂat direction with a Ricci-ﬂat manifold), which means that from

equation (6.2) we must have

ι F,ι F 0 , 〈 X Y 〉 = F 2 0 | | = for all vector ﬁelds X ,Y .

Considering the possible non-vanishing components that F can have in this case,

it is not hard to show that in fact F 0. In a pseudo-orthonormal frame e ,ei , the = ±

possible components of F are F i j ,F i jk ,F i jk ,Fi jkl . The condition ιX F,ιY F 0 +− − + 〈 〉 = then implies that (with summation over repeated indices implied):

i jk F i jk F 0 + + = i jk F i jk F 0 − + = i j F i j F 0 +− +− = F F i jkl 0 . i jkl = Since these are sums of squares, each term must in fact vanish separately and thus

all components of F vanish.

In other words, F 0 and any solutions of the SCKS equation are again geometric = conformal Killing spinors. R8 R The remaining possibility is that Vχ is null and Hol(M) Spin(7) ! . In this ⊂ × case it can be shown [12] that ! "

V χ 0 . (6.17) χ · = Since χ is parallel, we have χ 0 for any vector ﬁeld X , and iterating this equation ∇X = we ﬁnd the following integrability condition:

R(X ,Y ) χ 0 (6.18) · = for any vector ﬁelds X ,Y . Taking the Clifford trace of this equation and using the al-

gebraic Bianchi identity (2.21), we ﬁnd that or, using the Ricci map deﬁned in (2.23),

Ric(X ) χ 0 , (6.19) · = 61 which in turn implies that Ric(X ) 2χ 0. Since g (Ric(X ),Ric(Y )) 0 for all vector | | = = ﬁelds X ,Y , metrics of this type are often called Ricci-null.

The most general local metric in eleven dimensions admitting a parallel null spinor

is given by [36] :

2 9 2 i j g 2dx+dx− a(dx−) (dx ) h dx dx , (6.20) = + + + i j

where i, j 1...9 and and a is a function satisfying ∂ a 0 but otherwise arbitrary. = + =

hi j is now an x−-dependent family of metrics with holonomy contained in Spin(7) and with the property

∂ Υ λΥ Ψ , (6.21) − = +

where Υ is the self-dual Spin(7)-invariant Cayley 4-form, λ a smooth function of

i (x−,x ) and Ψ is an anti-selfdual 4-form. Bryant calls such metrics conformal anti- selfdual and shows that any one-parameter family of Spin(7)-metrics can be made to satisfy this property using diffeomorphisms[36].

Following [37], we want to couple the metric (6.20) to a four-form F satisfying the

M-theory equations of motion (6.1), (6.2) in addition to the SCKS integrability con-

ditions 25. Note that for the metric (6.20), the vector ∂ is parallel and null and in +

fact Vχ ∂ . ∝ + The only nonzero component of the Ricci tensor is R , which follows from the −− Ricci-null propety[37] and the fact that ∂ is parallel. Satisfying the Einstein equa- + tion then requires that the four-form F must be null. It is easy to see that the only

nonzero components of F are F i jk ; in other words, F must be of the form −

F dx− Θ , (6.22) = ∧

9 i where Θ is a 3-form on the transverse space N9 with coordinates x ,x : we refer to

[37] for details. The requirement dF 0 is satisﬁed provided that ∂ Θ 0 and that = + =

Θ is closed as a three-form on N9; note that it may still have x−-dependence. The

Maxwell equation (6.1) is satisﬁed [37] if d + Θ 0 on N , where + is the Hodge 9 = 9 9

dual operator on N9. 62 Finally, we want to make sure that the integrability conditions (6.15), necessary for the SCKS equation to admit solutions, are satisﬁed. Using the condition ι F χ 0 X · = and contracting F with ∂ implies that −

Θ χ 0 . · =

In addition, since Vχ ∂ and Vχ χ 0, we know that χ KerΓ . ∝ + · = ∈ + In summary:

Theorem 27. Let (ψ,χ) Ker℘ deﬁne a (non-Killing, non-geometric) SCKS on a M- ∈

theory background (M,g ,F). Then the vector Vχ is null, (M,g ) is Ricci-null and

Hol(M) Spin(7) !R8 R. Furthermore, locally the metric g can be written in ⊂ × ! " the form (6.20). The four-form F can be written as F dx− Θ, with Θ an x−- = ∧

dependent family of 3-forms which are closed and coclosed on N9. In addition, χ Γ ∈ + and Θ χ 0. · =

There are plentiful examples of this type[37], so provided that we equip them with suitable four-forms satisfying the integrability conditions (6.15), we expect to ﬁnd examples of non-Killing, non-geometric supergravity conformal Killing spinors out of which we also hope to construct a supergravity version of the conformal Killing superalgebra.

6.5 Conformal Killing spinors of Hpp-waves

We are now ready to solve the SCKS equation P ψ 0 on Hpp-waves, that is, the X = equations

∂ ψ 0 , (6.23) + = 1 i 1 1 ∂ ψ x Ai j Γi Γ ψ Θ ψ Γ+Γ−Θψ Γ χ 0 , (6.24) − + 2 + + 6 · + 12 + − = 1 1 ∂k ψ Γ ΘΓk Γ Γk Θ ψ Γk χ 0 , (6.25) + 8 + + 24 + · + =

where χ KerΓ is a constant spinor that also satisﬁes Θ χ 0. The ﬁrst equation ∈ + · = i simply implies that ψ ψ(x−,x ). It is convenient to decompose ψ as ψ ψ ψ , = = + + − 63 where ψ KerΓ . Thus we can rewrite the last equation as ± ∈ ±

1 1 ∂k ψ ∂k ψ ΘΓk Γ ψ Γk ΘΓ ψ Γk χ 0 . + + − + 8 + − + 24 + − + =

It immediately follows that ∂k ψ 0. Taking ∂j of the previous equation also shows − =

that ∂j ∂k ψ 0, which means that ψ depends at most linearly on the transverse + = + coordinates xi . We can therefore decompose it as

i 1 1 ψ (x−,x ) ϕ(x−) Θ x⊥ Γ ψ x⊥ ΘΓ ψ x⊥ χ , (6.26) + = − 8 · · + − − 24 · + − − ·

i where x⊥ : x Γi and ϕ KerΓ . · = ∈ + Next, we look at the KerΓ -component of the ∂ ψ equation. We ﬁnd − −

∂ ψ 1 Θ ψ 1 Γ Γ Θ ψ Γ χ Γ χ , − − = − 6 · − − 12 − + · − − − = − −

because Γ Γ ψ 2ψ . We can immediately integrate this equation and ﬁnd − + − = − −

ψ ζ0 x−Γ χ , − = − −

where ζ0 KerΓ is a constant spinor. We substitute the expressions for ψ ,ψ into ∈ − + − the remaining equation and obtain

1 1 1 i 0 ϕ6 Θ ϕ Θ x⊥ χ x Ai j Γi Γ ζ0 = + 6 · − 4 · · + 2 + i 1 2 1 2 x Ai j Γj x−χ Θ x⊥ Γ ζ0 Θ x⊥ x−χ + − 48 · · + − 24 · 1 1 Θ x⊥ ΘΓ ζ0 Θ x⊥ χ . − 144 · · + − 6 · ·

This equation contains three kinds terms: those depending only on x−, those de-

pending on xi and those depending on both. Taking derivatives, it is easy to see that

all three must vanish separately. The terms depending solely on x− give an equation

for ϕ,

1 ϕ6 Θ ϕ , = − 6 ·

which is readily solved to give

x− Θ ϕ e− 6 ϕ , = 0

where ϕ0 KerΓ is a constant spinor. ∈ + 64 The vanishing of the terms depending on x− and x− both implies that

A Γ χ 1 Θ2Γ χ , (6.27) i j j = 24 i

and the vanishing of the remaining terms requires that

5 1 2 1 ΘΓi χ Ai j Γj Γ ζ0 Θ Γi Γ ζ0 ΘΓi ΘΓ ζ0 0 . (6.28) − 6 + + − 24 + − 72 + =

Recall that the M-theory equations of motion are satisﬁed when Tr A 1 Θ 2. Tak- = − 2 | | ing the trace of equation (6.27), we ﬁnd that

Θ 2χ 1 Θ Θ Γ Γ Γ χ | | = 96 jkn nlm i jklm i (6.29) 1 Θ Θ Γ χ , = − 96 jkn nlm jklm where we have used a C&(9) Fierz identity Γ Γ Γ Γ . But recall that Θχ i jklm i = − jklm = 0. This means that the Clifford square of Θ acting on χ must vanish as well, which

implies

Θ 2χ 1 Θ Θ Γ , | | = 4 jkn nlm jklm

since Θ is a 3-form. Clearly, this contradicts (6.29), unless either Θ 0 or χ 0. We = = have thus estabilished the following somewhat disappointing result:

Proposition 28. Let (M,g ,F dx− Θ) be a supersymmetric Hpp-wave solution of = ∧ M-theory and (ψ,χ) deﬁne a SCKS. Then one of the following holds.

(a) Θ 0 and ψ is a geometric conformal Killing spinor. =

(b) χ 0 and ψ is a (supergravity) Killing spinor (Dψ 0). = =

65 Chapter 7

Conformal Killing superalgebras in M-theory

We now turn to the question of deﬁning a supergravity analogue of the conformal

Killing superalgebra introduced in Chapter 5. In particular, we will deﬁne a supercovariant conformal Killing superalgebra associated to a M-theory background

(M,g ,F) consisting of supergravity conformal Killing spinors and so-called super-

normal conformal Killing vectors of M.

7.1 M-theory conformal Killing spinors

As in the geometric case, we would also like to construct a map from S2(S S) to sec- ⊕ tions of mo(T M) which maps parallel sections with respect to ℘ to parallel sections

of the Geroch connection D.

We can easily formulate the analogue of Proposition 15 with slight modiﬁcations

involving F-dependent terms.

Proposition 29. The map [ , ] : (ψ,χ) (X , A,β) is deﬁned by the following equa- − − 0→ tions

g (X ,Y ) (ψ,Y ψ) = · g (A Z ,Y ) 2(ψ,Y Z χ) 2(ψ,Y Ω ψ) X = · · + · Z · β (Y ) 2(Y χ,χ) L(X ,Y ) . X = · −

Proof. Let (ψ,χ) Ker℘. As before, we take X to be the Dirac current of ψ and dif- ∈ 66 ferentiate it, obtaining

g ( X ,Y ) ( ψ,Y ψ) (ψ,Y ψ) ∇Z = ∇Z · + ·∇Z (Z χ,Y ψ) (ψ,Y Z χ) (Ω ψ,Y ψ) (ψ,Y Ω ψ) . = − · · − · · − Z · · − · Z ·

and using the properties of the symplectic spinor inner product gives the result.

Now h 2(ψ,χ), and thus we ﬁnd the one-form α by differentiating X = − X

α (Y ) h 2(Y χ,χ) 2(Ω ψ,χ) , X = ∇Y x = · + Y ·

since χ is parallel, and in fact taking the Clifford adjoint of ΩY and using the in-

tegrability condition in Corollary 25, we see the last term in the previous equation

vanishes. Thus, α (Y ) 2(Y χ,χ) and using the deﬁnition of β, we arrive at the X = · desired result.

7.2 Supernormal conformal Killing vectors

In order to be able to construct a well-deﬁned superalgebra from M-theory conformal Killing spinors, we must again show that there is a special ideal of conformal

Killing vectors X for which C(X ,Y ) 0 for any vector ﬁeld Y — recall that this is = required for L to be a homomorphism from the algebra of vector ﬁelds to a subal-

gebra of mo(T M). To get an idea of what we should require from these vectors, we now determine if Dirac currents of M-theory conformal Killing spinors satisfy this condition.

Recall that the conformal Killing transport equations (3.16) imply that if X is a conformal Killing vector and Y is any vector ﬁeld,

C(X ,Y ) β L(Y ) A . (7.1) = −∇Y X − ◦ X

Now suppose (ψ,χ) Ker℘ deﬁne a conformal Killing spinor and X is the Dirac ∈

67 current of ψ. Then we have

C(X ,Y )Z 2(Z χ,χ) L(X , Z ) L(X , Z ) L(Y , A Z ) = −∇Y · − − ∇Y − X 2 3 ( L)(X , Z ) L( X , Z ) L(Y , X ) = ∇Y + ∇Y + ∇Z ( L)(X , Z ) (A L)(Y , Z ) = ∇Y + X (L L)(Y , Z ) ( L)(X , Z ) ( L)(Y , Z ) , = X + ∇Y − ∇X

where we recognise the last two terms as C(X ,Y )Z . Thus, we see that −

C(X ,Y )Z 1 (L L)(Y , Z ) . (7.2) = 2 X

This suggests that a natural choice for the analogue of normal conformal Killing

vectors in supergravity context would be the CKVs for which L L 0 . We can state X = this requirement in a slightly more M-theoretic way by rewriting expression (6.3) in

local coordinates as

1 mn pq r s 7 kl mn pq r s Lab g g g Fmpr aFnqsb g g g g Fkmnpr Flnqs , (7.3) = 108 − 4! 360 × where g ab are the components of the inverse metric. Since L g ab 2h g ab (which X = X 1 is easy to see by taking the Lie derivative of Id g g − ), the Lie derivative of L = ab vanishes if L F 3h F. This motivates the following deﬁnition. X = − X

Deﬁnition 30. Let (M,g ,F) be a M-theory background. If X is a conformal Killing

vector of (M,g ) and in addition,

L F 3h F , (7.4) X = − X

we call X a supernormal conformal Killing vector.

Note that a supernormal conformal Killing vector is not necessarily normal in the

same sense as in the geometric case, since it might not correspond to a parallel sec-

tion with respect to the normal conformal Cartan connection (even if it does deﬁne

a parallel section of mo(T M) with respect to the Geroch connection).

It is clear that supernormal conformal Killing vector ﬁelds form a subalgebra of the

conformal Lie algebra of M, since if X and Y are supernormal, L L (L L [X ,Y ] = X Y − 68 L L )L 0. Note, however, that unlike for normal conformal Killing vectors, their Y X = Lie bracket does not reduce to the algebraic one since W (X ,Y ) does not necessarily

vanish even when X and Y are supernormal.

We can motivate this deﬁnition further with the following proposition.

Proposition 31. Let (M,g ,F) be a M-theory background and X a conformal Killing

vector. Then X preserves the kernel of P, that is,

[L ,P ] P X Y = [X ,Y ] if and only if X is supernormal.

Proof. Using equation (9.6), we can rewrite PY as

P P 1 F Y 3 X F . (7.5) Y = Y + 8 · − 88 ·

We know that [L ,P ] P by equation (5.15). Now note that X Y = [X ,Y ]

[L ,F Y ] L F Y ψ 4h F Y ψ F L (Y ψ) X F L ψ X · = X · · + X · · + · X · − · · X L F Y ψ 3h F Y ψ F [X ,Y ] ψ , = X · · + X · · + · ·

where we have used the properties of the Kosmann-Schwarzbach Lie derivative from

Lemma 20. In other words, [L ,F Y ] F [X ,Y ] if and only if L F 3h F; sim- X · = · X = − X ilarly for [L ,Y F]. X ·

7.3 Jacobi identities in the M-theory conformal Killing superalgebra

As before, we need to check the Jacobi identities of the conformal Killing superalgebra we have construced. Again, the even-even-odd Jacobi identity follows from the fact that for supernormal conformal Killing vectors, X L is a Lie algebra 0→ X homomorphism. Let Ψ (ψ,χ) Ker℘ deﬁne a SCKS and let X be a supernormal = ∈ conformal Killing vector. The even-odd-odd Jacobi identity can be written as

[X ,[Ψ,Ψ]] [L Ψ,Ψ] [Ψ,L Ψ] . (7.6) = X + X 69 Again, we have

g (A V , Z ) g (B X , Z ) g (V ,S Z ) h g (V , Z ) 2(ψ, Z X χ) 2(ψ, X Ω ψ) X ψ,ψ − Vψ,ψ = − ψ,ψ X + X ψ,ψ − · · − · Z · (ψ,S Z ψ) h (ψ, Z ψ) 2(ψ, Z X χ) 2(ψ, X Ω ψ) . = − X · + X · − · · − · Z ·

Note that when (ψ,χ)Ker℘, the spinorial Lie derivative L acts as follows:

ψ ψ L P +(X ) X χ = X + χ , - , - ! " 1 X ψ X χ σ 2 (AX )ψ X χ ∇1 + · −1 · = L(X ) χ +  1  , 2 · - βX ψ σ− 2 (AX )χ 2 · + 1   σ 2 (AX )ψ X χ ΩX ψ − · 1 − · . =  1  α ψ σ− 2 (A )χ 2 X · + X   We can thus write the right-hand side of equation (7.6) as

1 1 g (V , Z ) g (V , Z ) (σ 2 (A )ψ, Z ψ) (Ω ψ, Z ψ) (ψ, Z σ 2 ψ) (ψ, Z Ω ψ) LX ψ,ψ + ψ,LX ψ = X · − X · · + · − · X · (ψ,S Z ψ) h (ψ, Z ψ) 2(ψ, Z X χ) 2(ψ, X Ω ψ) , = − X · + X · − · · − · Z ·

which thus agrees with the left-hand side and the Jacobi identity is satisﬁed. As for

the odd-odd-odd Jacobi identity, the same comments as in the geometric case apply.

Since geometric conformal Killing spinors are special cases of M-theory conformal

Killing spinors, the fourth Jacobi identity does not vanish in general.

70 Chapter 8

Conformal Killing spinors in type IIA and IIB supergravities

One way to generate supergravity solutions starting from purely geometrical backgrounds is Kaluza-Klein reduction. In this procedure one exploits a symmetry of the background corresponding to a Killing vector ξ, which generates a 1-parameter subgroup Γ of the isometry group of M. If we take M to be a principal Γ-bundle, we can construct a metric on the base N M/Γ. This is a special case of a semi-Riemannian = submersion[22] . In addition to the metric, there will also be other fields on N, arising from the curvature of the bundle and the norm of the vector field ξ. For supergravity backgrounds without flux, the field equations amount to the Ricci-flatness of the metric on M, and the corresponding equations on N can be derived e.g. using standard formulas relating the Ricci curvatures of the total space and the base of a semi-Riemannian submersion [38]. Any other objects on M (such as differential forms and Killing or conformal Killing spinors) left invariant by the action of ξ will also induce corresponding objects on N.

In particular, it is well known that the 10-dimensional type IIA supergravity can be obtained as a Kaluza-Klein reduction of M-theory. Many of the supersymmetric reductions of M-theory backgrounds to type IIA solutions have been classified recently, including reductions of flat space (leading to so-called fluxbranes), the M- wave, the Kaluza-Klein monopole and the M-branes[39, 40, 41, 42].

For our purposes, the utility of the Kaluza-Klein procedure lies in the fact that the

71 connection in the lower-dimensional theory now includes ﬂuxes even if we start with a purely geometric background of the higher-dimensional theory.

In this chapter we show that starting from geometric M-theory backgrounds, we can use the Kaluza-Klein procedure to construct 10-dimensional supergravity backgrounds with supergravity conformal Killing spinors. We also compute the associ-

ated conformal Killing superalgebras. Finally, we make a brief comment on the role

of conformal Killing spinors in type IIB supergravity.

8.1 The Kaluza-Klein ansatz

Let (M,g ) be an eleven-dimensional Lorentzian manifold that admits a Killing vec-

tor ξ which is everywhere spacelike. Note that if we regard (M,g ) as a M-theory background with F 0, it must actually be Ricci-ﬂat by virtue of the ﬁeld equations = (6.2).

Now suppose that ξ generates a 1-parameter group Γ. We think of M as a principal

Γ-bundle π M N M/Γ , −→ = where π is the projection hat maps points in M to their Γ-orbits. We also have the

derivative of this map: π : Tp M Tq N, where q π(p). ∗ → = For any point p M we have a split T M H V of the tangent space into hor- ∈ p = p ⊕ p

izontal and vertical subspaces, where Vp Kerπ . This split is orthogonal with re- = ∗ 1 ! spect to g , and the vertical subspace Vp is spanned by ξ. Now let α ξ 2 ξ . Clearly = 7 7 α(ξ) 1 and α(X ) 0 X ξ — that is, H Kerα. We call a vector ﬁeld X hori- = = ↔ ⊥ p = zontal if α(X ) 0. =

For every X Tq N, there exists a horizontal lift X˜ Hp , deﬁned via π X˜ X . There ∈ ∈ ∗ = is a unique metric h on N for which the map π is an isometry, deﬁned via h(X ,Y ) ∗ = g (X˜ ,Y˜ ). We can write g in the following form:1

2 g π∗h ξ α α . (8.1) = + 7 7 ⊗ 1 In string theory literature, this is often called the Einstein-frame Kaluza-Klein ansatz: this can be related to the standard IIA string-frame ansatz via a conformal rescaling.

72 We can characterise geometric objects induced on N with the help of the following

deﬁnition.

Deﬁnition 32. Let η be a p-form on M. We say that η is horizontal if ι η 0, and ξ = invariant if L η 0. If η satisﬁes both of these properties, we call it basic. A basic ξ = form is a pull-back of a form on N.

Applying this definition to the objects defined above, we find:

Proposition 33. Let (M,g ) be a M-theory background with a Killing vector ξ as above.

Then ξ 2 and dα are basic. 7 7 Proof. The squared norm of ξ is clearly basic since ξ is Killing: L g (ξ,ξ) 0. An- ξ = other natural basic form is dα. It is easy to show that it is horizontal. For any vector

ﬁeld X , ι dα dα(ξ, X ) ξ = ξα(X ) X α(ξ) α([ξ, X ]) = − − g (ξ, X ) g ([ξ, X ],ξ) ξ , = ξ 2 − ξ 2 7 7 7 7 which vanishes since L g (X ,ξ) 0. Since dα is closed, L dα dι α 0, so dα is ξ = ξ = ξ = invariant as well, and hence basic.

We express these ﬁelds as ξ 2 e2π∗φ, where φ is a function on N — usually called 7 7 =

the dilaton — and dα π∗F , where F is a 2-form on N. In the sequel we usually = 2 2 omit explicit pull-backs.

Deﬁnition 34. Let (M,g ) and (N,h,F2,φ) be as above. We say that (N,h,F2,φ) is a

Kaluza-Klein reduction of (M,g ).

In summary, via the Kaluza-Klein reduction we can identify (M,g ) with a back-

ground (N,h,F2,φ) of type IIA supergravity. The Ricci-ﬂatness of M naturally gives

rise to the IIA ﬁeld equations which can be readily derived by the standard submer-

sion formulas for the Riemann curvature [38, 22]. In particular, the Ricci tensor of

(N,h,F2,φ) can be written in our conventions as

r (X ,Y ) 1 e2φ ι F ,ι F 1 e2φ F 2h(X ,Y ) . (8.2) = 2 〈 X 2 Y 2〉 + 4 | | 73 We remark that it is possible to include the four-form F in this construction by

further demanding that L F 0. Then it is easy to show that we can write F ξ = =

α π∗H π∗H , where H , H a 3-form and a 4-form on N, respectively. However, in ∧ 3+ 4 3 4 this thesis we will only consider Kaluza-Klein reductions of M-theory backgrounds

with F 0. =

8.2 Kaluza-Klein reduction of the conformal Killing spinor equation

We begin by describing the connection that the Levi-Cività connection on (M,g ) induces on (N,h,F2,φ).

A natural coframe for the metric (8.1) is {e˜α,e˜i }, where e˜α e˜φα. We can deﬁne a = coframe on (N,h) via ei e˜i . Note that the 11-dimensional volume element dvol = M can be written as

dvol dvol eα, (8.3) M = N ∧

Since we’re focusing on the Clifford module on M for which the action of the center

of the Clifford algebra is non-trivial,

dvol ψ dvol eα ψ ψ , M · = N · · = −

Now dvol2 Id, so the last equation implies that eα ψ : Γαψ dvol ψ. In other N = − · = = N · words, eα acts on spinors like the 10-dimensional volume element. As a representa-

tion of Spin(1,9), the 11-dimensional spinor bundle S breaks up into S+ S m, 11 10 ⊕ 10

where S1±0 are distinguished by chirality. For the purposes of this section, we prefer not to break 11-dimensional spinors explicitly into 10-dimensional spinors, leaving the 10-dimensional volume element manifest in the expressions below.

Proposition 35. The spin connection on (M,g ) induces the following connection ∇ on (N,h,F2,φ). 8

D 1 eφα(X )gradφ dvol 1 eφι F dvol , (8.4) X = ∇X − 2 · N ·− 8 X 2 · N ·

74 where is the Levi-Cività connection of h. We call D the IIA supercovariant connec- ∇

tion on (N,h,F2,φ).

Proof. Recall that the spin connection acting on a spinor ψ is ˜ ψ X (ψ) 1 ω˜ (X )abΓ ψ, ∇X = + 4 ab ab where ω˜ are the connection one-forms on (M,g ). To determine the connection on8 (N,h) induced from the spin connection of M we look at the structure equations.

deα ω˜ α ei 0, + i ∧ = dei ω˜ i e j 0 . + j ∧ =

The latter equation implies that ωi ω˜ i . To determine the remaining connection j = j one-forms, we compute

deα d eφα = ! " eφdφ α eφdα, = ∧ +

and, writing F dα, 2 =

ω˜ α ei 2eφα dφ 1 eφ(F ) e j ei i ∧ = ∧ + 2 2 i j ∧ which becomes

ω˜ α 2eφ∂ φα 1 eφF e j . i = i + 2 i j

Now ι F ι 1 (F ) ei e j (F ) e j (X )ei , so X 2 = X 2 2 i j ∧ = − 2 i j ! " ω˜ αi (X ) 2eφ∂i φα(X ) 1 eφ (ι F )i , (8.5) = − 2 X 2

and the result follows.

Using Proposition 35, we can now work out what happens to the conformal Killing spinor equation under Kaluza-Klein reduction. To begin with, we can decompose the Dirac operator.

/ ψ Γα ψ Γi ψ (8.6) ∇ = ∇α + ∇i 8 Γα 8 ψ / ψ8 1 eφF dvol ψ, (8.7) = ∇α + ∇ − 4 2 · N · 8 75 where we have used (2.10).

Now suppose ψ is a conformal Killing spinor on M which is left invariant by ξ; that is, L ψ 0, so that it can be identiﬁed with a pullback of a spinor on N. We wish to ξ = know what equation it satisﬁes on N.

The vertical component of the conformal Killing spinor equation implies that

Γα ψ 1 / ψ. (8.8) ∇α = n ∇ 8 8 Inserting this into (8.6), we obtain

1 1 1 φ n / ψ n 1 / ψ 4(n 1) e F2 dvolN ψ . (8.9) ∇ = − ∇ − − · · 8 The horizontal component of the twistor equation can then be written as

1 φ 1 1 φ X ψ 2 e ιX F2 dvolN ψ n 1 X / ψ 4(n 1) e X F2 dvolN ψ 0 , (8.10) ∇ − · · + − ·∇ − − · · =

where X is any horizontal vector ﬁeld.

We have thus proven the following:

Proposition 36. Let (M,g ) be a vacuum M-theory background with a Killing vector ξ

and (N,h,F2,φ) its Kaluza-Klein reduction with respect to ξ. Furthermore, let ψ be a conformal Killing spinor satisfying L ψ 0. Then Pψ 0, where for any horizontal ξ = = vector ﬁeld X ,

1 φ 1 1 φ PX X 2 e ιX F2 dvolN n 1 X / 4(n 1) e X F2 dvolN = ∇ − · ·+ − ·∇− − · · ·

is the Penrose operator associated to the IIA supercovariant connection on (N,h,F2,φ).

8.3 Kaluza-Klein reduction and conformal Killing superalgebras

As we have seen, we can associate a conformal Killing superalgebra h˜ h˜ h˜ = 0 ⊕ 1 to (M,g ), constructed out of its normal conformal Killing vectors and conformal

Killing spinors. In this section we show that via Kaluza-Klein reduction we can construct a supergravity conformal Killing superalgebra h from h˜. 76 We deﬁne h1 to be the space of ξ-invariant conformal Killing spinors of M, that is

h {ψ KerP˜ L ψ 0} . 1 = ∈ | ξ =

As for h , suppose that h h˜ is a subalgebra. Let ψ h and X h Then it follows 0 0 ⊂ 0 ∈ 1 ∈ 0 from the even-even-odd Jacobi identity of h˜ that

L L ψ L ψ L L ψ . ξ X = [ξ,X ] + X ξ

But the last term vanishes, since L ψ 0. Therefore, for L ψ to lie in h , we must ξ = X 1 require that L ψ 0, which implies that [ξ, X ] leaves ψ invariant. Since we are [ξ,X ] = only considering Kaluza-Klein reductions with respect to one Killing vector, we as-

sume [ξ, X ] must be proportional to ξ. In other words, we must have X Norm(ξ), ∈

where Norm(ξ) is the normaliser of ξ in h˜0.

Furthermore, we can prove the following:

Proposition 37. Let (M,g ) be a vacuum M-theory background, ξ a Killing vector and

X˜ Norm(ξ) a normal conformal Killing vector with [X˜ ,ξ] cξ with c R a constant. ∈ = ∈

Then X˜ induces a conformal Killing vector X of (N,h,F2,φ), and furthermore it holds

that

L φ f c X = − X + L F cF . X 2 = − 2

1 Proof. Suppose that X˜ is a conformal Killing vector on M, with f ˜ div X˜ . Fur- X = 11 thermore, assume that X Norm(ξ), with [X˜ ,ξ] cX˜ for some constant c. We can ∈ = decompose X˜ as X˜ aξ X , where a is a function on M and X is the horizontal = + component of X˜ .

As a preliminary, let us compute the Lie derivative of ξ! along X˜ . Let Y be any vector

77 ﬁeld on N and Y˜ its horizontal lift. Then

! ! ! L ˜ ξ Y˜ X˜ ξ (Y˜ ) ξ [X˜ ,Y˜ ] X = − ! " . / ! " g ˜ ˜ ξ,Y˜ g ξ, ˜ ˜ Y˜ g ξ,[X˜ ,Y˜ ] = ∇X + ∇X − ! " ! " ! " g ˜ X˜ ,Y˜ g ξ, ˜ ˜ X˜ cg ξ,Y˜ = ∇ξ + ∇Y + ! " ! " ! " 2f ˜ c g ξ,Y˜ = − X + ! " !! " 2f ˜ c ξ (Y˜ ) , = − X + ! " where we have used the Koszul formula and the conformal Killing equation (3.1).

Another useful quantity is the Lie derivative of ξ 2: this is simply 7 7

2 L ˜ ξ X˜ g (ξ,ξ) X 7 7 = ! " ! " 2g ˜ ˜ ξ,ξ = ∇X ! " 2g ˜ X˜ ,ξ 2cg (ξ,ξ) = ∇ξ + ! " 2 f ˜ c g (ξ,ξ) = − X + ! " 2 2 f ˜ c ξ . = − X + 7 7 ! " From this we can immediately see the action of X on the dilaton φ, for

X˜ g (ξ,ξ) 2e2φX (φ) , (8.11) = ! " and combined with the previous result this implies that L φ X (φ) f c. X = = − X + It remains to compute the action of X˜ on α and F dα. Using the above, 2 =

1 ! L ˜ α L ˜ ξ X = X ξ 2 ,7 7 - 1 2 1 ! L ˜ ξ L ˜ ξ = − ξ 4 X 7 7 + ξ 2 X 7 7 7 7 2 f ˜ !c " 2f ˜ c − X + ! − X + ! 2 ξ 2 ξ = − ! ξ " + ! ξ " c 7 7 7 7 ξ! = − ξ 2 7 7 cα . = −

78 What about dα? The last equation immediately gives

L ˜ dα dι ˜ ι ˜ d dα X = X + X ! " d ι ˜ dα = X ! " d L ˜ α dι ˜ α = X − X ! " cdα = − cF , = − 2

2 since L ˜ α cα and d 0. X = − = With these observations in mind, we use the Kaluza-Klein ansatz (8.1) to determine

the action of X on the lower-dimensional metric h:

2 L ˜ g L h L ˜ ξ α α X = X + X 7 7 ⊗ ! 2 " 2 L h L ˜ ξ α α 2 ξ L ˜ α α = X + X 7 7 ⊗ + 7 7 X ⊗ ! " 2 ! "2 L h 2f ˜ 2c ξ α α 2c ξ α α = X + − X + 7 7 ⊗ − 7 7 ⊗ ! 2 " L h 2f ˜ ξ α α = X − X 7 7 ⊗

2f ˜ g . = − X

For this equality to hold, we must have L h 2f ˜ h 2f h, so X is indeed a X = − X = − X conformal Killing vector of (N,h).

We can in fact do slightly better and show that conformal Killing vectors on N induced by normal conformal Killing vectors of the M-theory background satisfy a property analogous to Deﬁnition 30.

Deﬁnition 38. Let X be a conformal Killing vector ﬁeld of (N,h,F2,φ). Then we say

that X is IIA supernormal if

L L 0 , X = where L is the Schouten tensor of (N,h).

Proposition 39. Let (N,h,F2,φ) be a Kaluza-Klein reduction of a vacuum M-theory

background (M,g ) and X a conformal Killing vector ﬁeld induced by a normal con-

formal Killing vector of (M,g ). Then X is IIA supernormal. 79 Proof. As an immediate consequence of Proposition 37 we have L (eφF ) f eφF . X 2 = − X 2 Using a similar argument as in Section 7.2, Chapter 7, it is clear from equation (8.2)

that L r 0. It follows that L L must vanish as well. X = X

Finally, we show that a conformal Killing vector X on N inherited from X˜ Norm(ξ) ∈ on M preserves the space of supergravity conformal Killing spinors on N.

Proposition 40. Let (N,h,F2,φ) be a Kaluza-Klein reduction of a vacuum M-theory

background (M,g ) and X a conformal Killing vector ﬁeld induced by a normal con-

formal Killing vector of (M,g ). Then LX preserves the space of superconformal Killing

spinors on N, that is

[L ,P ] P . X Y = [X ,Y ]

Proof. Obviously P can be writen as P Φ F ,φ , where + 2 ! " 1 φ 1 φ ΦY 2 e ιY F2 4(n 1) e Y F2 , (8.12) = − − − ·

Then we have

[L ,P ] P [L ,Φ ] . (8.13) X Y = [X ,Y ] + X Y Using Proposition 37 and the properties of Kosmann-Schwarzbach Lie derivative,

we compute

L ,eφι F ψ L eφι F ψ eφι F L ψ f eφι F ψ eφι F L ψ X Y 2 · = X Y 2 · + Y 2 · X − X Y 2 · − Y 2 · X 2 3 ! " 2eφX (φ)ι F ψ eφι F ψ eφι L F ψ f eφι F ψ = Y 2 · + [X ,Y ] 2 · + Y X 2 · − X Y 2 · eφι F ψ , = [X ,Y ] 2 ·

since L F cF and X (φ) f c. A similar calculation for the remaining term X 2 = − 2 = − X + yields

L ,eφY F ψ L eφY F ψ eφY L F ψ f eφY F ψ eφY F ψ X · 2 · = X · 2 · + · X 2 · − X · 2 · − · 2 · 2 3 ! " ! " 2eφX (φ)Y F ψ eφY L F ψ eφ [X ,Y ] F ψ f eφY F ψ = · 2 · + · X 2 · + · 2 · − X · 2 · eφ [X ,Y ] F ψ , = · 2 ·

because L F ψ L F ψ 2h F ψ. It follows that [L ,Θ ] Θ . X 2 · = X 2 · + X 2 · X Y = [X ,Y ] ! " 80 The results of this section guarantee that we can associate a well-deﬁned conformal

Killing superalgebra h to a IIA background (N,h,F2,φ) obtained as a quotient of the

geometric conformal Killing superalgebra of a vacuum M-theory background (M,g ).

We devote the rest of this chapter to presenting a number of explicit examples.

8.4 Conformal Killing spinors and Kaluza-Klein reductions of ﬂat space

A generic Killing vector of Minkowski space R1,n can be written in the form

ξ τ λ , (8.14) = +

where τ is a translation and λ is a Lorentz transformation. Requiring that a confor-

mal Killing spinor ψ ψ x χ of the ﬂat space given in equation (5.30) is invariant = 0 + · under ξ implies that

λ χ 0 , (8.15) · = τ χ 1 λ ψ 0 . (8.16) · − 2 · 0 =

By imposing the requirement that ξ be everywhere spacelike (so that its action on

Minkowski space is free), it can be shown [39] that there are two families of space-

like Killing vectors which give rise to smooth quotients and preserve some spinors.

i There exists a coordinate system (z,x ,x±) in which the ﬂat metric takes the form

8 i i 2 gE1,10 2dx+x− dx dx dz , (8.17) = + i 1 + += and in this metric (up to a scale), the normal forms of the relevant Killing vectors are

given by

ξ ∂ R (β ) R (β ) R (β ) R (β ) , (8.18) = z + 12 1 + 34 2 + 56 3 + 78 4

βi 0 (8.19) i = + and by

ξ ∂z N 1(u) R34(β16 ) R56(β26 ) R78(β36 ) , (8.20) = + + + + +

βi6 0. (8.21) i = + 81 Here Ri j (β) are rotations in the i j plane with parameters β and N 1(u) is a null ro- + tation in the ith direction with parameter u. In the second case (8.18) the condition

(8.15) can be written as

(N S)χ 0 , (8.22) + = Γ χ 1 (N S)ψ 0 , (8.23) z − 2 + 0 =

where N is nilpotent and S is semisimple (since the Ri j ’s all commute and are there-

fore simultaneously diagonalisable); furthermore, N and S commute. The ﬁrst con-

dition then implies that Nϕ and Sχ 0 separately. In addition, N and S commute 2 = with Γ and Γ2 Id. z z = − Let us look at these possibilities separately. First, note that it follows from equation

(8.15)that a conformal Killing spinor with ψ 0 can never be invariant, since equa- 0 =

tion (8.15) then implies that χ must vanish as well: therefore, both ψ0,χ must be

nonzero, constrained by the invariance condition.

Consider the normal form (8.18). There are several possibilities depending on which

of the β’s vanish. At most two of the βi ’s can be zero, since if three vanish, the

remaining one must vanish as well. Therefore, let us consider the case when two

are nonzero; without loss of generality we can choose them to be β1,β2. Equations

(8.15) then become

β Γ χ β Γ χ 0 , (8.24) 1 12 + 2 34 = Γ χ 1 β Γ β Γ ψ 0 . (8.25) z − 2 1 12 + 2 34 0 = ! " Since R β Γ β Γ commutes with Γ , Clifford multiplying the last equation = 1 12 + 2 34 z again by R we obtain

R R ψ 0 , (8.26) · · 0 = which implies that

2β β Γ ψ β2 β2 ψ , 1 2 1234 0 = 1 + 2 0 ! " 2 and using the fact that β β 0 and writing β2 β2 β β 2β β , we get 1 + 2 = 1 + 2 = 1 + 2 − 1 2 ! " Γ ψ Γ ψ , 12 0 = 56 0 82 in other words, R ψ 0, which implies that χ 0 and the conformal Killing spinor · 0 = = ϕ is simply a parallel spinor invariant under the action of ξ.

In fact, one can show in general that for a semisimple element S, S S ϕ implies · · that S ϕ 0. Complexifying the spinors, if necessary, we can diagonalise the matrix · = S by which we mean the matrix that gives the Clifford action of S on spinors, not · the usual rotation matrix. The eigenvalues of S2 are the squares of the eigenvalues · of S , so the zero eigenvalues of both matrices coincide. This means that the only · conformal Killing spinors of ﬂat space left invariant by the action of a Killing vector

of the form (8.18) are (a subset of) the parallel spinors.

Next, let us consider the case involving the null rotation. Now (8.15) becomes

Γ χ 1 (N S)ψ 0 , (8.27) z − 2 + 0 =

from which we can deduce that

N S ψ 0 , · · 0 = S2ψ 0 , 0 =

since N is nilpotent. Using a similar argument as before, we can again conclude that

S ϕ 0. Now we can solve (8.27) to obtain · 1 =

χ 1 Γ N ψ , (8.28) = 2 z · 0

whence the conformal Killing spinors invariant under the action of a Killing vector

of the form (8.20) are given by

ψ(ζ) ζ 1 x Γ N ζ , (8.29) = + 2 · z ·

and ζ is an arbitrary constant spinor.

8.5 Conformal superalgebras of nullbranes

In [39] the reductions of ﬂat space with respect to Killing vectors involving a null rotation are called nullbranes, which have recently attracted much interest in the context of time-dependent string solutions and cosmological toy models [43, 44, 45]. 83 In the previous section we showed that in addition to parallel spinors, certain non- parallel conformal Killing spinors are also invariant under the action of the Killing vector used in the reduction. Therefore, it is possible to associate a conformal Killing superalgebra to each nullbrane solution. There are three distinct cases: the nullbrane solution and two solutions which interpolate between the nullbrane and ﬂuxbrane solutios: we consider each in turn, although it turns out that the form of the conformal Killing superalgebras of the latter follow largely from the computation of the former.

8.5.1 The nullbrane

When the Killing vector (8.20) used in the reduction is a pure null rotation (i.e. all the β’s vanish, with u 0), one obtains a IIA solution which the authors of [39] call a 4= 1 nullbrane. This is a 2 -BPS solution with the following metric, dilaton and Ramond- Ramond 2-form:

1 1 1 2 2 2 7 1 1 2 g Λ 2 2dx+dx− x (dx−) ds E Λ− 2 dx x x−dx− , = − + + + 2 9 1 ! " ! ": ! " F2 dx− dx , = Λ2 ∧ φ 3 logΛ , = 4

2 where Λ 1 (x−) . = + Note that the metric does not depend on the parameter u: it has been absorbed by

1 a rescaling x± u± x±. →

It is straightforward to calculate the normaliser of ξ ∂z uM 1. Norm(ξ) is gener- = + + ated by

1 1 X : M1z uK P , = − 2 + − u −

Y : P1 uM z , = + + Z : M D , = +− −

Pi , P , Mi j , Mi , Pz , M 1 , + + +

where i 2...8. As with the metric, it is possible to absorb the parameter u into the = 84 translation generators P . Also, because we’re quotienting by ξ and set ξ 0 in the ± =

quotient algebra, we can eliminate one generator, leaving Pz uM 1 : W , say. = − + = We can also compute the Dirac current of two arbitrary ξ-invariant twistor spinors,

u u ψ1 (ζ) ζ x Γz 1ζ and ψ2 (υ) υ x Γz 1υ. = + 2 · + = + 2 · +

V ψ1,ψ2 (Γz ζ,υ)ξ (ζ,Γ1υ)Y (Γ ζ,υ)X = + + + ! " i j i (Γz 1ζ,Γ υ)Mi j (Γ Γz1ζ,Γ υ)M i + i,j + + i +− + + + 9 i (ζ,Γ υ)P (ζ,Γ υ)Pi . (8.30) + − + + i 2 += For the even-odd bracket of the superalgebra we also need to compute the Lie deriva-

tive of ψ(ζ) with respect to elements of Nξ. This yields

u 1 LP ψ(ζ) ψ Γi z 1ζ LM ψ(ζ) ψ Γi j ζ i = 2 + i j = − 2 ! " ! 1 " LP ψ(ζ) 0 LX ψ(ζ) ψ 2 (Id Γ Γ )Γ1z ζ + = = + + − ! 1 " LY ψ(ζ) ψ(uΓz ζ) LZ ψ(ζ) ψ Γ Γ ζ = + = − 2 + − u ! " LW ψ(ζ) ψ Γ 1ζ , = 2 + ! " where we have used (5.31). To exhibit the form of the nullbrane conformal Killing

superalgebra, it is convenient to write ζ ζ ζ , where ζ KerΓ . To write down = + + − ± ∈ ± the algebra in a form that is more in line with notation used in the physics literature,

we introduce “fermionic” generators Q ,S , which generate inﬁnitesimal shifts in + − the direction of ζ and ζ , respectively. Note that Q corresponds to the supersym- + − + metry generator of the usual nullbrane supersymmetry algebra.

Thus, the (non-trivial) brackets of the nullbrane conformal Killing superalgebra are:

[X ,Y ] ξ ( 0) [Z ,Y ] Y = = = [W, Z ] W [W, X ] Y = − =

[W,Y ] uP Mi j ,Pk δjk Pi δik P j = − + = − 2 3 1 Mi ,P j δi j P [Mi , X ] Pi + = − + + = − u 2 3 [Mi , Z ] Mi [P , Z ] 2P + = − + + = − + 85 u 1 [Pi ,Q ] Γi z 1S Mi j ,Q Γi j Q + = 2 + − + = − 2 + 1 2 3 1 Mi j ,S Γi j S [Mi ,Q ] Γi S − = − 2 − + + = − 2 + − 2 3 1 1 [X ,Q ] Id Γ1z Q [X ,S ] S + = 2 − + − = 2 − ! " [Y ,Q ] uΓz S [Z ,Q ] Q + = + − + = + u 1 [W,Q ] Γ 1S [Q ,Q ] Γ C − P + = 2 + − + + = − +

and

1 i j 1 [S ,S ] Γ C − X Γ Γz 1C − Mi j − − = + + + 1 1 i 1 i 1 [Q ,S ] ΓzC − ξ Γ1C − Y Γ Γz1C − M i Γ C − Pi , + − = + + i + + i + + where C is the “charge conjugation matrix” used to deﬁne the spinor inner product

as (ψ,χ) ψt Cχ, notation often favored by physicists. Note that the even part of the =

algebra contains a natural iso(8) subalgebra generated by Mi j ,Mi ,Pi and P . + +

8.5.2 Interpolating solutions

When the Killing vector also contains rotations, one obtains solutions that interpolate between the nullbrane and the supersymmetric ﬂuxbrane solutions described in [39]. There are two possible cases:

1. β 0, β β β 1 = 2 = − 3 =

2. β ,β ,β 0, β β β 0 1 2 3 4= 1 + 2 + 3 =

The explicit metrics can be found in [39]. The conformal Killing superalgebras of

these solutions are actually subsuperalgebras of the nullbrane superalgebra.

86 In case (1) an iso(4) subalgebra remains. The brackets are2

[X ,Y ] ξ ( 0) [W, X ] Y = = =

[W,Y ] uP Mi j ,Pk δjk Pi δik P j = − + = − 2 3 1 Mi ,P j δi j P [Mi , X ] Pi + = − + + = − u 2 3 u 1 [Pi ,Q ] Γi z 1S Mi j ,Q Γi j Q + = 2 + − + = − 2 + 1 2 3 1 Mi j ,S Γi j S [Mi ,Q ] Γi S − = − 2 − + + = − 2 + − 2 3 1 1 [X ,Q ] Id Γ1z Q [X ,S ] S + = 2 − + − = 2 − ! " u [Y ,Q ] uΓz S [W,Q ] Γ 1S + = + − + = 2 + − 1 [Q ,Q ] Γ C − P + + = − +

and

1 i j 1 [S ,S ] Γ C − X Γ Γz 1C − Mi j − − = + + + 1 1 i 1 i 1 [Q ,S ] ΓzC − ξ Γ1C − Y Γ Γz1C − M i Γ C − Pi , + − = + + i + + i + + where index i can now only take values 2,3,4.

In the case (2) we only have an iso(2) subalgebra: the only possible value of i is 2,

otherwise the form of the algebra remains the same.

8.6 Conformal Killing spinors in HLW massive IIA supergravity

It has been shown by Howe, Lambert and West [46] that any M-theory background

(M,g ,F) can be viewed as a Weyl structure (M,g ,F,θ), where the Weyl one-form θ = dl is exact. The M-theory equations of motion written using the Weyl connection

are then equivalent to the standard ones, and spinors are taken to be sections of 1 S[ 2 ]. This fact has been used in [47, 46, 48] to produce a variant of the Kaluza-Klein construction we presented in Section 8.1.

2Note that this is the algebra in the quotient, so we set ξ 0. = 87 Lemma 41. Suppose (M,g ,F) is an M-theory background that admits a homothety

ξ (in other words, L g 2mg for some constant m) which is also supernormal, ξ = − i.e. L F 3m. Then there exists a M-theory background with a Weyl connection ξ = − (M,g ,F ,θ dl) for some function l for which L g 0 and L F 0. = ξ = ξ =

Proof. We simply perform a Weyl transformation g g e2l g and in addition rescale 0→ = the four-form: F F e3l F. Provided that ξ(l) m, the above conditions are then 0→ = = satisﬁed. We can always ﬁnd such an l, for instance by solving the equation ξ(l) m = in adapted coordinates where ξ ∂ . = z

We can now use the usual Kaluza-Klein procedure and ﬁnd that (M,g ,F ,dl) gives

rise to the data (N,h,F2, H2, H3,φ,m). The submersion formulas for curvature again

give rise to 10-dimensional equations of motion which are distinct from those of

the IIA theory since they now involve m-dependent terms. These equations deﬁne

a theory called the HLW massive IIA supergravity. Note that m is actually a free pa-

rameter since we can send it to any value by rescaling ξ, and if m 0 we recover the = usual IIA equations of motion.

We call (N,h,F2, H2, H3,φ,m) a homothetic Kaluza-Klein reduction of (M,g ,F). As

before, we will only consider reductions of M-theory backgrounds for which F van-

ishes. 1 Since acting on S[ 2 ], P D P, we immediately have =

Proposition 42. Let (M,g ) be a vacuum M-theory background with homothety ξ so

that L g 2mg and let (N,h,F ,φ,m) be the corresponding homothetic Kaluza- ξ = − 2 Klein reduction. The Penrose operator P˜ on M induces the following Penrose-type

operator on N:

1 φ 1 1 φ PX ψ X ψ 2 e ιX F2 dvolN ψ n 1 X / ψ 4(n 1) e X F2 dvolN ψ , (8.31) = ∇ − · · + − ·∇ − − · ·

We remark that although P agrees formally with (8.10), the ﬁelds appearing in the

expression now satisfy different equations of motion.

88 8.7 A HLW massive IIA conformal Killing superalgebra

In this section we construct an example of a conformal Killing superalgebra associated to a background of HLW massive IIA supergravity as a quotient of eleven- dimensional Minkowski conformal Killing superalgebra we introduced in Chapter

5, Section 5.5.

We proceed in a fashion similar to [47]. Consider a homothety

ξ D R, = + where D xa ∂ is the dilation vector ﬁeld and R Rb xa ∂ is a rotation. Let ψ = a = a b = ψ x χ be an arbitrary conformal Killing spinor. The action of ξ on ψ is given by 0 + ·

L ψ D(ψ) R(ψ) 1 dD! ψ 1 dR! ψ 1 ψ ξ = + − 4 · − 4 · − 2 x χ (Rx) χ 1 R ψ 1 R x χ 1 ψ 1 x χ = · + · − 4 · − 4 · · − 2 − 2 ·

Note that

R x χ R Γab xc Γ χ x R χ 4(Rx) χ , (8.32) · · = ab c = · · + ·

where we have used the commutation relation

[Γab,Γc ] 2ηac Γb 2ηbc Γa. (8.33) = −

The x-dependent part and the constant part of the expression must vanish sepa-

rately, so R must satisfy

R ψ 2ψ · = − R χ 2χ · =

Without loss of generality we can take R 2M . Half of the conformal Killing spinors = 0/ of the 11-dimensional Minkowski space satisfy the above condition. Hence, the

space of ξ-invariant conformal Killing spinors is 32-dimensional.

1 0 / For the following computations we prefer to use lightcone coordinates x± x x , = -2 ± where x/ is the 10th spacelike direction. This implies that ψ KerΓ and χ K!erΓ , " ∈ − ∈ + 89 so we write a conformal Killing spinor as ψ(ζ) ζ x ζ . In these coordinates = − + · + ξ D M . = + −+ We also want to exhibit the HLW solution itself. The reduction ansatz for the eleven-

dimensional metric g is usually written in the so-called string frame as [47]:

2φ (2mz ) 2φ 2 g e − 3 h e (dz A) , (8.34) = + + ! " and F d A. 2 = We start by writing the ﬂat eleven-dimensional ﬂat metric in the form

g (dx0)2 (dx/)2 dr 2 r 2dΩ2 (8.35) = − + + + 8 ! " 2 where dΩ8 is the metric on the 8-sphere. We choose new coordinates adapted to the vector ﬁeld ξ D 2M such that ξ ∂ . = + 0/ = z

0 1 2z 2y x y e e− 1 , = 2 2 + / 1 2 ! 2z 2y " x y e e− 1 , = 2 − z y!1 " r e − . =

In these coordinates, the eleven-dimensional metric becomes

2(z y1) 2 2 2 2 2 2 2 1 2 g e − [(dz (1 2(y ) )d y y d y ) 4y (1 y )(d y ) = − − 1 − + 2 − 2 (1 (y2)2)(d y2)2 4(y2)3d y1d y2 dΩ2 ] . − + + + 8 From the reduction ansatz (8.34) we can then read off the HLW solution:

9 y1 2 2 2 2 1 2 2 2 2 2 2 3 1 2 2 g e− 4 4(y ) (1 (y ) )(d y ) (1 (y ) )(d y ) 4(y ) d y d y dΩ , = − − + + + 8 ! " F 4y2d y1 d y2, 2 = − ∧ φ 3 y1. = 2

Note that the metric h has a singularity at y2 0. Given the form of the adapted = coordinates, at a ﬁrst glance one could imagine that this is a coordinate singular-

ity, but computation of the Ricci scalar R and the fully contracted Riemann tensor

R Rabcd shows that they both diverge as y2 0, so it is in fact a genuine curva- abcd → ture singularity. 90 We are now ready to calculate the structure of the superconformal algebra associated to this HLW IIA background. We proceed in a similar fashion as in Section 8.5.

It is a straightforward calculation to show that the normaliser Norm(ξ) is generated

by M ,D,Mi j ,P and K . In fact, this agrees with the centraliser of ξ. As we set ξ −+ − + to zero in the quotient, we can eliminate one of the generators, leaving D M = − −+ in the algebra.

As before, we calculate the Dirac current Vψ1,ψ2 associated to two arbitrary ξ-invariant

conformal Killing spinors ψ1 (ζ) ζ x ζ and ψ2 (υ) υ x υ . After some Γ- = − + · + = − + · + matrix algebra and repeated use of the antisymmetry of the spinor inner product

with respect to the Clifford multiplication by a vector ﬁeld (that is, (ψ , X ψ ) 1 · 2 = (X ψ ,ψ )) we obtain − · 1 2

Vψ ,ψ (ζ ,Γ υ )K [(ζ ,υ ) (ζ ,υ )](D M ) 1 2 = + − + + + + − − − + + −+ i j i j (ζ ,Γ υ ) (ζ ,Γ υ ) Mi j (ζ ,Γ υ )P . + − + − + − + − + − − . / We can also compute the action of the generators of Norm(ξ) on a ξ-invariant con-

formal Killing spinor ψ(ζ ,ζ ) ζ x ζ via the spinorial Lie derivative: − + = − + · +

LK ψ(ζ ,ζ ) ψ(0,Γ ζ ) LP ψ(ζ ,ζ ) ψ(Γ ζ ,0) + − + = + − − − + = − + 1 1 1 1 LM ψ(ζ ,ζ ) ψ Γi j ζ , Γi j ζ LD ψ(ζ ,ζ ) ψ ζ , ζ i j − + = − 2 − − 2 + − + = − 2 − 2 + ! " ! " To exhibit the structure of the superalgebra we again introduce fermionic generators

Q ,S which generate shifts along the spinors ζ ,ζ , respectively. In terms of these, − + − + the non-trivial brackets of the algebra are

[P ,K ] ξ ( 0) [P ,D] P − + = = − = − [K ,D] K [P ,S ] Q , + = − + − + = − 1 [K ,Q ] S , Mi j ,Q Γi j Q , + − = + − = − 2 − 1 2 3 1 Mi j ,S Γi j S , [D,Q ] Q , + = − 2 + − = − 2 − 2 3 1 1 [D,S ,] S , [S ,S ] Γ C − K , + = 2 + + + = − + i j 1 1 [S ,Q ] ξ Γ C − Mi j , [Q ,Q ] Γ C − P , + − = + − − = + − 91 where C is the charge conjugation matrix as before.

8.8 Conformal Killing spinors in type IIB supergravity

The spinors in type IIB supergravity [34] are real representations of Spin(1,9) SL(2,R). × It is convenient to consider them as sections of the bundle S ∆, where S is the + ⊗ + 16-dimensional positive chirality representation of Spin(1,9) and ∆ is the standard

representation of SL(2,R). The IIB backgrounds are given by the data (M,g , H),

where H is a self-dual closed 5-form that also satisﬁes an Einstein-type equation

along with the Lorentzian metric g . The full theory admits other ﬁelds, but we are

only interested in this truncated version.

Acting on the sections of S S ∆, the supercovariant connection is = + ⊗

D ι H J, X = ∇X + X ⊗

where J is a complex structure on ∆. We can consider S as a complexiﬁcation of S + and write this as

D iι H X = ∇X + X

Now as in the M-theory case, consider the twistor operator deﬁned using the super-

covariant connection:

P D 1 X D/ , X = X + 10 ·

where D/ e D . = i i · i

As observed5by Leitner [49], PX actually agrees with the geometric Penrose operator

PX due to a happy accident of Clifford product identities involving the self-dual ﬁve- form in ten dimensions, provided that only the self-dual 5-form ﬂux is turned on. It holds that

X H 2ι H · = − X +(X ! H) ι H, ∧ = − X

where the forms in these identities are understood to be acting on spinors via the

92 Clifford product. Now

P D 1 X D/ X = X + 10 · 1 X / iι H 1 X i H = ∇X + 10 ·∇ + X + 2 · P iι H 1 X i H = X + X + 2 · P , = X

so KerP KerP. In particular, supergravity Killing spinors are geometric conformal = Killing spinors.

The simplest example of a IIB solution that admits conformal Killing spinors is again the flat space R1,9. Studying the existing classification of Lorentzian manifolds admitting conformal Killing spinors [12], we can find other examples. Perhaps the most interesting is the IIB conformally flat pp-wave that has received much attention recently in the context of BMN correspondence and the Freund-Rubin solution

AdS S5 — in fact, the former is the Penrose limit of the latter [50]. The confor- 5 × mal Killing superalgebras of them both are isomorphic to the Minkowski conformal

Killing superalgebra we described in Chapter 5.

It has been suggested by [49] that this curious identity between geometric and supergravity Penrose operators might be useful in ﬁnding new supersymmetric backgrounds of type IIB supergravity, perhaps among the 10-dimensional pseudo-Hermitian

Einstein spaces described in [51].

93 Chapter 9

Killing spinors, discrete quotients and spin structures

The relationship between the choice of the spin structure of a Lorentzian symmetric space and the dimension of the space of its conformal Killing spinors was analysed in detail in [14]. In the case of non-conformally ﬂat symmetric spaces,the conformal

Killing spinors actually agree with parallel spinors, corresponding to supergravity

Killing spinors when F 0. It is a natural question to ask whether the dimension of = KerD also depends on the choice of spin structure in the case of nonzero four-form

ﬂux.

The authors of [52] observed that it is possible to construct examples of non-simply connected isometric M-theory backgrounds that have the same geometry and four- form F but admit different fractions of supersymmetry depending on the choice of the spin structure. Therefore, it would seem necessary to include the choice of spin structure in the data deﬁning a M-theory background as well.

In this chapter we illustrate this point further by considering backgrounds that are

Lorentzian symmetric spaces (Cahen-Wallach spaces), as opposed to the Freund-

Rubin solutions involving spherical space forms that were treated in [52]. We will show that at least for known symmetric M-theory backgrounds with with more than

16 Killing spinors the choice of spin structure that preserves any supersymmetry appears to be unique. In particular, this includes symmetric discrete quotients of

M-theory pp-wave solutions.

94 Orbifolds of 11-dimensional pp-wave solutions have also been considered previously in e. g. [53], but only for a very particular solution with 26 supersymmetries.

9.1 Discrete quotients and spin structures

Let M,g be a simply connected n-dimensional Lorentzian spin manifold. We de-

note! its is"ometry group by I (M,g ). Suppose D I M,g is a discrete, orientation- ⊂ preserving subgroup, and let e , B 0...n 1 be a!pseu"do-orthonormal frame on B = − M. Then for any γ D at a point x M, dγ SO(1,n 1) corresponds to the linear ∈ ∈ x ∈ −

map that transforms e A(x) to e A γ(x) ). There are now two possible lifts of dγx to

Spin(1,n 1) since the covering m! ap S"pin(1,n 1) SO(1,n 1) is two-to-one: we − − → − denote these by Γ(x). ± Now let E(D) be the set of all left actions of D on M Spin(1,n 1) satisfying × −

.(γ)(x,a) γ(x),.(γ,x) a , (9.1) = · ! " where .(γ,x) Γ(x). = ± Elements of E(D) correspond to spin structures on N M/D. The spinor bundle = corresponding to . E(D) is given by ∈

S. M ∆1,n 1 /. , (9.2) = × − ! " Here ∆1,n 1 is the spinor module and .(γ) x,ψ(x) γ(x),.(γ,x) ψ(x) . It follows − = · that the spinor ﬁelds ψ on N are the spinor!ﬁelds o"n M! that satisfy "

ψ γ(x) .(γ,x) ψ(x) . (9.3) = · ! " In particular, when M is a M-theory background and D also preserves the four-form

F, the Killing spinors on N M/D are the .-invariant Killing spinors of M. =

9.2 Hpp-waves in M-theory

The M-theory Hpp-waves are supersymmetric solutions of eleven-dimensional supergravity equipped with the metric of a Lorentzian symmetric space of the Cahen-

Wallach type [18] and a null homogeneous four-form. In the light-cone coordinates 95 i x±,x ,i 1...9 the Cahen-Wallach metric can be written as =

2 i j 2 i g 2dx+dx− Ai j x x (dx−) dx (9.4) = + i,j + i + +. / where A is a real 9 9 symmetric matrix. If A is nondegenerate, (M,g ) is inde- i j × i j composable; otherwise it decomposes to a product of a lower-dimensional inde-

composable CW space and an Euclidean space. If Ai j is zero, (9.4) is simply the

metric on ﬂat space. .

The moduli space of indecomposable CW metrics agrees with the space of unordered

8 eigenvalues of Ai j up to a positive scale: this space is diffeomorphic to S /Σ9, where

Σ9 is the permutation group of nine objects[18]. In particular, a positive rescaling of

Ai j can always be absorbed by a coordinate transformation. It is, of course, also pos-

sible to exhibit (M,g ) as a symmetric space by constructing its transvection group

G A for which (9.4) is the invariant metric. We refer the reader to [18] for details.

A natural choice of F is a four-form preserved by the symmetries of the CW metric that also satisﬁes the ﬁeld equations. As explained e.g. in [54, 18], the natural choice is a parallel form

F dx− Θ , = ∧

where Θ is a 3-form with constant coefﬁcients on R9. The equations of motion (6.2)

are satisﬁed iff Tr A 1 Θ 2. = − 2 | | We will make use of a global pseudo-orthonormal frame:

e ∂ , + = + e ∂ , i = i 1 i j e ∂ 2 Ai j x x ∂ − = − − i,j + + and the corresponding coframe

1 i j e+ dx+ A x x dx− = + 2 i j

e− dx− = ei dxi . = 96 For our purposes, there is no need to distinguish between coordinate and frame indices.

The only nonzero connection forms for the metric (9.4) are

i i j ω+ A x dx− . (9.5) = j

To solve the SCKS equation for the Hpp-waves, we will also need the explicit form of

their parallel spinors that satisfy the integrability conditions (6.15). Observing that

χ 0 implies that ∇a =

0 ∂ χ ∂i χ , = + = 1 i ∂ χ x Ai j Γj χ , − = − 2 +

and keeping in mind that since F χ 0, we must have χ KerΓ , this implies that · = ∈ + χ is a constant spinor in the kernel of Γ . We also require that Θ χ 0. It is clear + · = that there are many possible choices of Θ for which this condition is satisﬁed. For

example, we could choose Θ dx129 dx349, for which the integrability condition = − would be satisﬁed if

Γ χ χ , 1234 = −

a condition which is generally satisﬁed by half of the spinors, since Γ1234 squares to

the identity. In summary, the possible χ lie in KerΓ KerΘ. + ∩ Before tackling the SCKS equation, we also determine the amount of supersymme-

try preserved by the Hpp-wave solutions. As previously mentioned, supergravity

Killing spinors are parallel sections of the supercovariant connection

D 1 ι F 1 X ! F : Ω , X = ∇X + 6 X + 12 ∧ = ∇X + X

It is convenient to rewrite Ω involving Clifford products as

Ω 1 F X 1 X F . (9.6) X = 8 · − 24 ·

97 Thus, we ﬁnd that for the Hpp-waves,

Ω 0 , + = 1 1 Ω Θ dx+ dx− Θ , − = 6 + 12 ∧ ∧ 1 1 Ω Γ dx− Θ dx− Θ Γ . i = − 24 i · ∧ + 8 ∧ · i

For convenience, we denote the fraction of supersymmetry preserved by ν 1 dimKerD. = 32

For the generic Hpp solution with arbitrary Ai j and Θ one ﬁnds ([18]) that the solu-

tions to the Killing spinor equation

Dε 0 (9.7) =

take the form

1 i jk ε(ψ ) exp Θi jk Γ ψ , (9.8) + = 24 + . / where ψ KerΓ is a constant spinor. In other words, for the generic Hpp solution, + ∈ + ν 1 . = 2 There is, however, a special point in the moduli space with

Θ µdx1 dx2 dx3 , (9.9) = ∧ ∧ 1 µ2δ i, j 1,2,3 − 9 i j = Ai j 1 2 (9.10) = ; µ δi j i, j 4...9 − 36 = which preserves all supersymmetry. The explicit expression for the Killing spinors

of this background was given in [18]:

µ µ εψ ,ψ (x) cos x− Id sin x− I ψ + − = 4 − 4 + . . / .µ / / µ cos x− Id sin x− I ψ + 12 − 12 − . . / . / / 1 i 1 i µ µ 6 µ x Γi 2 x Γi sin x− Id cos x− I Γ ψ , (9.11) − 6i 3 − i 4 7 12 − 12 + − +≤ +≥ . . / . / / 2 where I Γ123, I Id and ψ KerΓ are arbitrary constant spinors. = = ± ∈ ± 1 In addition to the maximally supersymmetric Hpp-solution and the generic 2 -BPS solution with arbitrary A, there are a number of other interesting loci in the Hpp-

wave moduli space. In [55] Gauntlett and Hull constructed Hpp-solutions admit-

ting “exotic” values of ν 9 , 5 , 11 , 3 . These solutions possess Killing spinors that = 16 8 16 4 98 lie in KerΓ (often referred to as supernumerary Killing spinors) in addition to the − “generic” ones given in (9.8). As we will see, in these cases the 3-form Θ takes a very

particular form.

We brieﬂy outline how one obtains the form of the Killing spinors in these cases.

Since Ω Ω 0 for all i, j, a Killing spinor ε can always be written in the form i j =

i εψ ,ψ (x) Id x Ωi ϕ , (9.12) + − = + . / 1 where Ωi (Γi Θ 3ΘΓi )Γ and = − 24 + + 1 ϕ exp x−Θ ψ , + = − 6 + (9.13) ! 1 " ϕ exp x−Θ ψ . − = − 12 − As in the generic case, ψ is an arbitra!ry consta"nt spinor annihilated by Γ . How- + +

ever, now the ψ Ker are not arbitrary. Since Ωi always involves Γ , substitut- − ∈ − + ing (9.12) into the Killing spinor equation (9.7) imposes no extra conditions on ϕ . + i But further analysis reveals that (9.7) can be split into independent x− - and x -

dependent parts, the former of which gives the form of ϕ and the latter can be − written as

2 144 A jk Γk X j Γj ϕ 0 , (9.14) 6− j + 7 − = + for each j, where X Γ ΘΓ 3Θ. Finding solutions with supernumerary Killing j = j j + spinors amounts to ﬁnding solutions to this equation.

Let us assume that A has been brought to a diagonal form via an orthogonal trans-

formation so that A diag(µ ,µ ...µ ), µ R. Then in order to ﬁnd solutions = 1 2 9 i ∈ 2 to equation (9.14), we must ensure that the action of X j on spinors is diagonalis-

able. Since Xi involves the 3-form Θ, the natural next step is to ﬁnd a diagonalisable ansatz for Θ.

The Lie algebra of SO(16) is isomorphic to Λ2R9 Λ3R9 so(9) 3 R9 [55]. In par- ⊕ ≡ ⊕ ticular, given a Cartan subalgebra c so(16) (generated by skew-diagonal< matrices ⊂ with real skew eigenvalues), there is a decomposition c c c , where c so(9) and = 2⊕ 3 2 ⊂ c 3 R9. Since the so(16) has rank 8 and so(9) has rank 4, we can associate n 4 3 ⊂ ≤ < 99 2-form generators and 8 n 3-form generators to every Cartan subalgebra c via the − isomorphism.

Now obviously

[c ,c ] c , 2 2 ⊂ 2 [c ,c ] c , 2 3 ⊂ 3

since elements of c2 act as inﬁnitesimal SO(9)-rotations. Note that this doesn’t nec-

essarily imply that c3 is commutative.

If we further assume that c is also a Cartan subalgebra (so that [c ,c ] 0 and hence 3 3 3 = c c 0 as well), a direct calculation [56] shows that only cases that occur are n 1 , 3 = = a2nd 3n 3, and a convenient choices for 2-form and 3-form generators in terms of = gamma matrix monomials are

Γ12, Γ34, Γ56, Γ78

Γ129, Γ349, Γ569, Γ789 in the n 1 case and =

Γ78

Γ123, Γ145, Γ167, Γ246, Γ257, Γ347, Γ356 in the n 3 case. = These two orbit types give rise to 3-form ansätze whose action on spinors can be diagonalised. The ansatz for Θ is then

Θ α dx129 α dx349 α dx569 α dx789 (9.15) = 1 + 2 + 3 + 4

Θ β dx123 β dx145 β dx167 β dx246 β dx257 β dx347 β dx356 , (9.16) = 1 + 2 + 3 + 4 + 5 + 6 + 7

where the α’s and β’s are real parameters. As pointed out in [57], if the α’s are set to

be equal, Θ is proportional to dx9 ω, where ω is the Kähler form on R8. Similarly, if ∧ 100 the β’s agree in the second ansatz, Θ is proportional to the G2-invariant associative

3-form on R7. In both cases each of the three-form terms Γ for i ,i ,i {1...9} i1i2i3 1 2 3 ∈ is a real structure on the spinor bundle, so when diagonalised they act as Id. The ± skew eigenvalues of Θ in the four-parameter case are λ ,a 1...8, where a =

λ α α α α 1 = 1 − 2 + 3 − 4 λ α α α α 2 = 1 + 2 − 3 − 4 λ α α α α 3 = 1 + 2 + 3 − 4 λ α α α α 4 = − 1 − 2 − 3 − 4 λ α α α α 5 = − 1 + 2 + 3 + 4 λ α α α α 6 = 1 − 2 − 3 + 4 λ α α α α 7 = 1 − 2 + 3 + 4 λ α α α α 8 = − 1 + 2 − 3 + 4

Similarly the skew eigenvalues in the seven-parameter case are given by λ6 ,a a = 1...8, where

λ6 β β β β β β β 1 = − 1 − 2 − 3 − 4 + 5 + 6 + 7

λ6 β β β β β β β 2 = − 1 + 2 + 3 + 4 − 5 + 6 + 7

λ6 β β β β β β β 3 = 1 + 2 − 3 − 4 − 5 − 6 + 7

λ6 β β β β β β β 4 = 1 − 2 + 3 + 4 + 5 − 6 + 7

λ6 β β β β β β β 5 = 1 − 2 + 3 − 4 − 5 + 6 − 7

λ6 β β β β β β β 6 = 1 + 2 − 3 + 4 + 5 + 6 − 7

λ6 β β β β β β β 7 = 1 + 2 + 3 − 4 + 5 − 6 − 7

λ6 β β β β β β β 8 = − 1 − 2 − 3 + 4 − 5 − 6 − 7

Note that by choosing suitable α’s or β’s, some of the eigenvalues can be made to vanish, i.e. Θ can have a nontrivial kernel. Equation (9.8) then implies that the subspace of Killing spinors that lies in KerΘ will be independent of x−.

101 We can now work out the possible A that can occur in these ansätze, using the procedure explained in [57]. In the 4-parameter case, X9 4Θ acting on χ , and thus = −

µ2 1 λ2 (9.17) 9 = 9 a

for some choice of λa . That is, a priori we can choose ϕ to be any eigenspinor of −

Θ, and this choice in turn determines the rest of the µi . For example, consider the

direction i 1. To determine µ1, we need to solve the equation (X1 κ1)Γ1ϕ 0. = − − = Substituting X Γ ΘΓ 3Θ, we ﬁnd that 1 = 1 1 +

λaϕ 3Γ1ΘΓ1ϕ κ1ϕ 0 . (9.18) − + − − − =

Looking at the form of Θ, it is easy to see that the eigenvalues of Γ1ΘΓ1 obtained

from those of Θ by reversing the signs of α2, α3 and α4 (since Γ1 anticommutes

with these terms). Applying this to λa is sufﬁcient to solve the previous equation.

Following this procedure we can solve the rest of the µi . A similar argument works

in the 7-parameter case: now X X 2Θ. The possible metrics that can occur can 8 = 9 = be found in Appendix A. Note that in the four-parameter case µ2 µ2, µ2 µ2, µ2 1 = 2 3 = 4 5 = µ2, µ2 µ2 and in the seven-parameter case µ2 µ2. 6 7 = 8 8 = 9 The degeneracy of eigenspinors of Θ satisfying (9.14) gives the dimension of super-

numerary Killing spinors. In the generic case where the coefﬁcients of Θ are arbi-

trary there are 2 supernumerary Killing spinors, but there can be more if the coefﬁ-

cients are chosen so that some of the λa ’s or λ6a’s agree. The conditions for degeneracy are worked out in detail in [55].

In both cases the supernumerary Killing spinors are independent of x− if and only

if µ 0. Furthermore, if one or more of the µ vanish, the Killing spinors will be 9 = i independent of the corresponding transverse coordinates xi . For these solutions

the metric will be decomposable: the product of a lower-dimensional pp-wave with

an Euclidean space.

102 9.3 Symmetric discrete quotients of Hpp-waves

Considering all possible quotients of Hpp-solutions by discrete subgroups of I(M,g ,F) ⊂ I (M,g ) (the subgroup of the isometry group of M,g that also preserves the four-

form F) is somewhat intractable since we have no classiﬁcation of the crystallo-

graphic subgroups of Hpp-wave isometry groups at hand. Therefore, we will restrict

ourselves to quotients that are also symmetric. It is known [58] that a quotient of a

symmetric space M G/H (where G is a Lie group and H is the stabiliser subgroup = of a point) by a discrete subgroup D I (M,g ) is also symmetric if and only if D lies ⊂ in the centraliser of I (M,g ) inside the transvection group G. In other words, we want

to study quotients by discrete subgroups D Z , where ⊂

Z [x I (M,g ) xh hx h G] . (9.19) = ∈ | = ∀ ∈

The isometries and conformal symmetries of Cahen-Wallach spaces were investi-

gated by Cahen and Kerbrat in [59]. They also give expressions for the centralisers

that can occur.

There are two possibilities:

µi Case 1. One of the eigenvalues µi 0 for some i, or Q for some (i, j). Then > µj 4∈

i i Z R [γ γ x+,x−,x x+ α,x−,x ] , (9.20) > = α | α = + . / . / where α R. ∈

µi 2 Case 2. All the eigenvalues µi are negative and Q for all (i, j). We write µi k µj ∈ = − i for all i. Then

i m i Z [γ γ x+,x−,x x+ α,x− β,( 1) i x ] , (9.21) = α,m | α,m = + + − . / . /

πmi where mi Z and β for all i. ∈ = ki

The ratio of mi and ki is the same for all i, and for any (i, j) we can write

ki m j mi . (9.22) = k j 103 The values of all m are determined by any one of them, so in fact Z Z R in this i > ⊕ 9 case. Also note that for N to be orientable, i 1 mi must be even: otherwise the = 1 9 volume form dvol dx+ dx− dx dx5 would not be left invariant. = ∧ ∧ ∧ ··· ∧ Qualitatively speaking, in all cases quotienting by the action of Z consists of peri-

odic identiﬁcations of the light-cone coordinates and Z2-orbifoldings of the trans-

verse coordinates. We observe that all the pp-wave solutions with supernumerary

supersymmetries are examples of Case 2, provided that the ratios of the coefﬁcients

appearing in Θ are also rational.

We are also interested in spinors (in particular Killing spinors) that are left invari-

ant by the quotient. In Case 2, a discrete subgroup D Z is generated by the α,m ⊂

elements γα,0 and γ0,m. Their derivatives acting on the frame bundle are

dγα,0 Id11 11 , = × 1 0 0 0 0 0 ··· 0 1 0 0 0 0  ···  0 0 ( 1)m1 0 0 0 dγ  − m2 ···  0,k =  0 0 0 ( 1) 0 0   − ···   . .. .   . 0 0 0 . .   ···   0 0 0 0 0 ( 1)m9   ··· −    Now suppose that ms1 ,ms2 ...,msr are the odd mi . The fact that we want Dα,m to

preserve orientation means that r is even, as we mentioned previously. Then the

quotient N M/D has four possible spin structures, corresponding to the two = α,m possible lifts of dγ : Γ Γ Spin(9) and Γ Id [14]. Using these 0,m 0,m = ± s1s2...sr ∈ α,0 = ± expressions and equation (9.3), we can then study the existence of invariant spinors

explicitly.

It is easy to see that quotients of solutions for which Case 1 applies are rather trivial:

there are only two possible spin structures and since generic Killing spinors do not

depend on x+, only the trivial lift of γα,0 will preserve any (and in fact all) Killing

spinors. Therefore, in the sequel we will focus on the solutions that admit supernu-

merary Killing spinors.

104 9.3.1 The maximally supersymmetric case

To begin with a simple example before studying the generaly supernumerary case in detail, let us analyse the symmetric discrete quotients of the maximally supersymmetric solution (9.9) and see which choices of spin structure preserve Killing spinors. Now Ai j is diagonal and all eigenvalues of are negative,so Case 2 above ap-

plies. To obtain a quotient isometric to (M,g ,F) as a supergravity solution, we want

to focus on a subgroup Z Z that also preserves the four-form F. Looking at the F ⊂ form of F in (9.9), we observe that an element γ Z will preserve F if and only if α,m ∈

none of (m1,m2,m3) are odd or if two of them are: γα,m acts trivially on dx−. Now

1 3 µ ,i 1...3 ki 1 = = ; µ ,i 4...9 6 = But since the k are equal for i 1...3, equation (9.22) implies that m m m . i = 1 = 2 = 3 Therefore, for γ to preserve F, m m m : 2k for some k Z. Equation 0,m 1 = 2 = 3 = ∈ (9.22) also implies that m ... m : k. We conclude that 4 = = 9 =

i 1,2,3 k 4,...,9 Z [γ Z γ (x+,x−,x ) x+ α,x− β,x ,( 1) x ] , (9.23) F = α,k ∈ | α,k = + + − . / where β 6πk and α R. = µ ∈

A discrete subgroup Dα,k of ZF is generated by elements γα,0 and γ0,1. Looking at

the condition (9.3), it is again obvious that if dγ lifts to Id, no spinors will be α,0 − left invariant. But provided that Γ Id, there are two possible spin structures α,0 = depending on the choice of sign for Γ . ± 0,1 The derivative of γ lifts to the spinor bundle as Γ . Using the familiar trigono- 0,1 ± 4...9 metric identities cos π θ sinθ and sin π θ cosθ, we obtain 2 − = − 2 − = ! " ! " µ µ εψ ,ψ γ(0,1)(x) cos 4 x− Id sin 4 x− I ψ + − = − + + ! " ! ! " µ ! "µ " cos x− I sin x− Id ψ + 4 + 4 − 1 i ! !1 "i ! µ " " µ 6 µ x Γi 2 x Γi cos 4 x− Id sin 4 x− I Γ ψ , (9.24) + 6i 3 + 4 7 + + − +≤ +≥ ! ! " ! " " Comparing this expression with (9.8) and noting that Γ0,1 anticommutes with Γi for

105 i 4...9, we ﬁnd that we can write this as =

εψ ,ψ γ0,1(x) ε I ψ ,I ψ (x) . + − = − + − ! " Thus, the action of γ(0,1) leaves ε invariant if

Γ0,1ψ I ψ , + = − +

Γ0,1ψ I ψ , − = −

Recall that we have chosen the spinor module for which the action of the centre of

the Clifford algebra (and thus the volume element) agrees with Id. Then −

Γ Γ1...9ψ ψ , (9.25) −+ + = − +

implies that (since Γ Γ Γ Id) −+ = − + +

Γ4...9ψ I ψ . + = +

Correspondingly, Γ4...9ψ I ψ . Thus, equation (9.24) is satisﬁed if and only if − = − − Γ Γ . We can therefore conclude that for all symmetric discrete quotients of 0,1 = − 4...9 the maximally supersymmetric Hpp-wave, out of the four possible choices of spin

structure there is precisely one that preserves any (and indeed, all) supersymmetry.

9.3.2 The four-parameter case

Going through all possible quotients of supernumerary pp-wave solutions listed in

Appendix A using the explicit form of the Killing spinors would be somewhat te- dious, so instead we use a method that can be implemented more easily using a computer program (in our case, a Mathematica notebook).

To study the moduli space of possible quotients, we could take the coefﬁcients of Θ as the data, allow them to vary and examine the consequences, as in [55]. However, for computational purposes it is actually more useful to take the 9-tuple (m1,...,m9)

and the eigenvalue λa0 (where λa0 is the eigenvalue chosen to appear in equation (9.17), that is, µ 1 λ2 ) as the data deﬁning a quotient. Looking at the different 9 = 9 a0 106 metrics appearing in Table A, we ﬁnd that it is always possible to express the α’s

— and thus the eigenvalues λa — in terms of ki (recall that the ki are related to

the eigenvalues of the matrix A by µ k2). In other words, we can write λ i = − i a =

i ci ki for each λa and for some coefﬁcients ci . Given (m1,...,m9), we can use

e5quation (9.22) to determine the ki and hence the coefﬁcients α1,...,α4. Knowing

m1 ...,m9,λa0 for the quotient is thus sufﬁcient to determine the original solution.

Restricting to ZF , the subgroup of Z that also preserves the four-form F, we observe

that m9 must always be even. Using the equation (9.22) and the remarks in section

9.2, we also know that m m , m m , m m and m m . To preserve 1 = 2 3 = 4 5 = 6 7 = 8

orientation, i mi has to be even as well, but in this case this imposes no further

restrictions, s5ince there is always an even number of odd mi .

It is convenient to express the Killing spinors using the eigenspinors of Θ as a basis.

1 Note that acting on the λa -eigenspace, Jλ Θ is a real structure. Thus, we can a = λa write the exponentials appearing in χ explicitly as ± 8 λa x− λa x− χ cosh 4 sinh 4 ψ (λa) , (9.26) + = a 1 + + += . . / . // λa0 x− λa0 x− χ cosh sinh ψ (λa ) (9.27) − = 12 + 12 + 0 . . / . //

where Θ ψ (λa) iλa ψ (λa). · ± = ± To determine the fraction of supersymmetry preserved by a symmetric discrete quo-

tient, we need to work out how γ0,m and Γ0,m act on χ . Recall that γ0,m(x−) ± = mi x− π for some i. Computing the action of this shift on χ is straightforward. + ki ±

As mentioned above, we can express the α’s — and thus the eigenvalues λa — in

terms of ki . Thus,

m j m j ki λa ci (9.28) k j k j = i + ci mi , (9.29) = i + so under the action of the isometry, λ x− λ x− π c m . Using this obser- a → a + i i i vation and usual trigonometric identities, we can work 5out how the trigonometric

functions in (9.26) transform under γ0,m. 107 We also need to know how Γ0,m acts on the eigenspinors. Since we’re taking the mi to be our data, it is not hard to enumerate the possibilities. In the four-parameter case, each of m1, m3, m4, m5 and m7 can be even or odd.

If we write the 3-form in this ansatz as Θ α I α I α I α I , we can express = 1 1 + 2 2 + 3 3 + 4 4 any λ as

λ . (λ)α . (λ)α . (λ)α . (λ)α , (9.30) = 1 1 + 2 2 + 3 3 + 4 4

where Ip ψ (λ) i.p (λ)ψ (λ), p 1...4 and .p (λ) 1. It is not hard to see that ± = ± = = ±

any Γ0,m can be written as a product of the Ip ’s or identiﬁed with such a product via

9 the identity Γ1...9ψ ψ that relates the Clifford action of a form on R to that of ± = ± its dual. We can thus always write

Γ .(q)I I ... I 0,m = p1 p2 · · pq acting on ψ and + Γ .(q)I I ... I 0,m = − p1 p2 · · pq acting on ψ , where 1 q 4 and .(q) 1 if q 1 and 1 otherwise. − ≤ ≤ = − =

Since the action of each of the Ip on ψ (λ) is ﬁxed by equation (9.30), the action of ±

Γ0,m on ψ (λ) is given by ±

q Γ0,m ψ (λ) .(q).p .p ....p i ψ (λ) , · ± = ∓ 1 2 q +

With these observations, the problem becomes essentially algorithmic and can be

easily implemented in a symbolic computation environment. We have written a

1 Mathematica notebook that goes through the elements γα,m that generate discrete

subgroups D and computes their action on the Killing spinors. It turns out that for

every quotient, the result is similar to what occurs in the maximally supersymmetric

case: out of the four possible spin structures, there is only one that preserves any of

the original Killing spinors.

1Available upon request from the author.

108 9.3.3 The seven-parameter case

The method we described in the previous section also works in the seven-parameter case, but now we must take care to ensure that the discrete subgroups D also pre-

serve the four-form F. The most convenient way to express this condition is to re-

quire that for each term Γ appearing in Θ, m m m must be even. In other i jk i + j + k words, we have the equations

m m m 0 1 + 2 + 3 = m m m 0 1 + 4 + 5 = m m m 0 1 + 6 + 7 = m m m 0 2 + 4 + 6 = m m m 0 2 + 5 + 7 = m m m 0 3 + 4 + 7 = m m m 0 . 3 + 5 + 6 =

modulo 2. This system of equations is not hard to solve over Z2, and thus we ﬁnd

that the possible forms that Γ0,m can take are:

Γ , Γ , Γ , ± 1247 ± 1256 ± 1346 Γ , Γ , Γ , Γ . ± 1357 ± 2345 ± 2367 ± 4567

Again, we note that all these terms can be written as products of terms in Θ, and

thus the method described in the previous section works, provided that we take the

above constraints into account.

Examining all the possible quotients yields the same result as in the four-parameter

case: for all possible quotients, there is only one choice of spin structure that pre-

serves any (and indeed all) of the supersymmetry of the original background. Fur-

thermore, we do not obtain any new fractions of supersymmetry in either case.

109 9.4 A conjecture

The results of the previous section lead to the curious observation that all symmetric quotients of known symmetric M-theory backgrounds with more than 16 Killing spinors possess a unique spin structure that preserves supersymmetry – in contrast to the supersymmetric space forms described in [52]. The only such backgrounds we haven’t yet considered are Freund-Rubin -type solutions of the form AdS S/Z , × 2 since the only symmetric spherical space form is the projective space[60], and it

is easy to see that there is no ambiguity about the choice of spin structure in this

case – this situation only arises if D 4, where D is the discrete group used in the | | ≥ quotient[52]. Thus we arrive at a

Conjecture. All symmetric quotients of symmetric M-theory backgrounds for which

ν 1 possess a unique spin structure which preserves all of the original supersym- > 2 metry.

We now show that the requirement ν 1 is in fact necessary. > 2 Provided that we drop the requirement of supernumerary Killing spinors, it is not

hard to exhibit examples of symmetric quotients of Hpp-waves that admit more

than one spin structure preserving some Killing spinors. For example, consider a

solution of the form

Θ µ dx129 dx349 (9.31) = 2 + ! µ δ , i, j 1",...4 − 9 i j = µ2 Ai j  δ , i, j 5...8 (9.32) = − 36 i j =  4µ2  , i j 9 . − 9 = =  The solution is obtained by takinga solution preserving 24 supersymmetries given

by taking α α , α α 0 in the four-parameter case and permuting the values 1 = 2 3 = 4 =

of the ki : equation (9.14) is no longer satisﬁed, and thus this solution only admits 16

supersymmetries.

Let us analyse the centraliser of the isometry group. Now k k k k µ , 1 = 2 = 3 = 4 = 3 k ... k µ and k 2µ . Equation (9.22) then implies that m ... m and 5 = = 8 = 6 9 = 3 1 = = 4 110 m ... m : m. Moreover, m 2m and m 4m : in other words, m ,...m 5 = = 8 = 1 = 5 9 = 5 1 4 and m9 will always be even, and since they correspond to the transverse directions

that appear in the form of Θ, all elements of Z will preserve F as well. Thus, Z is of

the form

i 1,2,3,4 k 5,...,8 9 Z [γ Z γ (x+,x−,x ) (x+ α,x− β,x ,( 1) x ,x )] . (9.33) = α,k ∈ | α,k = + + −

Again, Z is generated by γα,0 and γ0,1. These elements lift to the spinor bundle as

Γ Id and Γ Γ . The Killing spinors are of the form given in equation α,0 = ± 0,1 = ± 5678 (9.13). We could, of course, use the method described in the previous section and

decompose the spinorial parameter ψ into eigenspinors of Θ and work out pre- +

cisely how Γ0,1 acts on them, but it is sufﬁcient to observe that (9.3) becomes

εψ γ0,1(x) εΓ1234ψ (x) . + = + ! " For this equation to be satisﬁed we must have Γ5678ψ Γ1234ψ . In other words, + = ± +

ψ must lie in the -eigenspaces of Γ12...8, depending on which lift of γ0,1 we choose. + ± 2 Half of the ψ satisfy this additional condition; Γ12...8 commutes with X , so de- + i manding that its eigenspinors belong to the -eigenspace of Γ is an indepen- ± 12...8 dent constraint.

We conclude that out of the four possible spin structures on the quotient, two admit

no Killing spinors and two preserve 8 of the original sixteen supersymmetries, thus

showing that the inequality in our conjecture must be sharp.

111 Chapter 10

Conclusions

Heitän kirjan luotani Ennen kuin halu antautua Valoa nopeamman matkustamisen Pohdiskelemiseen kokopäiväisesti Käy ylitsepääsemättömäksi

I cast the book away Before the desire to be consumed By thoughts Of faster than light travel Becomes unbearable

— A. W. Yrjänä, Tesserakti

In this thesis we have explored a natural conformal invariant associated to a semi-

Riemannian spin manifold: the conformal Killing superalgebra. We constructed this

object from ﬁrst principles in a manifestly Weyl-invariant way and were naturally led

to introduce a spinorial Lie derivative. However, we were forced to conclude that the

resulting object is not in general a Lie superalgebra.

We have also seen that it is possible to generalise the concept of conformal Killing

spinors to eleven-dimensional supergravity and other supergravity theories. We

have also singled out a subspace of conformal Killing vectors of supergravity back-

grounds — the supernormal conformal Killing vectors — that can be used alongside

the conformal Killing spinors to construct a supergravity conformal Killing superal-

gebra. We showed that M-theory backgrounds that admit a supergravity conformal

Killing spinor distinct from Killing spinors and geometric conformal Killing spinors

112 must be of a very particular type: the metric must be one of the Bryant metrics and the four-form must satisfy a strong integrability condition. We have also exhausted

one possible class of examples, namely the supersymmetric Hpp-wave solutions of

M-theory. Nevertheless, we were able to ﬁnd examples of supergravity conformal

Killing superalgebras in type IIA and the HLW massive IIA supergravities, via Kaluza-

Klein reduction and homothetic Kaluza-Klein reduction, respectively.

Finally — deviating slightly from the main line of development — we saw that as in

the geometric conformal Killing spinor case, there is a relationship between the di-

mension of the space of Killing spinors of a non-simply connected M-theory back-

ground and its spin structure. In particular, we examined the symmetric discrete

quotients of all the known symmetric M-theory backgrounds with more than 16

Killing spinors. In all cases, we found that there is a unique spin structure that pre-

serves all of the original supersymmetry.

All the three threads in this thesis provide ample material for future work. For exam-

ple, it would be interesting to study the interplay between spin structure, symmetry

anda supersymmetry and ﬁnd a formal proof of the conjecture presented in section

9.4. Failing that, as we mentioned in section 9.2, the known Hpp-wave solutions

with supernumerary supersymmetries are very special and a more careful study of

the moduli space of these solutions might reveal loci for which the matrix A is not diagonal but which still admit more than 16 Killing spinors. It would be interesting to see if these solutions could provide counterexamples to Conjecture 9.4.

While we have met our primary goal, we appear to have failed to provide Nahm’s superconformal algebras with a geometric realisation. The algebras on Nahm’s list [2] certainly are Lie superalgebras, but in general conformal Killing superalgebras are not. This is somewhat puzzling since in analogue with the Killing supersymmetry algebra case (as mentioned in the introduction), one would expect at least some of

Nahm’s algebras to have a geometric orgin.

We now make a few speculative remarks to explain why this failure occurs. Namely, some of the superconfomal algebras appear to have a component which has no di-

113 rect geometric analogue, namely the so-called R-symmetry. The R-symmetry generators commute with the even part of the algebra but act nontrivially on the odd part.

R-symmetry does not seem to arise naturally in the context of conformal Killing superalgebras, and it is likely that it is this fact that is responsible for the failure of the odd-odd-odd Jacobi identity (5.18) to vanish.

An example that occurs in the context of so-called AdS/CFT duality [61] provides some hints as to how one might hope to remedy the situation. It is widely believed that type IIB supergravity on the Freund-Rubin background AdS S5 is dual to a 5 × conformal ﬁeld theory that admits a superconformal symmetry algebra living on

the conformal boundary of AdS — that is, R3 S1. In the IIB setting, the super- 5 × symmetries of the theory correspond to supergravity Killing spinors on AdS S5. 5 × In the Freund-Rubin ansatz, these are actually tensor products of geometric Killing

5 spinors on AdS5 and S [61, 62]. The generators of the odd part of the ﬁeld theory su-

perconformal algebra correspond to the conformal Killing spinors of the boundary,

which are geometric Killing spinors from the AdS5 point of view. The natural action

of so(6) (the isometry algebra of S5) on the S5 part of the IIB Killing spinors then

induces the R-symmetry in the four-dimensional CFT. Obviously, the so(6)-action

commutes with the action of isometries of AdS5, the latter generating the conformal

symmetries of the ﬁeld theory superalgebra.

Motivated by this example, one might imagine making a conformal Killing super-

algebra into a Lie superalgebra by adding a central extension to the even part and

tensoring the odd part with the (bundle of ) appropriate representations. Consider

a superalgebra h h h which is not a Lie superalgebra but satisﬁes all Jacobi = 1 ⊕ 2 identities apart from the odd-odd-odd one: that is, we have a nonzero map

J : S3h h (10.1) 1 → 1

deﬁned by the fourth Jacobi identity. Now let g g g , where g h k and = 0 ⊕ 1 0 = 0 × 0 g h V , where V is a k -module and [h ,k ] 0 and furthermore now the Jacobi 1 = 1 ⊗ 0 0 0 = map J : S3g g deﬁned by the fourth Jacobi identity vanishes so that g is a Lie 1 → 1 114 superalgebra. In this construction, k0 plays the role of R-symmetry. Unfortunately,

given the above assumptions for h, there does not appear to be a general recipe for

carrying out this construction as k0 and V have to be put in by hand.

We note that there is a geometric formalism in which one extends M to a superman-

ifold by introducing fermionic coordinates [63]. It is then possible [64, 65] to realise

some of Nahm’s superconformal algebras directly as algebras of superisometries on

the superspace: the R-symmetries then correspond to rotations of the fermionic co-

ordinates. However, introducing supermanifold formalism and associated machin-

ery is beyond the scope of the present treatment.

In spite of the R-symmetry problem,we have presented a variety of conformal Killing

superalgebras in this thesis with the hope that they could perhaps be extended to Lie

superalgebras using the procedure outlined above.

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121 Appendix A

Metrics of pp-waves with supernumerary supersymmetries

122 µ µ µ , i 1...7 8 = 9 i = 1 ( β β β β β β β )2 µ 1 ( 2β 2β 2β β β β β )2 − 36 − 1 − 2 − 3 − 4 + 5 + 6 + 7 1 = − 36 − 1 − 2 − 3 + 4 − 5 − 6 − 7 µ 1 ( 2β β β 2β 2β β β )2 2 = − 36 − 1 + 2 + 3 − 4 + 5 − 6 − 7 µ 1 ( 2β β β β β 2β 2β )2 3 = − 36 − 1 + 2 + 3 + 4 − 5 + 6 + 7 µ 1 (β 2β β 2β β 2β β )2 4 = − 36 1 − 2 + 3 − 4 − 5 + 6 − 7 µ 1 (β 2β β β 2β β 2β )2 5 = − 36 1 − 2 + 3 + 4 + 5 − 6 + 7 µ 1 (β β 2β 2β β β 2β )2 6 = − 36 1 + 2 − 3 − 4 − 5 − 6 + 7 µ 1 (β β 2β β 2β 2β β )2 7 = − 36 1 + 2 − 3 + 4 + 5 + 6 − 7 1 ( β β β β β β β )2 µ 1 ( 2β 2β 2β β β β β )2 − 36 − 1 + 2 + 3 + 4 − 5 + 6 + 7 1 = − 36 − 1 + 2 + 3 − 4 + 5 − 6 − 7 µ 1 ( 2β β β 2β 2β β β )2 2 = − 36 − 1 − 2 − 3 + 4 − 5 − 6 − 7 µ 1 ( 2β β β β β 2β 2β )2 3 = − 36 − 1 − 2 − 3 − 4 + 5 + 6 + 7 µ 1 (β 2β β 2β β 2β β )2 4 = − 36 1 + 2 − 3 + 4 + 5 + 6 − 7 µ 1 (β 2β β β 2β β 2β )2 5 = − 36 1 + 2 − 3 − 4 − 5 − 6 + 7 µ 1 (β β 2β 2β β β 2β )2 6 = − 36 1 − 2 + 3 + 4 + 5 − 6 + 7 µ 1 (β β 2β β 2β 2β β )2 7 = − 36 1 − 2 + 3 − 4 − 5 + 6 − 7 1 (β β β β β β β )2 µ 1 (2β 2β 2β β β β β )2 − 36 1 + 2 − 3 − 4 − 5 − 6 + 7 1 = − 36 1 + 2 − 3 + 4 + 5 + 6 − 7 µ 1 (2β β β 2β 2β β β )2 2 = − 36 1 − 2 + 3 − 4 − 5 + 6 − 7 µ 1 (2β β β β β 2β 2β )2 3 = − 36 1 − 2 + 3 + 4 + 5 − 6 + 7 µ 1 ( β 2β β 2β β 2β β )2 4 = − 36 − 1 + 2 + 3 − 4 + 5 − 6 − 7 µ 1 ( β 2β β β 2β β 2β )2 5 = − 36 − 1 + 2 + 3 + 4 − 5 + 6 + 7 µ 1 ( β β 2β 2β β β 2β )2 6 = − 36 − 1 − 2 − 3 − 4 + 5 + 6 + 7 µ 1 ( β β 2β β 2β 2β β )2 7 = − 36 − 1 − 2 − 3 + 4 − 5 − 6 − 7 1 (β β β β β β β )2 µ 1 (2β 2β 2β β β β β )2 − 36 1 − 2 + 3 + 4 + 5 − 6 + 7 1 = − 36 1 − 2 + 3 − 4 − 5 + 6 − 7 µ 1 (2β β β 2β 2β β β )2 2 = − 36 1 + 2 − 3 + 4 + 5 + 6 − 7 µ 1 (2β β β β β 2β 2β )2 3 = − 36 1 + 2 − 3 − 4 − 5 − 6 + 7 continued on next page

123 continued from previous page µ µ µ , i 1...7 8 = 9 i = µ 1 ( β 2β β 2β β 2β β )2 4 = − 36 − 1 − 2 − 3 + 4 − 5 − 6 − 7 µ 1 ( β 2β β β 2β β 2β )2 5 = − 36 − 1 − 2 − 3 − 4 + 5 + 6 + 7 µ 1 ( β β 2β 2β β β 2β )2 6 = − 36 − 1 + 2 + 3 + 4 − 5 + 6 + 7 µ 1 ( β β 2β β 2β 2β β )2 7 = − 36 − 1 + 2 + 3 − 4 + 5 − 6 − 7 1 (β β β β β β β )2 µ 1 (2β 2β 2β β β β β )2 − 36 1 − 2 + 3 − 4 − 5 + 6 − 7 1 = − 36 1 − 2 + 3 + 4 + 5 − 6 + 7 µ 1 (2β β β 2β 2β β β )2 2 = − 36 1 + 2 − 3 − 4 − 5 − 6 + 7 µ 1 (2β β β β β 2β 2β )2 3 = − 36 1 + 2 − 3 + 4 + 5 + 6 − 7 µ 1 ( β 2β β 2β β 2β β )2 4 = − 36 − 1 − 2 − 3 − 4 + 5 + 6 + 7 µ 1 ( β 2β β β 2β β 2β )2 5 = − 36 − 1 − 2 − 3 + 4 − 5 − 6 − 7 µ 1 ( β β 2β 2β β β 2β )2 6 = − 36 − 1 + 2 + 3 − 4 + 5 − 6 − 7 µ 1 ( β β 2β β 2β 2β β )2 7 = − 36 − 1 + 2 + 3 + 4 − 5 + 6 + 7 1 (β β β β β β β )2 µ 1 (2β 2β 2β β β β β )2 − 36 1 + 2 − 3 + 4 + 5 + 6 − 7 1 = − 36 1 + 2 − 3 − 4 − 5 − 6 + 7 µ 1 (2β β β 2β 2β β β )2 2 = − 36 1 − 2 + 3 + 4 + 5 − 6 + 7 µ 1 (2β β β β β 2β 2β )2 3 = − 36 1 − 2 + 3 − 4 − 5 + 6 − 7 µ 1 ( β 2β β 2β β 2β β )2 4 = − 36 − 1 + 2 + 3 + 4 − 5 + 6 + 7 µ 1 ( β 2β β β 2β β 2β )2 5 = − 36 − 1 + 2 + 3 − 4 + 5 − 6 − 7 µ 1 ( β β 2β 2β β β 2β )2 6 = − 36 − 1 − 2 − 3 + 4 − 5 − 6 − 7 µ 1 ( β β 2β β 2β 2β β )2 7 = − 36 − 1 − 2 − 3 − 4 + 5 + 6 + 7 1 ( β β β β β β β )2 µ 1 ( 2β 2β 2β β β β β )2 − 36 − 1 + 2 + 3 − 4 + 5 − 6 − 7 1 = − 36 − 1 + 2 + 3 + 4 − 5 + 6 + 7 µ 1 ( 2β β β 2β 2β β β )2 2 = − 36 − 1 − 2 − 3 − 4 + 5 + 6 + 7 µ 1 ( 2β β β β β 2β 2β )2 3 = − 36 − 1 − 2 − 3 + 4 − 5 − 6 − 7 µ 1 (β 2β β 2β β 2β β )2 4 = − 36 1 + 2 − 3 − 4 − 5 − 6 + 7 µ 1 (β 2β β β 2β β 2β )2 5 = − 36 1 + 2 − 3 + 4 + 5 + 6 − 7 µ 1 (β β 2β 2β β β 2β )2 6 = − 36 1 − 2 + 3 − 4 − 5 + 6 − 7 µ 1 (β β 2β β 2β 2β β )2 7 = − 36 1 − 2 + 3 + 4 + 5 − 6 + 7 continued on next page

124 continued from previous page µ µ µ , i 1...7 8 = 9 i = 1 (β β β β β β β )2 µ 1 (2β 2β 2β β β β β )2 − 36 1 + 2 + 3 − 4 + 5 + 6 + 7 1 = − 36 1 + 2 + 3 + 4 − 5 − 6 − 7 µ 1 (2β β β 2β 2β β β )2 2 = − 36 1 − 2 − 3 − 4 + 5 − 6 − 7 µ 1 (2β β β β β 2β 2β )2 3 = − 36 1 − 2 − 3 + 4 − 5 + 6 + 7 µ 1 ( β 2β β 2β β 2β β )2 4 = − 36 − 1 + 2 − 3 − 4 − 5 + 6 − 7 µ 1 ( β 2β β β 2β β 2β )2 5 = − 36 − 1 + 2 − 3 + 4 + 5 − 6 + 7 µ 1 ( β β 2β 2β β β 2β )2 6 = − 36 − 1 − 2 + 3 − 4 − 5 − 6 + 7 µ 1 ( β β 2β β 2β 2β β )2 7 = − 36 − 1 − 2 + 3 + 4 + 5 + 6 − 7 Table A.1: Metrics associated to different eigenvalues of Θ for the 7-parameter ansatz.

125 µ µ ,i 1...8 9 i = 1 (α α α α )2 µ µ 1 (α α α α )2 − 9 1 + 2 + 3 + 4 1 = 2 = − 36 1 − 2 − 3 − 4 µ µ 1 (α 2α α α )2 3 = 4 = − 36 1 − 2 + 3 + 4 µ µ 1 (α α 2α α )2 5 = 6 = − 36 1 + 2 − 3 + 4 µ µ 1 (α α α 2α )2 7 = 8 = − 36 1 + 2 + 3 − 4 1 ( α α α α )2 µ µ 1 (2α α α α )2 − 9 − 1 − 2 + 3 − 4 1 = 2 = − 36 1 − 2 + 3 − 4 µ µ 1 (α 2α α α )2 3 = 4 = − 36 1 − 2 − 3 + 4 µ µ 1 (α α 2α α )2 5 = 6 = − 36 1 + 2 + 3 + 4 µ µ 1 (α α α 2α )2 7 = 8 = − 36 1 + 2 − 3 − 4 1 (α α α α )2 µ µ 1 (2α α α α )2 − 9 1 + 2 − 3 − 4 1 = 2 = − 36 1 − 2 + 3 + 4 µ µ 1 (α 2α α α )2 3 = 4 = − 36 1 − 2 − 3 − 4 µ µ 1 (α α 2α α )2 5 = 6 = − 36 1 + 2 + 3 − 4 µ µ 1 (α α α 2α )2 7 = 8 = − 36 1 + 2 − 3 + 4 1 (α α α α )2 µ µ 1 (2α α α α )2 − 9 1 + 2 + 3 − 4 1 = 2 = − 36 1 − 2 − 3 + 4 µ µ 1 (α 2α α α )2 3 = 4 = − 36 1 − 2 + 3 − 4 µ µ 1 (α α 2α α )2 5 = 6 = − 36 1 + 2 − 3 − 4 µ µ 1 (α α α 2α )2 7 = 8 = − 36 1 + 2 + 3 + 4 1 ( α α α α )2 µ µ 1 (α α α α )2 − 9 − 1 + 2 + 3 + 4 1 = 2 = − 36 1 + 2 + 3 + 4 µ µ 1 (α 2α α α )2 3 = 4 = − 36 1 + 2 − 3 − 4 µ µ 1 (α α 2α α )2 5 = 6 = − 36 1 − 2 + 3 − 4 µ µ 1 (α α α 2α )2 7 = 8 = − 36 1 − 2 − 3 + 4 1 (α α α α )2 µ µ 1 (2α α α α )2 − 9 1 − 2 − 3 + 4 1 = 2 = − 36 1 + 2 + 3 − 4 µ µ 1 (α 2α α α )2 3 = 4 = − 36 1 + 2 − 3 + 4 µ µ 1 (α α 2α α )2 5 = 6 = − 36 1 − 2 + 3 + 4 µ µ 1 (α α α 2α )2 7 = 8 = − 36 1 − 2 − 3 − 4 1 (α α α α )2 µ µ 1 (2α α α α )2 − 9 1 − 2 + 3 + 4 1 = 2 = − 36 1 + 2 − 3 − 4 µ µ 1 ( α 2α α α )2 3 = 4 = − 36 − 1 − 2 − 3 − 4 µ µ 1 ( α α 2α α )2 5 = 6 = − 36 − 1 + 2 + 3 − 4 µ µ 1 ( α α α 2α )2 7 = 8 = − 36 − 1 + 2 − 3 + 4 1 ( α α α α )2 µ µ 1 ( 2α α α α )2 − 9 − 1 + 2 − 3 + 4 1 = 2 = − 36 − 1 − 2 + 3 − 4 µ µ 1 (α 2α α α )2 3 = 4 = − 36 1 + 2 + 3 − 4 µ µ 1 (α α 2α α )2 5 = 6 = − 36 1 − 2 − 3 − 4 µ µ 1 (α α α 2α )2 7 = 8 = − 36 1 − 2 + 3 + 4 Table A.2: Metrics associated to different eigenvalues of Θ for the 4-parameter ansatz

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