UUniversidad Consejo Superior deAut´onoma de Madrid Investigaciones Cient´ıficas
Departamento de F´ısica Te´orica Instituto de F´ısica Te´oricaFacultad de Ciencias K ILLING S PINORS - B EYOND S UPERGRAVITY
Alberto R. Palomo Lozano a r X i v : . [ h e p - t h ] M a r niversidad Consejo Superior deAut´onoma de Madrid Investigaciones Cient´ıficas Departamento de F´ısica Te´orica Instituto de F´ısica Te´oricaFacultad de Ciencias E SPINORES DE K ILLING - M ´AS A LL ´A DE LA S UPERGRAVEDAD
Memoria de Tesis Doctoral presentada ante el Departamento de F´ısica Te´oricade la Universidad Aut´onoma de Madrid para la obtenci´on del t´ıtulo de Doctor en CienciasTesis Doctoral dirigida por:
Dr. D. Patrick A. A. Meessen
Investigador
Ram´on y Cajal , Universidad de Oviedoy tutelada por:
Dr. D. Tom´as Ort´ın Miguel
Cient´ıfico Titular, Consejo Superior de Investigaciones Cient´ıficas2012
For the things we have to learnbefore we can do,we learn by doing” `Αριστοτέλης
El trabajo de un f´ısico te´orico es probablemente uno de los trabajos m´as solitarios queexisten. La elaboraci´on de esta memoria es el resultado de numerosas horas en soledaddedicadas primero a entender/ reproducir literatura y c´alculos previos, frutos del esfuerzo deotras personas, y sobre los que se cimentan los trabajos que forman el n´ucleo de mis a˜noscomo doctorando, que han sido finalmente plasmados en estas p´aginas. Dicho esto, hubierasido imposible llegar hasta aqu´ı de haber trabajado exclusivamente por mi cuenta, sin habercontado con la ayuda de varias personas. Quisiera por tanto agradecerles su apoyo duranteestos a˜nos.Un lugar privilegiado ocupan mis supervisores de tesis, Tom´as y Patrick, a quienes quieroagradecer primero el haberme ofrecido la oportunidad de cumplir uno de mis sue˜nos, ytambi´en el haberme dedicado un tiempo y esfuerzo sin el que no podr´ıa haber culminadoesta tesis. Patrick, con qui´en he trabajado m´as directamente, ha sido mucho m´as que undirector acad´emico; me ha instruido en aspectos t´ecnicos relacionados con las construccionesmatem´aticas que figuran en estas p´aginas, pero tambi´en hemos hablado de la vida en laAcademia, y las implicaciones personales que conlleva esta profesi´on. Es una de las personasacad´emicamente m´as capacitadas que jam´as he conocido , y tiene adem´as un sentido delhumor y una capacidad de auto-desaprobaci´on que encuentro muy refrescante, y de lo queespero haberme empapado un poco. De Tom´as he aprendido, o al menos as´ı creo que lo haintentado ´el, a ser m´as pr´actico, y no andarme por las nubes.Este trabajo representa el final de mi andadura como estudiante de posgrado, pero noquisiera olvidarme de tres buenos amigos que me orientaron en los comienzos en Inglaterra, ya quienes debo en gran parte la que ser´ıa mi futura vinculaci´on con la Universidad Aut´onomade Madrid, Daniel Cremades, Manuel Donaire y Javier Fdz. Alcazar.As´ı mismo, debo dar las gracias a mis colaboradores, con quienes he compartido ideas ye-mails estos a˜nos, y de los que he aprendido otra forma de trabajar. Estos son: Jai Grover,Jan Gutowski, Carlos Herdeiro y Wafic Sabra. No puedo tampoco olvidarme de Ulf Gran,bajo cuya direcci´on e instrucci´on pas´e seis fant´asticos meses en la Chalmers Tekniska H¨ogskola de Gotemburgo, Suecia, y que me instruy´o en el m´etodo espinorial de atacar la clasificaci´onde soluciones a teor´ıas de Supergravedad.Aunque he hecho poca vida social en el IFT durante estos a˜nos, he llegado a entablarunos v´ınculos que sobrepasaban lo puramente acad´emico con mis compa˜neros de doctorado,a los que quisiera dar las gracias por los buenos momentos pasados. Tambi´en a Paco, de
Limcamar , con quien he compartido reflexiones sobre f´utbol y el devenir de la Liga 2011-2012. As´ı mismo, no podr´ıa haber llevado a buen culmen toda la parte log´ıstica sin la ayudade un fant´astico equipo administrativo, encabezado por Isabel P´erez. ¡Muchas gracias atodos! Quisiera tambi´en agradecer al director del IFT, Alberto Casas, el haber sido siempre Hace algunos a˜nos dije de organizar una fiesta cuando descubriera un fallo significativo en alguno de losarticulos en que hemos trabajado juntos. Aunque logr´e encontrar uno, seguramente era de transcripcci´on, yno fundamental como yo hubiese querido. an coagente con mis peticiones relacionadas con el nuevo edificio, y a Carlos Pena lo f´acil queresulta hablar con ´el, que aprend´ı cuando estaba organizando el minicurso sobre branas. Esun placer tambi´en mencionar la ayuda inform´atica recibida por parte de Giovanni Ram´ırez,Andr´es D´ıaz-Gil y Charo Villamariz, as´ı como de la familia Carton en los ´ultimos d´ıas antes dela entrega de esta tesis, porque prefiero tirarme por la ventana antes de que falle el ordenador.En este sentido, me gustar´ıa hacer menci´on de las comunidades de open source
GNU/ Linux/L A TEX, que se encargan de producir y mantener software de muy alta calidad, gracias al cualme ha resultado mucho m´as f´acil desarrollar mis tareas. Si todav´ıa usas un sistema operativopropietario, realmente vale la pena hacer la transici´on a una distribuci´on Linux. Juan Joverme ayud´o cuando yo la hice a
Ubuntu , all´a por la release 7.10
Gutsy Gibbon .Debo tambi´en agradecer el apoyo econ´omico recibido durante estos a˜nos por el
ConsejoSuperior de Investigaciones Cient´ıficas , en el contexto de las ayudas para la investigaci´on‘Junta para la Ampliaci´on de Estudios’ JAE 2007.Probablemente la persona m´as feliz por este trabajo sea mi padre, que lleva cuatro a˜nospregunt´andome si todav´ıa no hab´ıa acabado la tesis. “¡Tesis, tesis, tesis!”, ha repetido ennumerosas ocasiones. Es pues que quiero dedicarle estas p´aginas, al igual que a mi madre yhermana, por proveeer el balance adecuado, y por soportar mi tendencia a estar despierto aaltas horas de la noche, as´ı como a mi t´ıa Luz, por estar siempre ah´ı , y a mi re-t´ıo Mat´ıas,que ahora D.E.P. junto a su Se˜nor.Por ´ultimo, quisiera terminar recordando una peque˜na frase que se atribuye a Arist´oteles,pero que a mi me transmiti´o el que fuera mi profesor de
M´etodos num´ericos , el Dr. VentzeslavValev, que dec´ıa que “aprendemos haciendo, no viendo” . Es una m´axima que en su momentono llegu´e a comprender del todo, pero que durante el proceso de este doctorado he llegadoa asimilar y reconocer. ´Este ha sido para mi un viaje que me ha instruido m´as all´a de lopuramente acad´emico, tambi´en acerca de mis propias capacidades, mis l´ımites, e incluso mismecanismos epistemol´ogicos, que espero me sirvan en el futuro. Madrid, junio 2012. ontents N = 2 d = 4 gauged fakeSUGRA coupled to non-Abelian vectors 35 N = 2 Einstein-Yang-Mills . . . . . . . . . . . . . . . . . . . . . . . . . 363.2 Analysis of the timelike case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3 Analysis of the null case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4 Non-BPS solutions to N = 2 SUGRA . . . . . . . . . . . . . . . . . . . . . . 523.5 Summary of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 N = 1 d = 5 minimal fakeSUGRA 59 d = 4 fakeSUGRA . . . . 664.5 Summary of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 N = 1 d = 5 fakeSUGRA coupled to Abelian vectors 69 N = 1 d = 5 fSUGRA and Killing spinors . . . . . . . . . . . . . . . . . . . . 695.2 Analysis of gravitino Killing spinor equation . . . . . . . . . . . . . . . . . . . 715.3 Some simple examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4 The Berger sphere provides a solution . . . . . . . . . . . . . . . . . . . . . . 795.5 Solutions with a recurrent vector field . . . . . . . . . . . . . . . . . . . . . . 835.6 Summary of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 N = 1 d = 4 solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.4 Null N = (1 , d = 6 solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 936.5 Remaining null cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.6 Summary of the chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 NNEXE
I Introducci´on 99II Resumen 101
APPENDICES
A Conventions 105
A.1 Tensor conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.2 Spinorial structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109A.3 Spinor bilinears and Fierz identities . . . . . . . . . . . . . . . . . . . . . . . 112
B Scalar manifolds 117
B.1 K¨ahler geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118B.2 Special geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125B.3 Hyper-K¨ahler and quaternionic-K¨ahler geometry . . . . . . . . . . . . . . . . 129B.4 Real special geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
C Geometrical data for null case solutions 137
C.1 Spin connection and curvatures in d = 4 fSUGRA . . . . . . . . . . . . . . . . 137C.2 Spin connection and curvatures in minimal d = 5 fSUGRA . . . . . . . . . . . 138C.3 Kundt metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 D Weyl geometry 141
D.1 Einstein-Weyl and Gauduchon-Tod spaces . . . . . . . . . . . . . . . . . . . . 142
E Similitude group and holonomy 145
E.1 Sim and ISim as subgroups of the Caroll and Poincar´e groups . . . . . . . . . 145E.2 Holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146E.3 Recurrency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
F The Lorentz and the Spin groups 149References 153 hapter 1
Introduction
These pages are the result of the research undertaken in the
Grupo de Gravitaci´on y Cuerdas ,led by Prof. Tom´as Ort´ın at the Instituto de F´ısica Te´orica UAM/ CSIC, during the tenureof my doctoral degree. They are intended to give an account of the kind of problems attackedduring this time, as well as an illustration of why they can prove of interest, and the methodsconsidered to solve them. In particular, it is a study of various constructions which arisein the context of supergravity theories, and along those lines we have made use of methodsdeveloped during the programme of classification of supersymmetric solutions to said theories,in order to obtain our results.In a very concise manner, Supergravity (SUGRA, for short) is a mathematical constructionmade from adding the algebra of Supersymmetry (which gives the
Super in SUGRA) to theframework of General Relativity. This gives a field theory, which is interesting because onecan recognise in it a spin-2 massless particle, identified with the graviton (the postulatedparticle mediating the gravitational force). In this way the ‘Physics’ behind the mathematicalconstruction appears quite naturally in the construction. For the interested reader, there areseveral available books and review articles which give a detailed account of these topics.The outline of this work is the following. We start in section 1.1 with a quick introductionto Killing spinors and their use in the study of Supergravity. In particular, we comment on theinterest for supersymmetric solutions to supergravity theories, from where we transition tothe core topic of this thesis, which is the application of techniques arising in the search for thelatter to other contexts. We begin by discussing fakeSupergravity in detail. This theory can beunderstood as a spin-off from genuine (regular) SUGRA, which arises by introducing a changeof sign on the potential of the latter, and can be used as a scaffolding from where to obtainbackgrounds with a positive cosmological constant. At last, we comment on possibilities ofusing these supersymmetric techniques to tackle problems of interest in Mathematics.Chapter 2 summarises the programme of classification of supersymmetric solutions to su-pergravity theories. This was initiated in 1983 by Paul Tod, and was pushed to within itscalculational limits in the last decade, where many interesting characterisations were pro-duced. The chapter gives details of the spinor bilinear method pioneered by Gauntlett etal. in [1], as well as the spinorial geometry approach of Gillard et al. [2], including the rˆolethat the mathematical concept of G-structures plays. Chapter 3 gives the full classification ofsolutions of N = 2 d = 4 gauged fakeSUGRA, by considering bilinears formed out of spinors.The solution includes both the case where the vector bilinear’s absolute value is larger thanzero, dubbed the timelike case, and that when it is zero, i.e. the null case. Chapter 4 consid-90 INTRODUCTION ers the problem of characterising the solutions to five-dimensional minimal fSUGRA, also bymeans of the spinor-bilinear formalism. Chapter 5 generalises this work, by presenting theclassification of N = 1 d = 5 fSUGRA coupled to Abelian vector multiplets, but now usingthe spinorial geometry approach. Chapter 6 is an application of the techniques considered inprevious sections to the mathematical investigation of Einstein-Weyl spaces. The problem ofclassifying this kind of spaces has largely been ignored, and only few certain examples havebeen proposed as of present. We add to the spectrum of solutions by considering a modifiedconnection and using the techniques that were useful in classifying fSUGRA solutions, onlythis time we obtain a characterisation of Einstein-Weyl (EW) spaces.Annexes I-II give a summary and conclusions of the work presented, in Spanish language.The last part of the thesis has the appendices, which contain information used during themain chapters of the work. Appendix A has the preferred conventions. It includes thedefinitions for the tensorial calculus used extensively in both four and five dimensions, givenin appendix A.1. Appendix A.2 provides the conventions for the spinorial geometry structures.Appendix A.3 gives account of the spinor bilinears and Fierzs identities, which are relevantto the classification method of chapters 3 and 4. Appendix B defines the geometry of thescalar manifolds which appear naturally in the theories considered. We discuss K¨ahler-Hodge,special K¨ahler, quaternionic-K¨ahler and real special K¨ahler geometries, which appear in thecontext of pure SUGRA theories with vector and hyperscalar multiplets in both four andfive dimensions. Appendix C has some geometric information useful in the characterisationof the null cases in both d = 4 and minimal d = 5 fSUGRA, as well as a little note onKundt waves, which recurrently show up as a solution to this class. Appendix D contains ascholium on Weyl geometry, including annotations on Einstein-Weyl spaces and their subclassof Gauduchon-Tod, which appear naturally as the base-space geometry in the timelike casein d = 4 and in the null case of d = 5. Appendix E includes some basic information on theSimilitude group, which appears when studying null case scenarios. Finally, appendix F isa short introduction to the Lorentz and the Spin groups, and the 2-1 relationship existentbetween them. Spinorial fields parallelised by some connection have been studied in both Mathematics andPhysics for some time now. In Mathematics, this idea is succinctly expressed in the languageof Killing spinors (see e.g. [3] for a review). These are spinors (cid:15) such that ∇ X (cid:15) = λX · (cid:15) , (1.1)for a vector field X , where λ ∈ C is usually called the Killing number , and · is the Cliffordproduct. Moreover, we say that a spinor is covariantly constant (or parallel) if it is Killingwith Killing number λ = 0, so that ∇ (cid:15) = 0 . (1.2)A very remarkable result is the correspondence between Riemannian manifolds admittingconnections with special holonomy and parallel spinors [4]. In hindsight, this is clear sinceif the spin manifold M d carries a Killing spinor, then M is Einstein and has Ricci scalar R = 4 d ( d − λ . (1.3) See appendix E.2 for some information on the holonomy group and special holonomy manifolds. .1. KILLING SPINORS e.g. [8]). This is because they serve to describe spinorial fields parallelw.r.t. a supercovariant derivative which encodes the vanishing of the supersymmetric variationof the fermionic superpartners. In other words, solutions having unbroken SUSY admit Killingspinors.Supersymmetric solutions to SUGRA theories are prescriptions for the bosonic fields ofthe latter such that they solve the equations of motion arising from its action. In principle,a solution to a SUGRA theory, i.e. a theory whose action is invariant under the SUSYtransformations δ (cid:15) e aµ = ¯ (cid:15) γ a ψ µ , δ (cid:15) ψ µ = D µ (cid:15) + . . .. . . (1.4)need not be supersymmetric itself. A supersymmetric solution is defined as one whichpreserves a certain amount (all, or a fraction) of the original supersymmetry. This meansthat if we act on the solution fields with the (corresponding) elements of the SUSY group,these remain invariant. Nothing prevents us from trying to find non-BPS solutions to aSUGRA theory, but solving the equations of motion will in general not prove an easy task.Looking for solutions which preserve at least some fraction of the original SUSY will be moremanagable, and it is in this sense that supersymmetric solutions to supergravity theories havebeen of interest in the last couple of decades.For one, if Nature were to hold a symmetry between fermions and bosons at some highenergy level, it is presumed that its vacuum would be described by some kind of supersym-metric solution. Furthermore, SUGRA has been notably relevant to string theorists ever sinceit was interpreted as a low-energy limit of Superstring Theory, providing with the frameworkof an effective field theory where one has the ability to calculate, and whose solutions arebone fide backgrounds which capture many of the gravitational aspects present in the fullpicture. In this sense, supersymmetric solutions offer a very convenient scenario where to testand develop ideas for the latter. For instance, the study of the quantum regimes where stringtheories purportedly live is by no means an easy task; supersymmetric solutions, on the otherhand, have further stability properties granted by the presence of Supersymmetry, which con-strains the equations of motion that need to be solved, and this can be used in certain casesto prove their non-renormalisability under quantum corrections (see e.g. [9 - 13]).An example of this is the microscopic interpretation of black hole (BH) states, that ledto the renowned matching of macroscopic and microscopic black hole entropy [14]. This wasachieved by considering an asymptotically-flat five-dimensional extremal black hole, wherethe non-renormalisability of the mass-charge relation for supersymmetric bound states allowsfor the counting of microstates. Moreover, the analysis makes use of the so-called attractormechanism of [15 - 17], which was discovered in the study of supergravity models, and saysthat the matter fields on the event horizon of a supersymmetric black hole depend only onthe electric and magnetic charges of the solution, and not on the asymptotic values of thefields. It is also customary to refer to those solutions preserving some residual SUSY as BPS solutions, since theysaturate the Bogomol’nyi-Prasad-Sommerfeld bound. INTRODUCTION
Another instance of foremost importance is that which led to the very famous AdS/CFTconjecture, that relates a string theory/ gravity theory to a quantum field theory (with-out gravity) on its boundary. In [18], Maldacena described the correspondence between anextremal black hole composed of N D3-branes, whose near-horizon geometry is given bythe maximally-symmetric and maximally-supersymmetric
AdS × S , and the N = 4 superYang-Mills theory on R × S , to which the string theory drifts as the string scale goes tozero. This AdS geometry is actually the only maximally-supersymmetric ground state forfive-dimensional minimal gauged SUGRA, and it is presumed that other less-supersymmetricsolutions would be of interest in further studies of the conjecture, as well as in other brane-world constructions of a similar nature. In particular, these BPS solutions with smaller resid-ual SUSY may correspond to states of the conformal field theory expanded around operatorswhich have a non-zero vacuum expectation value.In view of this, it is clear that the connection between the study of supersymmetricsolutions and our understanding of String Theory has proven very fruitful. However, itshould be noted that most of the advances in the topic have been concerned with the studyof asymptotically-flat or anti-De Sitter solutions. Very few advances have been produced inthe area of De Sitter solutions, and little is known about them at present. We shall return tothis topic in the following section. In this section we introduce the core topic of this thesis, which is the application of techniquesarising from the classification of supersymmetric solutions to supergravity theories to othercontexts. We start with a description and motivation to the theory of fakeSupergravity, whichis considered in chapters 3, 4 and 5. Later on we will see that the arguments can be refinedto also classify geometries.As commented above, not much is known about the behaviour of De Sitter solutions.From a Supergravity perspective, this is because there are few models allowing for a positivecosmological constant, and the ones that do are very complicated, which hinders our ability tofind non-trivial solutions. Even though some solutions do exist, they are not general enough,and thus far the achievements obtained with the likes of the attractor mechanism, or theAdS/CFT correspondence, are unmatched. It is in this context that the study of solutions tofakeSupergravity is motivated.Roughly speaking, fakeSupergravity (fSUGRA) arises by performing a Wick rotation onthe Fayet-Iliopoulos (FI) term [19] of a standard SUGRA, which modifies the gauge groupof the theory. Its connection with cosmological solutions is given by the works of Kastorand Traschen (KT). In [20], they created an asymptotically-De Sitter charged multi-blackhole solution by observing that the extreme Reissner-Nordstr¨om-De Sitter black hole solutionwritten in spherical coordinates could be transformed to the time-dependent conforma-staticform ds = Ω − dτ − Ω d(cid:126)x with Ω = Hτ + mr , (1.5)where 3 H is the cosmological constant. As the r -dependent part of Ω is a spherically-symmetric harmonic function, the multi-BH solutions can be created by changing it to amore general harmonic function.Seeing the similarity of the above solution and the supersymmetric solutions to minimal N = 2 d = 4 Supergravity [21], whose bosonic part is just E-M theory, Kastor and Traschen .2. BEYOND SUPERGRAVITY ∇ a (cid:15) I = − iH γ a ε IJ (cid:15) J + H A a (cid:15) I + iF + ab γ b ε IJ (cid:15) J . (1.6)This fermionic rule can be derived from the supersymmetry variations of minimal gauged N = 2 d = 4 Supergravity, which has an anti-De Sitter type cosmological constant Λ = − g ,by Wick-rotating g → iH . Eq. (1.6) looks a lot like a Killing spinor equation, however, unlikea proper KSE, this one does not arise from SUSY considerations. Therefore it is common torefer to an equation like this as a fake-Killing spinor equation (fKSE) [23].In this direction, further works were produced by London in [24], where he generalised theKT solutions to produce higher-dimensional multi-black holes, by showing that his solutionssolved too a suitable fKSE. Also, Shiromizu included spinning solutions in a stringy theoryin [25]. Similarly, the first solutions to five-dimensional E-M-dS theory were given in [26, 27],which were based on proposed Ans¨atze. A little later, Behrndt and Cvetiˇc generalised in[28] the KT backgrounds to solutions to four- and five-dimensional SUGRA theories coupledto vector multiplets, by noting that the difference between the cosmological solutions of [20]and the usual supersymmetric solutions (see e.g. eq. (2.6) in [21]) is given by the linear τ -dependence in Ω. They thus proposed a substitution rule, in which they called for adding apiece (linear in the time-coordinate) to the harmonic functions which recurrently appear inthe study of supersymmetric solutions to SUGRA theories.Furthermore, they showed that their solutions solved fKSEs that could be obtained fromthe KSEs of gauged supergravity with vectors, by Wick-rotating the coupling constant. Theypointed out that this is equivalent to considering an R -gauged symmetry. This can be seenexplicitly e.g. in the construction of gauged N = 2 d = 4 Supergravity coupled to vectormultiplets, which calls for the inclusion of a U (1) Fayet-Iliopoulos term. In terms of the KSEof theory, this FI term is gauged, proportional to the coupling constant (see e.g. [29]). ByWick-rotating the coupling constant, this is equivalent to performing a Wick-rotating on thegauge group, which now becomes R .In summary, the work of Kastor and Traschen opened up a whole new window of research,where new backgrounds can be obtained from the study of theories allowing for parallelspinors w.r.t. a new supercovariant derivative, different from the one prescribed by SUSY.In this sense, Grover et al. pioneered in [30] the programme of systematic classification ofsolutions to theories admitting fKSEs. They used the spinorial geometry techniques of [2]to characterise the timelike solutions of minimal N = 1 d = 5 fakeSupergravity. One ofthe novel results they obtained is that the geometry contains a base-space which is a hyper-K¨ahler manifold with torsion (HKT), whereas it is of hyper-K¨ahler type in ungauged N = 1 d = 5 SUGRA [1], and K¨ahler in the gauged version [31]. Moreover, ungauged and gaugedSupergravities permit the embedding of gravitational models with a vanishing or negativecosmological constant, respectively. Since the procedure involved in fSUGRAs generates achange of sign in the potential of the theory, the resulting cosmological constant is positive,and fSUGRA is often referred to as De Sitter Supergravity.However, a word of caution should be issued here: it has long been known that although ac-tions with local dS Supersymmetry exist, they violate the positive-defineteness of the Hilbertspace, so the corresponding supergravity theory contains ghosts [32, 33]. This of course posesa problem with unitarity, and thus one has to be particularly careful with the interpretationof physical results coming from studies of such a setting. In this sense, the idea of usingspinors that are not parallel under the standard covariant derivative prescribed by SUSY was4 INTRODUCTION (to the best of our knowledge) first introduced in [23, 34]. These fields were later employed toestablish a duality between supersymmetric domain-wall solutions of a supergravity theoryand supersymmetric cosmologies [35], in the context of the Domain Wall/ Cosmology corre-spondence (cf. [36 - 40]). In their construction, Skenderis, Townsend and Van Proeyen requirethe closure of the SUSY algebra, on top of the rotation of the coupling constant. This theydo by introducing a further reality condition on the spinorial and matter fields of the theory,which produces the expected ghost fields. They have indeed a SUGRA action invariant underthe De Sitter group.The situation for our course of study is different, in that fSUGRA is considered as asolution-generating technique. We only perform a Wick-rotation on the coupling constant(FI term) of the corresponding SUGRA theory, and solve the condition of parallelity of thespinorial field under the (resulting) supercovariant connection. There is no demand for theSUSY algebra to hold; this does away with the ghosts, and hence the solutions to fSUGRAare faithful physical backgrounds. Furthermore, starting from a fakeSupergravity theory andtaking the limit of vanishing FI term, one recovers an ordinary Supergravity theory withvanishing coupling constant (Minkowski SUGRA).We have just seen some instances of the link between Killing spinors and certain classesof geometrical spaces. This suggests that the techniques used in the classification of fSUGRAsolutions can be extrapolated to also classify geometries. A case study of this is given inchapter 6, where we have considered a non-vanishing spinorial field fulfilling a particularcondition of parallelity. The condition has been chosen so that its integrability conditiongives rise to the equation defining Einstein-Weyl spaces, thus offering a way to classify these.By way of the bilinear method, we come to characterise all possible EW manifolds which arisein this SUSY-like manner. hapter 2
Classification of supersymmetricSUGRA solutions
In the previous chapter, we have commented on the interest of finding BPS solutions tosupergravity theories. A lot of efforts have been devoted to this end throughout the lastdecades. The classical approach is to take some motivated Ansatz for the bosonic fields,and seek examples that admit the vanishing of the fermionic superpartners, see e.g. [41 - 49].While this approach has given some results, it is of course useful to obtain a more systematicmethod for finding supersymmetric solutions. In particular, given a supergravity theory, it isnatural to ask whether one can obtain all supersymmetric solutions of that theory. This wasdone in 1983 by Tod for the case of minimal N = 2 d = 4 Supergravity [21]. He obtained themost-general background which would saturate the positivity bound for the ADM mass of anasymptotically-flat spacetime that satisfies the dominant-energy condition [50]. Tod’s workwas a completion of an analysis that Gibbons and Hull had started in [51], where they noticedthat a solution saturating the bound would imply that a Dirac spinor (cid:15) i (for i = 1 ,
2) wouldserve as a supersymmetry transformation. Many years later, Gauntlett and collaboratorstook the method one step further by classifying all the supersymmetric solutions of minimalfive-dimensional SUGRA [1]. One then naturally asks why did it take so long, from theoriginal article by Tod, to the feverish activity that started in 2002. The reason is that theoriginal article by Tod employed the Newman-Penrose formalism for General Relativity [52],which is inherently limited to four dimensions. Gauntlett et al. surpassed that limitationby considering a novel techique in the characterisation of solutions. They asummed theexistence of a Killing spinor, and constructed bilinears out of them. These bilinears satisfy anumber of algebraic and differential equations, and their analysis gives a characterisation ofthe supersymmetric configurations .Roughly, the process of characterisation is the following: we consider the vanishing ofthe variations (with respect to a SUSY parameter) of the supersymmetric partners of thebosonic fields of the theory, which gives rules for these fields. We shall generically referto these rules as the Killing spinor equations (KSEs). They are prescribed in terms of therelevant bosonic supergravity fields, as well as the assumed Killing spinor . Among them, When we refer to a configuration, we mean the prescription for the fields of the theory arising from theexistence of the Killing spinor. It will become a solution once the field equations of the theory are also satisfied. Notice that because supersymmetric partners have spin either 1 / /
2, the variation also has to behalf-spin-valued, which a combination of integer-valued bosonic fields and a spinor will respect.
CHAPTER 2. CLASSIFICATION OF SUGRA SOLUTIONS there is one that can be interpreted as a rule for the parallel propagation of the spinor, in theMathematical sense. One then uses these rules to obtain the most general form of the bosonicfields, of course compatible with the KSEs and the existence of a preserved spinor. They forma supersymmetric configuration, since they are consequence of assuming the existence of anon-vanishing spinorial field, which is interpreted as the supersymmetric transformation.Moreover, the configurations thus obtained need to also solve the dynamical field equa-tions. Since the rules obtained above are linear in derivatives, and the equations of motion(EOMs) are second order, one cannot hope to obtain a recipe for solutions to the EOMsstraight away. Instead, one uses the supersymmetric configurations as Ans¨atze from which toobtain supersymmetric solutions, by solving the field equations. In this sense, an observationmade by Gauntlett et al. in [1] (originating from the work in [53], and further formalised in[54]) reduces the amount of work necessary to find the conditions that a supersymmetric fieldconfiguration needs to fulfill. The key idea is that the existence of Killing spinors ( i.e. thesolutions preserving some supersymmetry) implies relations between the EOMs.This was already seen in [9], where they called these relations Killing spinor identities(KSIs), although at that time they used them to prove invariance under quantum correctionsof certain supersymmetric black holes solutions. In particular, these KSIs relate equationsfor fields of spin differing by 1 /
2. In turn, this implies that one only needs to check explicitlya certain number of the components of the field equations, since others are automaticallyfulfilled by means of these relations between them. The KSI is based on the fact that theinvariance of an action under a (super)symmetry implies the following gauge identity δ S = (cid:90) δ Φ A δ (cid:0) √ g S (cid:1) √ gδ Φ A = (cid:90) δ Φ A E A −→ δ Φ A E A (Φ) , (2.1)where we are using a superset of fields Φ A = { B a , F α } ), and we have introduced the notationin which the equation of motion for a field Φ A is written as E A (Φ) = 0. If one then considersthe functional derivative of the last equation in (2.1) w.r.t. some fermion field, and evaluatethe resulting identity for purely bosonic configurations that solve the Killing spinor equations, i.e. F α = δ (cid:15) F α | F =0 = 0, ones sees that0 = δδF β [ δ (cid:15) B a ] (cid:12)(cid:12)(cid:12)(cid:12) F =0 E a . (2.2)This equation is the Killing Spinor Identity , and must hold for any supersymmetric system.Equivalently, the KSIs can be seen as a subset of the integrability conditions of the KSEs.We now proceed to describe the bilinear formalism of [1]. Spinorial geometry techniques,first introduced in [2], are discussed in section 2.2.
The bilinear formalism is a method for characterising supersymmetric solutions. Thus, itstarts with assuming at least one Killing spinor; this is an (cid:15) such that D µ (cid:15) = 0 , (2.3)where D is the covariant derivative that results from demanding the vanishing of the gravitinovariation (w.r.t. (cid:15) ) for the theory in question, i.e. δ (cid:15) ψ iµ = 0. To be specific, lets focus on the .1. BILINEAR FORMALISM N = 1 d = 5 theory of [55], where we shall repeat the calculations presentedin [1]. The bosonic field content of this theory is given by a F¨unfbein (the graviton) e aµ and the graviphoton A µ ; the fermionic content by the gravitinos ψ iµ , which are taken to besympletic-Majorana. There are two of them ( i = 1 , Sp (1) as thegroup defining the condition on the spinors.The action of the theory is S = 14 πG (cid:90) (cid:18) − R dvol − F ∧ (cid:63)F − √ F ∧ F ∧ A (cid:19) , (2.4)and the field equations are0 = R µν + 2 ( F µρ F ν ρ − g µν F τσ F τσ ) , (2.5)0 = d (cid:63) F + 2 √ F ∧ F . (2.6)The supersymmetric transformation rules for the bosonic fields are given by [55] δ (cid:15) e aµ = i (cid:15) i γ a ψ iµ (2.7) δ (cid:15) A µ = i √ (cid:15) i ψ iµ . (2.8)and they shall be used in the context of the KSIs. The gravitino KSE is δ (cid:15) ψ iµ = D µ (cid:15) i = (cid:18) ∇ µ + 14 √ γ µνρ − η µν γ ρ ) F νρ (cid:19) (cid:15) i = 0 , (2.9)where (cid:15) i are again sympletic-Majorana spinors, and, as the KSE is linear in them, we cantake them to be classical commuting spinors.Continuing with the formalism, we proceed to construct bilinears out of the two spinors(see also appendix A.3.2), ˜ f = i ¯ (cid:15) i (cid:15) i (2.10) V a = i ¯ (cid:15) i γ a (cid:15) i (2.11)Φ r ab = ( σ r ) ij ¯ (cid:15) j γ ab (cid:15) i . (2.12)The key step in the method is relating them to the fields in the theory through the KSE(2.9). In order to do this, we now summon the very important Fierz identities, cf. appendixA.3.3. These allow us to obtain several algebraic identities which shall play a major part inthe analysis. The five-dimensional identities are given in eqs. (A.69)-(A.76), and in particular(A.69) says V a V a = ˜ f , (2.13)This implies that V a is either timelike, null or vanishing . The reasoning needed in these twocases are different, and thus we shall treat them sequentially. Its solutions, just as in Tod’sanalysis, will hence vary depending on the case. The possibility of having a zero V is excluded in regions where the spinor is non-vanishing, and we shallonly differentiate among timelike or null case. CHAPTER 2. CLASSIFICATION OF SUGRA SOLUTIONS
We now differentiate the bilinears and use the KSE to obtain the following differentialequations d ˜ f = − √ ι V F (2.14) ∇ ( µ V ν ) = 0 (2.15) dV (cid:91) = 4 ˜ f √ F + 2 √ (cid:63) ( F ∧ V (cid:91) ) (2.16) ∇ µ (Φ r ) νρ = 2 √ F στ (cid:18) g σ [ ν ( (cid:63) Φ r ) ρ ] µτ − g µ [ σ ( (cid:63) Φ r ) τ ] νρ − g µ [ ν ( (cid:63) Φ r ) ρ ] στ (cid:19) , (2.17)where ι V F is the interior product of F and V defined as ι V F = ι V µ ∂ µ ( F νρ dx ν ∧ dx ρ ) = V µ F µν dx ν , and V (cid:91) is the one-form obtained from V through the musical isomorphism (cid:91) : T M → T ∗ M . In components, this is the action of raising/ lowering indices with the metric.Notice eq. (2.15) is saying that V is a Killing vector . Furthermore, taking the exteriorderivative of eq. (2.14), one gets 0 = d ( ι V F ) , which, by the Bianchi identity of the field strength and the definition of the Lie derivative,implies that L V F = 0 . (2.18)This result, plus eq. (2.15), means that V is a symmetry of the putative solution ( g, F ). Also,by totally antisymmetrising eq. (2.17) over the free indices we see that d Φ r = 0 , (2.19)hence the Φ r are closed 2-forms.Before splitting the analysis in the two possible cases, let us consider a subset of theintegrability condition on the supersymmetry transformation of the gravitino. This will givean equation which Kallosh and Ort´ın first called the Killing spinor identity. It is0 = 4 γ ν ∇ [ µ δ (cid:15) ψ iν ] = (cid:26) ( E µσ − g µσ E ρρ ) γ σ + γ µ √ M σ γ σ + 23 B σ γ σ ) − √ M µ + 23 B µ ) (cid:27) (cid:15) i , (2.20)where E µν is, when equal to zero, the Einstein equation, M µ the Maxwell equation and B µνσ the Bianchi identity, such that E µν = e µa E aν , E aν = − √ g δSδe aν , M µ = 1 √ g δSδA µ , B µνσ = ( dF ) µνσ . (2.21)We can get an identity for E ρρ by acting on (2.20) with γ µ (from the left); plugging it backinto (2.20) one gets the KSI relating Einstein, Maxwell and Bianchi equations0 = (cid:40)(cid:16) E µσ − √ (cid:63) B ) µσ (cid:17) γ σ − √ M µ (cid:41) (cid:15) i . (2.22) This will not be the case in the study of fakeSUGRA solutions. .1. BILINEAR FORMALISM i ¯ (cid:15) i γ ν , one gets again the structure of the bilinears, which upon recalling theFierz identities gives 0 = ˜ f (cid:16) E µν − √ (cid:63) B ) µν (cid:17) − √ M µ V ν . (2.23)Acting instead with i ¯ (cid:15) i , one arrives at0 = (cid:16) E µν − √ (cid:63) B ) µν (cid:17) V ν − √
34 ˜ f M µ , (2.24)which in the timelike ˜ f (cid:54) = 0 case is equivalent to eq. (2.23), by means of the Fierz identity(A.69).One can also act on eq. (2.22) with ¯ (cid:15) j ( σ r ) ij γ ν [56], which gives0 = (cid:16) E µσ − √ (cid:63) B ) µσ (cid:17) Φ r νσ . (2.25)In the timelike case we can separate eq. (2.23) in symmetric and antisymmetric parts,which read E µν = √
34 ˜ f M ( µ V ν ) (2.26)( (cid:63)B ) µν = −
14 ˜ f M [ µ V ν ] . (2.27)Notice that this says that if the Maxwell equantion is satisfied, i.e. M µ = 0 ∀ µ , thenboth the Einstein equation and the Bianchi identity are also automatically satisfied. Theseequations will be used in section 2.1.2, when we analyse the equations of motion of the Ansatzfor the timelike supersymmetric configurations.In the null ( ˜ f = 0) case, the tensorial equation (2.23) can be expressed as M µ V ν = 0 , (2.28)which implies that the Maxwell equation is identically satisfied.To proceed, we shall distinguish between the two possible cases. In this case V = ˜ f = 0 and hence ˜ f = 0. Eq. (2.16) expresses that V (cid:91) ∧ dV (cid:91) = 0, i.e. V (cid:91) ishypersurface-orthogonal. The Frobenius theorem of differential geometry then implies that V (cid:91) can be written as V (cid:91) = f du , where f and u are functions. Furthermore, since ι V dV (cid:91) = 0 , (2.29) V is tangent to surfaces of constant u ; this allows us to introduce coordinates ( u, v, y m ) ( m =1 , ,
3) such that the surfaces tangent to V are parametrised by v . Hence V = ∂ v . The metricthen has the form ds = 2 f du ( dv + Hdu + w ) − f − h mn dx m dx n , (2.30)0 CHAPTER 2. CLASSIFICATION OF SUGRA SOLUTIONS where the function f , the function H and the three-dimensional metric h mn do not dependon v , because by eq. (2.15) V is Killing. This is a Kundt metric, as given in eq. (C.28). Usingidentities (A.71) and (A.72) one easily arrives atΦ r = Φ rum du ∧ dx m . (2.31)Since the Φ r forms were previously determined to be closed, cf. eq. (2.19), this implies that( d Φ r ) uvn = 0, whence Φ r is v -independent, and also that ( d Φ r ) umn = 0, which means that byPoincar´e’s lemma we can locally write Φ r as an exact form, i.e. Φ rum = ∂ m y r , (2.32)for some function y r = y r ( u, x m ), as Φ r should not depend on v . Precisely because of thecoordinate dependence of the new functions y r , one can perform a coordinate transformationthat respects the metric (2.30), and rewrite Φ r as Φ r = du ∧ dx r . (2.33)Moreover, we can use the Fierz (A.73) to find h mn Φ rum Φ sun = h rs = δ rs , (2.34)so the base-space metric h mn is Euclidean, i.e. flat.We now proceed to obtain the form of F . To do so, it is easier to work with an orthogonalframe where the full metric is flat; a convenient one is given by e + = f du , θ + = f − ( ∂ u − H∂ v ) ,e − = dv + Hdu + w , θ − = ∂ v ,e i = f − e im dx m , θ i = f e mi ( ∂ m − w m ∂ v ) , (2.35)where e im = δ im , e mi = δ mi because of the flatness of the base-space; the metric is consequentlygiven by ds = e + ⊗ e − + e − ⊗ e + − e i ⊗ e i , (2.36)where i = 1 , ,
3. An inmediate thing to see is that due to eq. (2.14), ι V F = 0, and just as inthe timelike case we use it to symplify the field strength to F = F + i e + ∧ e i + 12 F ij e i ∧ e j . (2.37)Likewise, using eqs. (2.15) and (2.17) one obtains the full form, which reads F = 2 √ f du ∧ (cid:63) (3) ˜ dw + 2 √ f − (cid:63) (3) ˜ df , (2.38)where ˜ d is the derivate w.r.t. to the base-space coordinates x i , i.e. ˜ d = dx i θ i . The Fierzidentity eq. (A.75) in the tangent-space base reads γ + (cid:15) = 0 . (2.39) Observe that the coordinate transformation sends the x i (in the metric and in Φ r ) to y i , which we relabelagain to x i for convenience. .1. BILINEAR FORMALISM γ + has rank 2, this entails that at least half ofthe supersymmetry is preserved.We have so far obtained supersymmetric configurations to the theory in question, but wewould now like to obtain solutions. For this we need to impose the Einstein field equation, aswell as Maxwell’s equation and the Bianchi identity on the gauge field strength. The analysisof these will also give us the final pieces in the characterisation, the descriptions of f and H above. Recall that, because of the KSIs, Maxwell’s equation (2.6) is identically satisfied. TheBianchi identity, dF = 0, gives the following constraints0 = ∇ f − (2.40)0 = ˜ d ( f (cid:63) (3) ˜ dw ) + 3 (cid:63) (3) ∂ u (cid:63) (3) ˜ d ( f − ) . (2.41)Eq. (2.40) implies that f − = f − ( u, x i ) is a harmonic function, and eq. (2.41) can be recastas − (cid:15) ijk ∇ j w k = f − ∇ i φ , (2.42)for an undetermined function φ = φ ( u, x i ). Its integrability condition implies that it isgenerically solved in terms of another harmonic function L = L ( u, x i ).We now analyse the gravity field equation. In view of eq. (2.25), it is easily seen that thecomponents E + − , E + i , E −− , E − i , E ij (2.43)all vanish, thus only E ++ needs to be explicitly solved. The explicit form of this equation ofmotion is not too enlightening and we shall content ourselves with the knowledge that it canbe solved by a function H = H ( u, x i ) that is harmonic w.r.t. the x i coordinates.Observe that we still have remaining gauge freedoms, e.g. shifts to the coordinate v → v + g ( u, x i ), which can lead to simplification of the results. We will, however, refrain fromdoing so, since the purpose of this section is solely to introduce the method. Interested readerscan of course refer to [1] for details, as well as the solution to the problem by means of spinorialgeometry techniques, in section 2.2.1. The end result is that the general supersymmetric nullsolution to d = 5 SUGRA is a geometry given by the metric (2.30) and the field strength(2.38). The functions f and H are harmonic, and fulfill the eq. E ++ = 0, which is solved byexpressing H in terms of a harmonic function H . The 1-form w is given by eq. (2.41), alsosolved by means of another harmonic function, in this case L . Moreover, the three-dimensionalbase-space is flat. In the timelike case, since V is larger than zero, eq. (2.13) implies that ˜ f >
0. The twopossible cases are positive and negative ˜ f . We focus solely on the positive one, for illustrativepurposes; the other case is obtained analogously, and grants the same solution modulo someminus signs and replaces the self-duality conditions by antiself-duality, and vice versa.Because V is timelike, we introduce a time coordinate adapted to its flow, i.e. V = ∂ t .The metric can then be decomposed in conforma-stationary form ds = f ( dt + w ) − f − h mn dx m dx n , (2.44)where h mn is the metric on the four-dimensional base-space M (4) , which is four-dimensional( m, n = 1 , ..., w = w m dx m is a 1-form. Because V is a Killing vector field, the function2 CHAPTER 2. CLASSIFICATION OF SUGRA SOLUTIONS f , the metric h mn and the 1-form w are time-independent. Notice that the presence of the f − factor multiplying the base-space metric guarantees that the Laplacian operator actingon time-independent fields can be expressed solely in terms of h mn .We choose the following Vielbein e = f ( dt + w ) , θ = f − ∂ t ,e i = f − / e im dx m , θ i = − f / w i ∂ t + f / e mi ∂ m , (2.45)where e a and θ a are canonically-dual, i.e. e a ( θ b ) = δ ab , and we have defined e im e in ≡ h mn , w i ≡ w m e mi and e im e mj ≡ δ ij , for i, j = 1 , . . . ,
4. The metric is then given by ds = e ⊗ e − e i ⊗ e i . (2.46)The Fierz identity (A.71) shows that the two-forms Φ r live in the base-space, and by(A.72) they are also antiself-dual. (A.73) can then be written as(Φ r ) mn (Φ s ) np = − δ rs δ pm + ε rst (Φ t ) mp . (2.47)This is the algebra of imaginary unit quaternions (see appendix B). By eq. (2.17) we cansee that Φ r is covariantly constant w.r.t. the Levi-Civit`a connection on the base-space, in-duced from the five-dimensional one. Since Φ r is also compatible with the metric (they are2-forms and thus antisymmetric in their indices), we can say that the base-space has an inte-grable quaternionic structure with three closed K¨ahler forms (given by (Φ r ) mn ), and whence( M (4) , h mn ) is a hyper-K¨ahler manifold.Since, as commented above, ω is time-independent, its exterior derivative dω is a 2-formthat lives on the base-space M (4) . We can therefore decompose f dw = G + + G − , (2.48)where G ± are (anti)self-dual 2-forms on the basespace, i.e. (cid:63) (4) G ± = ± G ± . We can use thisdecomposition to study the field strength. This can be generically be written as F = F i e ∧ e i + 12 F ij e i ∧ e j . (2.49)Then after some calculations with eqs. (2.14) and (2.16) one obtains (respectively) the formof F i and F ij , such that F = 12 √ (cid:16) − f − V (cid:91) ∧ df + G + + 3 G − (cid:17) . (2.50)We now show that the necessary conditions obtained so far are also sufficient to satisfy theKSE for a non-vanishing (cid:15) i , and hence the configuration has some unbroken SUSY. Eq. (A.75)in a Vielbein basis implies that γ (cid:15) i = (cid:15) i . (2.51)Notice this equation also says that (cid:15) i is chiral with respect to the associated spin structuredefined on the base-space, i.e. (cid:15) i = γ (cid:15) i = γ γ (cid:15) i = γ (cid:15) i , (2.52) .1. BILINEAR FORMALISM (cid:15) = 1. Invoking again (A.43), oneobtains that the antisymmetric product of two gamma matrices acting on the spinor is antiself-dual w.r.t. the base-space metric, i.e. ( (cid:63) (4) γ ) jk (cid:15) i = − γ jk (cid:15) i ; (2.53)whence the product G + ij γ ij (cid:15) = 0 (2.54)identically. This is useful for analysing the time-component of the KSE (2.9), which resultsin the statement that the spinor is time-independent, i.e. (cid:15) = (cid:15) ( x ). The spatial componentscan be generically solved by taking an epsilon of the form (cid:15) ( t, x m ) = (cid:15) ( x m ) = f / η ( x m ) , (2.55)where η is such that ∇ m η = 0. In other words, ( M (4) , h mn ) admits a parallel spinor. Fur-thermore, as the base-space is hyper-K¨ahler, we are assured that such a spinor exists [4]. Theprojector ( γ − ) has rank 2, and consequently the configurations presented preserve atleast 1 / ∇ m ∇ m f − = 29 ( G + ) mn ( G + ) mn . (2.56)The Bianchi identity dF = 0, in turn, says that G + is closed, i.e. dG + = 0 . (2.57)The remaining EOMs do not impose any further conditions, as by the KSIs all of thecomponents of the Einstein field equation are automatically satisfied. To summarise, wehave that supersymmetric timelike solutions to minimal d = 5 SUGRA are given by themetric (2.44), where the base-space is time-independent and described by a hyper-K¨ahlergeometry ( M (4) , h mn ) and antiself-dual K¨ahler 2-forms. Also, there is a globally-defined time-independent function f , and a time-independent 1-form w locally-defined on M (4) , such that f dw = G + + G − . Moreover, the field strength is given by F = 12 √ (cid:0) − dt + w ) ∧ df + 3 f dw − G + (cid:1) , (2.58)and eqs. (2.56) and (2.57) ought to be satisfied. So far in this chapter we have described the bilinear method for characterising supersymmetricsolutions to SUGRA theories. One of its foremost ingredients is the construction of formsout of the supersymmetric parameters (spinors) of the theory. By means of the KSE, theseforms are analysed and prescribe the resulting geometry. While we did not comment on itabove, historically the construction of these forms was strongly suggested by the mathematicalconcept of G -structures, and in fact one can reinterpret the method from their point of view.We proceed to give a brief description of these structures, and their relevance to our study.4 CHAPTER 2. CLASSIFICATION OF SUGRA SOLUTIONS
Pure gravity supersymmetric solutions (where the only turned-on field is the vielbein) aregiven by Ricci-flat metrics with special holonomy. By this we mean metrics on a manifold M d whose torsion-free, metric connection has holonomy group smaller than the most-general Gl ( d ) group , and such that it paralellises a spinor, i.e. ∇ (cid:15) = 0 . (2.59)The classification of such possible metrics, in Euclidean signature, was famously given byBerger in 1955 (see appendix E.2 for some more detail). Alas, here we are interested in aLorentzian signature, which is much less studied. Furthermore, we are dealing with theorieswhich also have matter, and hence the spinor will in general no longer be parallel w.r.t. theLevi-Civit`a connection. It is in this sense that G -structures come to our aid, as they provideus with a framework with which to generalise the concept of special holomomy.We define G -structures in terms of the frame bundle F ( M ) on our manifold M d [57, 58].Generically, any element of GL ( d ) will take one choice of frame into another. However, assoon as there is any additional structure defined on M , the group G preserving such structurewill be strictly smaller than GL( d ). This smaller group will serve as the structure group for aframe sub-bundle, and it defines a G -structure. This is similar to SO (1 , d ) being the largestpossible holonomy group on a manifold with a connection that respects lenghts. Thus, theexistence of a G -structure implies that the possible change of frames of F ( M ) is given by G .There is an equivalent definition of G -structures using tensors (see e.g. [58]). In particular,any tensor can be decomposed into representations of the group G . We want to consider thosetensors that are non-vanishing, globally-defined and invariant under G . Precisely because ofthe global aspect, if there is such an invariant tensor, it means that the structure groupof F ( M ) is no longer GL ( d ), but rather G (or a subgroup of it). So we can establish acorrespondence between the existence of a G -invariant tensor, which is something that caneasily be calculated, and having a G -structure.Additionally, there is a relation between G -invariant tensors and torsionful connections[59], which is what is needed for our desired generalisation. We consider the standard covariantderivative ∇ of a G -invariant form Ω; it can be decomposed into irreducible G -modules W i ∇ Ω → ⊕ W i (cid:39) Λ ⊗ g ⊥ , (2.60)where g ⊥ are the elements in so (1 , d ) which are not in the Lie algebra g of G . The right handside is so because Ω is G -invariant, and hence g ◦ Ω = 0. Equation (2.60) can be explicitlyexpanded as ∇ m Ω n ...n r = − K mn p Ω pn ...n r − K mn p Ω n p...n r − . . . − K mn r p Ω n ...n r − p , (2.61)where the K mnp ∈ ⊕ W i label the modules. The decomposition allows us to define a newcovariant derivative ∇ (cid:48) ≡ ∇ + K with torsion K . Since ∇ (cid:48) Ω = 0 by construction, theholonomy group of this new connection is inside of G . Note that this serves as the desiredgeneralisation, since when all the modules W i vanish, it is equivalent to ∇ Ω = 0, i.e. havingspecial holonomy inside of G . Having the modules turned on implies deviating away fromspecial holonomy, which is what we want for classifying supersymmetric solutions to SUGRAtheories with matter. Having chosen a metric-compatible connection, SO ( d ) is in fact the largest possible holonomy groupallowed. .2. SPINORIAL GEOMETRY TECHNIQUES G -invariant tensors, and they encode the existence of a G -structure, where the K mnp aregiven by the matter content of the theories studied. This G -structure is defined in terms ofthe supersymmetry parameters (cid:15) i , which are solutions to the KSE of (2.9). The bilinearsconstructed in the timelike case determine an SU (2)-structure, while in the null case its an R -structure. In this section we introduce the
Spinorial Geometry technique for classifying supersymmetricsolutions to supergravity theories. In [2] Gillard et al. characterised the solutions of d = 11SUGRA for N = 1 , , , Spin (1 ,
10) togreatly simplify the calculations. This approach stands on the shoulders of G-structures, andproves very adequate to treat problems involving a high number of dimensions and severalpreserved supersymmetries, where the bilinears method presented in the previous sectionbecomes increasingly unmanageable. In this sense, this method has been used to extensivelyclassify solutions in many scenarios (see e.g. [60 - 69]). The study of chapter 5 has beenproduced considering these techniques , which we now review.The starting idea behind the formalism is to consider the spinors as forms [70 - 72]. Thisgives us the means to introduce an explicit basis of spinors, which (and this is where theeffectiveness of the method lies) due to the gauge symmetry of the theory, we can reduce toa simple expression/s. These canonical spinors are then introduced into the Killing spinorequation for an explicit evaluation, which translates into a linear system of algebraic anddifferential equations. These equations no longer have Gamma matrices in them, and can besolved to obtain the form of the resulting geometry.From an abstract point of view, the spinorial fields for different theories come in differentirreducible representations of the Spin group, depending on the particular algebraic structureassociated to the signature of the theory in question, which allows or not for certain restrictionson the spinors. A summary of this is given e.g. in [73] or [74, appendix B]. These spinorialrepresentations each have a number of R independent components, with the minimal numbercorresponding to the smaller representation in a given dimension d .Generically, the space of Dirac spinors has 2 (cid:98) d/ (cid:99) (complex) components, and one canrecast them in terms of the complexified space of forms on R (cid:98) d/ (cid:99) , which is denoted mathe-matically by ∆ = Span C (1 , e , . . . , e (cid:98) d/ (cid:99) ) = Λ ∗ ( R (cid:98) d/ (cid:99) ⊗ C ). The canonical basis for ∆ is thengiven by ξ i = { , e i , e i i , e i i i , . . . , e i ...i (cid:98) d/ (cid:99) } , (2.62)where each index i takes values in { , . . . , (cid:98) d/ (cid:99)} and e i i ...i p ≡ e i ∧ e i ∧ . . . e i p . Thedimensionality of the space of forms is thus given by the sum of the number of p -forms, where p = 0 , . . . , (cid:98) d/ (cid:99) . This can be shown to equal (cid:98) d/ (cid:99) (cid:88) d =0 (cid:18) (cid:98) d/ (cid:99) d (cid:19) = 2 (cid:98) d/ (cid:99) , (2.63) The results can also be obtained by employing the bilinear method. CHAPTER 2. CLASSIFICATION OF SUGRA SOLUTIONS thus agreeing with the (complex) dimensionality of the space of Dirac spinors. A generalDirac spinor can then be written as (cid:15) = µ (0) µ (1) i e i + µ (2) i i e i i + . . . + µ ( (cid:98) d/ (cid:99) ) i ...i (cid:98) d/ (cid:99) e i ...i (cid:98) d/ (cid:99) , (2.64)where all the µ ( n ) s are complex-valued functions, antisymmetric in all their n -indices.The given isomorphism also needs to carry a recipe for the action of the Lorentz algebraon the spinors. In this sense, the representation chosen for the action of the Gamma matriceson ∆ will depend on the theory in question, since these need to respect the Clifford algebradefining-relation { γ a , γ b } = 2 η ab , (2.65)where η in our case is diagonal mostly-minus. For odd-dimensional M (1 ,d ) theories , one can e.g. choose γ i (cid:15) = i ( e i ∧ (cid:15) + ι e i (cid:15) ) , γ i +( d/ (cid:15) = − e i ∧ (cid:15) + ι e i (cid:15) , (2.66)where i = 1 , . . . , ( d/
2) and (cid:15) is a Killing spinor that can be expressed in term of forms, as ineq. (2.64). Observe that the action of the γ i s sends the space of even forms to the space ofodd form, and vice versa. The additional gamma matrix is given by γ = k γ . . . γ d , (2.67)where k = k ( η, d ) is a constant that equals either i or 1, depending on the metric’s signatureand the dimensionality of the theory, so that γ squares to , in accordance with our choiceof metric.To study smaller representations of the Spin group, we need a definition for the innerproducts on the space of forms. We do so by using the charge conjugation matrix C = γ ( d/ . . . γ d , (2.68)and an extension of the Euclidean inner product on R d/ to ∆ (cid:104) z i ξ i , w j ξ j (cid:105) = − (cid:98) d/ (cid:99) (cid:88) i =1 ( z i ) ∗ w i , (2.69)where ( z i ) ∗ is the complex conjugate of z i , and only identical forms (up to reordering ofindices) have non-vanishing product. This inner product is not invariant under the Spingroup, but we can define the Dirac inner product on ∆ as D ( η , η ) = (cid:104) γ η , η (cid:105) = ¯ η η , (2.70)which is so by construction. Notice that (¯ η ) α = ( η † γ ) α = ( η β ) ∗ ( γ ) βα is the Dirac conjugateof η α . We also define the Majorana inner product as B ( η , η ) = (cid:104)C η ∗ , η (cid:105) = η c η , (2.71) Interested readers can consult [60] for an explicit representation suitable for type IIB theory. In (1 , d odd ),one can essentially use the equations in (2.66) to represent d − d th one. Note that on even-dimensional theories there are actually two Spin-invariant Majorana inner products:one defined as ˜ B ( η , η ) = (cid:104) γ γ ( d/ . . . γ ( d − η ∗ , η (cid:105) , and one defined as ˜ A ( η , η ) = (cid:104) γ . . . γ ( d/ η ∗ , η (cid:105) [60]. .2. SPINORIAL GEOMETRY TECHNIQUES B ( − , − ) is antisymmetric by virtue of the form of the gamma matrices (2.66). TheMajorana conjugate of η α is given by η c α = − ( η T C ) α = − η β C βα . The Majorana conditionon η is then given by demanding that the Dirac and Majorana conjugation are equal, i.e. η ∗ = γ C η . (2.72)Spinors satisfying this condition are called Majorana spinors, and are used e.g. in the studyof M-theory backgrounds [2, 61].For the purpose of establishing a connection between the geometrical structures of thetangent and curved spaces, note that one can also construct k -forms from the spinors (where I, J go from 1 to N ) in the following way [2] α IJ ≡ α ( η I , η J ) = 1 k ! B ( η I , γ a ... a k η J ) e a ∧ . . . ∧ e a k . (2.73)Observe that by using Hodge duality one can find the higher order forms Ω ( d/ , . . . , Ω d to be dual or antiself-dual to the lower order ones Ω , . . . , Ω ( d/ . Also due to the symmetryproperty of B ( − , − ) , α JI = − α IJ for 0, 3 and 4-forms and α JI = α IJ for 1,2 and 5-forms, soone only needs to compute α IJ for pairs of spinors ( η I , η J ), where I ≤ J . This and otheranalogue constructions of forms give a method to obtain Killing vector fields, along whosedirections to introduce adapted curved space coordinates.This completes the description of spinors in terms of forms. The identification allows usto explicitly construct a basis of spinors for any given theory, and furthermore cast them incanonical form. Moreover, it will permit us to find the stability subgroups of the spinors inside Spin (1 , n ), which will in general be related to the G-structure of the solution backgrounds.The stability groups define classes of spinors (given by their orbits under such groups), sowe can use a representative of each orbit to substitute into the KSE and operate on. Thiswill generate a system of equations for the geometry and fluxes of the theory, all of whichare locally expressed in terms of functions that are originally undetermined. The solution tothese equations will give the resulting geometry, which is shaped by supersymmetry, henceserving to fully classify the theory’s vacua. As an example, we repeat the analysis done on thesupersymmetric solutions to minimal N = 1 d = 5 SUGRA, this time using these techniques. d = 5 SUGRA
In five dimensions, the minimal spinor used to construct the theory is a symplectic-Majorana(SM) spinor . The isomorphism outlined above is then to the complexified space of formson R , which we again denote by ∆ = Λ ∗ ( R ⊗ C ). A basis of ∆ is given by ξ i = { , e , e , e } , (2.74)where e ≡ e ∧ e . The action of the Gamma matrices on ∆ is represented by γ i (cid:15) = i ( e i ∧ (cid:15) + ι e i (cid:15) ) , γ i +2 (cid:15) = − e i ∧ (cid:15) + ι e i (cid:15) , (2.75) Throughout this thesis we will be employing the notation that a symplectic-Majorana spinor is defined asa pair of Dirac spinors on which one imposes a reality condition. We say we have a single spinor since the spinstructure group is Sp (1), in our notation. CHAPTER 2. CLASSIFICATION OF SUGRA SOLUTIONS where i = 1 ,
2. The zeroth element of the Clifford algebra is given by γ = γ ; its actionon 1 and e is given by the identity operator and on e and e by minus the identity γ , γ e = − e ,γ e = e , γ e = − e , (2.76)and thus it squares to .A generic Dirac spinor can then be written as (cid:15) = λ µ e + µ e + σe , (2.77)where λ , µ , µ and σ are complex functions, representing the eight real degrees of freedomof an unconstrained complex spinor in five dimensions. One can see that imposing the realitycondition (2.72) on (cid:15) does not work, which was to be expected since there are no Majoranaspinors in d = 5. Instead, we are lead to consider the symplectic-Majorana representation ona pair of Dirac spinors (cid:15) and (cid:15) . This is given by (cid:15) i ∗ = ε ij B (cid:15) j , (2.78)where B = − γ C = − γ (cf. appendix A.2.2). Thus, we can generically write (cid:15) = −B (cid:15) ∗ ,where the dim C ( (cid:15) ) = 2 (cid:98) / (cid:99) = 4.We now consider the null case, leaving the timelike one for section 2.2.2. Precisely, beingin the null class implies that we need to fulfill that (see eq. (A.65)) which can be recast interms of the following nullity condition [75] B ( (cid:15) , (cid:15) ) = B ( (cid:15) , γ (cid:15) ∗ ) = 0 , (2.79)where the inner product is defined in (2.71). Since (cid:15) is a generic Dirac spinor, given byeq. (2.77), the condition then reads | λ | − | µ | − | µ | + | σ | = 0 . (2.80)We thus have seven real degrees of freedom, that make up the general null supersymmetricparameter. Furthermore, we can reduce the form of (cid:15) by using the Spin (1 ,
4) gauge freedom ofthe theory. This will make the calculations for solving the KSE much simpler. In particular,one can show that starting from any spinor of the form eq. (2.77) which solves (2.80), byacting on it with
Spin (1 , (cid:15) = 1 + e .Alternatively, one can see eq. (2.80) as saying that there is a null complex 4-vector living on M (2 , C , whose inner product is invariant under U (2 , SU (2 , i.e. the nullity condition. We can use this in our searchfor a simpler form of a null (cid:15) with the same generality as that of eq. (2.77). From an algebraicpoint of view, SU (2 , (cid:39) Spin (2 , ⊃ Spin (1 , v in the vector representation of SU (2 ,
2) goes to a 4 s in Spin (2 , s irrep in Spin (1 ,
4) [76], the minimal spinor in d = 5 being symplectic-Majorana. Thisimplies that there is only one orbit of Spin (1 ,
4) on (cid:15) , and hence starting from any elementof the form (2.77), satisfying eq. (2.80), one can generate the most-general seven DOFs.Moreover, the solutions we are about to find preserve at least half of the original supersym-metry. We show this by using the operator γ C ∗ , which is invariant w.r.t. the supercovariantderivative eq. (2.9), i.e. γ C ∗ γ µ = γ µ γ C ∗ , (2.81) .2. SPINORIAL GEOMETRY TECHNIQUES µ = 0 , i , and where ∗ is the complex conjugation operator, i.e. ∗ η = η ∗ ∗ . We then seethat if (cid:15) = 1 + e is a Killing spinor, then so is γ C ∗ (cid:15) = − e + e .In order to perform the calculation of the KSE, it is useful to cast the Gamma matricesin terms of a light-cone basis { Γ a } Γ + = 1 √ γ − γ ) , Γ − = 1 √ γ + γ )Γ = 1 √ γ − iγ ) = √ i e ∧ , Γ ¯1 = 1 √ γ + iγ ) = √ i ι e , Γ = γ . (2.82)In terms of these coordinates, the metric is thus given by ds = 2 e + ⊗ e − − e ⊗ e ¯1 − e ⊗ e . (2.83)Imposing the KSE gives a linear system of equations, that can be solved to give a relationbetween the geometry (spin connection) and the field strength, i.e. F + − = F +1 = F +2 = 0 ,F − = i √ ω − , , F − = − i √ ω − , , F = − i √ ω − , +2 , F = − i √ ω , + − , (2.84)as well as the following restrictions on the spin connection components ω + , + − = ω + , +1 = ω + , +2 = ω + , = ω + , = ω − , + − = ω , +1 == ω , +¯1 = ω , +2 = ω , = ω , +1 = ω , +2 = ω , = 0 , (2.85) ω , ¯12 = 2 ω − , +2 = − ω , + − = ω ¯1 , , (2.86) ω , = − ω − , +1 = 2 ω , + − = ω , , (2.87)and those related by complex conjugation. The 2-form is thus given by F = iq √ ω − ,jk (cid:15) ijk e − ∧ e i − i √ q (cid:15) ijk ω k, + − e i ∧ e j , (2.88)where q (cid:15) = 1.We now introduce the basis of vector fields { θ a } dual to that of 1-forms. In particular,we perform the following calculations considering the vector field θ + ≡ N = ∂ v , which isnormalised as ι N ( e a ) = e a ( θ + ) = δ a + , and defines the direction of v , a coordinate on thecurved manifold. Consider now the following Lie derivatives of the Funfbein along N L N e + = ( ω a, + − + ω + , − a ) e a , L N e − = ω + , + a e a , L N e i = − ( ω a, +¯ i + ω + , ¯ ia ) e a , (2.89) The constant q is used to tune the field strenghts in the two approaches. CHAPTER 2. CLASSIFICATION OF SUGRA SOLUTIONS for a = { + , − , i } , i = { , ¯1 , } and ¯2 = 2, which imply( L N e a ) + = 0 , ( L N e + ) − = 0 , ( L N e + ) i = ω i, + − + ω + , − i , ( L N e − ) − = 0 , ( L N e − ) i = 0 , ( L N e i ) − = ω + , − ¯ i − ω − , +¯ i , ( L N e i ) j = 0 . (2.90)Furthermore, one can use the residual gauge freedom (those spinorial transformations respect-ing the chosen representative for the spinor (cid:15) = 1 + e ) to simplify some of these results. Inparticular (see e.g. [77, appendix B]), one can set( L N e i ) − = 0 , (2.91)This, along with eqs. (2.86) and (2.87), gives ω + , − i = ω − , + i = − ω i, + − , (2.92)which implies that L N e + = 0, and allows us to express de a as de + = ω − , − i e − ∧ e i + 12 ( ω i, − j − ω j, − i ) e i ∧ e j ,de − = 2 ω − , + i e − ∧ e i ,de i = − ( ω − , ¯ ij + ω j, − ¯ i ) e − ∧ e j + 12 ( ω j,k ¯ i − ω k,j ¯ i ) e j ∧ e k , (2.93)We now focus on de − , as it is hypersurface orthogonal ( i.e. de − ∧ e − = 0), which we shallnow use in the analysis of the problem. The Frobenius theorem allows us to recast e − as e − = f du , (2.94)where f is a generic real function, which by the system of eqs. (2.90) we know it is independentof v , and where u is another (curved) coordinate.To continue with the analysis, we integrate L N e + = 0 to obtain e + = dv + α , (2.95)where α = Hdu + w m dx m is a 1-form independent of v , and the normalisation condition e + ( θ + ) = 1 has also been considered. The three remaining curved coordinates are given as e i = f − e im dx m , (2.96)where the du terms have been eliminated by using again the residual gauge freedom of thetheory , and dv -terms are prohibited by the condition e i ( θ + ) = 0. Furthermore, L N e i = 0implies that the e im are v -independent, i.e. e im = e im ( u, x n ).Consider now other Lie derivatives on the Vielbein in terms of curved space coordinates.These establish relations that allow us to describe the field strength and its field equation. In The induced effect on e + can be eliminated by considering a redefined α function. .2. SPINORIAL GEOMETRY TECHNIQUES L θ − e a and L θ i e a implies that ∂ u w m = ω − , − i e im + ∂ m H , (2.97) ∂ m f = 2 ω i, + − e im , (2.98) f − e im ∂ u f = ∂ u e im + f e jm ( ω − , ¯ ij + ω j, − ¯ i ) , (2.99)( ˜ dw ) mn = 2 f − e i [ m e jn ] ω i, − j , (2.100) ∂ [ m e in ] = 0 , (2.101)where for the last equality we have used eqs. (2.86), (2.87). By using the defining antisym-metry of the spin connection, one can then show that the base-space is flat, in accordancewith the bilinears’ analysis above. Thus, the solutions belonging the null case have a metricgiven by ds = 2 f du ( dv + Hdu + w m dx m ) − f − h mn dx m dx n , (2.102)where h mn = e im e in = e im e in = δ mn . Furthermore, by making the gauge choice ω − ,ij = − ω i, − j (2.103)and using e j ( θ i ) = δ ji = e jm e mi , as well as eqs. (2.98) and (2.100), we have a field strengthgiven by F = − iq √ f du ∧ (cid:63) (3) ˜ dw − i √ q f − (cid:63) (3) ˜ df , (2.104)Choosing (cid:15) = 1 /
4, we arrive at the same result for the field strength as in section 2.1.1. d = 5 SUGRA
We now focus in the class of timelike solutions. Because we are still in the minimal five-dimensional case, the first two paragraphs of section 2.2.1 still hold, which describe thespinorial structure of the theory. For this analysis, however, it will prove useful to follow [30]and cast the Gamma matrices in terms of a different basis { Γ , Γ α , Γ ¯ α } , where the 0-directionwill be associated with physical time, and α = { , } , ¯ α = { ¯1 , ¯2 } Γ = γ , Γ α = √ i e α ∧ , Γ ¯ α = √ i ι e α . (2.105)The flat metric is thus given by ds = e ⊗ e − e α ⊗ e ¯ α − e ¯ α ⊗ e α . (2.106)As in the previous analysis, we want to use the Spin (1 ,
4) symmetry of the theory to find asimple canonical form for the spinor. One can show that it can be reduced to (cid:15) = f
1. Theanalysis of the KSE (2.9) with this spinor then gives the following set of equations θ ff − ω ,µµ + 12 √ F µµ = 0 , (2.107) ω , µ − √ F µ = 0 , (2.108) (cid:18) − ω , ¯ α ¯ β + 1 √ F ¯ α ¯ β (cid:19) (cid:15) ¯ α ¯ β = 0 , (2.109)2 CHAPTER 2. CLASSIFICATION OF SUGRA SOLUTIONS θ α ff − ω α,µµ + √ F α = 0 , (2.110) − ω α, β + √ F α ¯ β + 1 √ F µµ δ βα = 0 , (2.111) − ω α, ¯ µ ¯ ν (cid:15) ¯ µ ¯ ν − √ F µ (cid:15) αµ = 0 , (2.112) θ ¯ α ff − ω ¯ α,µµ + 12 √ F α = 0 , (2.113) ω ¯ α, β − √ F ¯ α ¯ β = 0 , (2.114) ω ¯ α, ¯ µ ¯ ν (cid:15) ¯ µ ¯ ν = 0 . (2.115)Their analysis gives F α = √ ω , α , (2.116) F αβ = √ ω ,αβ , (2.117) F α ¯ β = 1 √ (cid:0) ω α, β + ω µ, µ δ α ¯ β (cid:1) , (2.118)as well as the following restrictions on the spin connection components ω α, β = ω ,αβ , (2.119) ω α, β = − ω ¯ β, α , (2.120) ω µ, µ = ω ,µµ , (2.121)( θ f ) /f = 0 , (2.122) ω ¯ µ,µα = − ω α,µ ¯ µ = 12 ω , α , (2.123) ω α,βγ = 0 , (2.124)along with their complex conjugates. Furthermore, ω , α = − θ α ff (2.125)As before, we use the conditions just found to analyse the geometry of the problem. Weintroduce curved-space coordinates { t, x m } , where m = (1 , , , e = g ( dt + w m dx m ) , e α = g − / e αm dx m , e ¯ α = g − / e ¯ αm dx m , (2.126)where g is a generic function. The canonically-dual vector fields are given by θ = g − ∂ t , θ α = − g / e mα w m ∂ t + g / e mα ∂ m , θ ¯ α = − g / e m ¯ α w m ∂ t + g / e m ¯ α ∂ m , (2.127) .2. SPINORIAL GEOMETRY TECHNIQUES e αm e mβ = δ αβ , e αm e m ¯ β = 0, e ¯ αm e m ¯ β = δ ¯ α ¯ β . As expressed in [78], supersymmetricsolutions to SUGRA theories will always have a Killing vector field, which we can use tosimplify the problem. This was for example implicitly used in section 2.1.2 above, whenthe coordinate t was chosen along the flow of Killing vector V , which guarantees that thefields contained in the metric are all t -independent. Similarly, we can consider the vector V = f θ = f g − ∂ t . Because of eqs. (2.122) and (2.125), V is Killing; we then choose g = f so that the function g , the 1-form w = w m dx m and the four-dimensional base-space( B ) metric ds B = ˜ e α ⊗ ˜ e ¯ α + ˜ e ¯ α ⊗ ˜ e α = h mn dx m ⊗ dx n , (2.128)where ˜ e α = g / e α and h mn = e αm e ¯ αn + e ¯ αm e αn = e αm e αn + e ¯ αm e ¯ αn , are all time-independent.Consider now the Lie derivative of the F¨unfbein along θ L θ e = ω , a e a , L θ e α = − ( ω , ¯ αa − ω a, ¯ α ) e a , L θ e ¯ α = − ( ω ,αa − ω a,α ) e a , (2.129)for a = (0 , α, ¯ α ). The first implies that ∂ t w m = g − ( ω , α e αm + ω , α e ¯ αm ) + g − ∂ m g . (2.130)The two last read( L θ e α ) = 0 , ( L θ e ¯ α ) = 0 , ( L θ e α ) β = − ( ω , ¯ αβ − ω β, ¯ α ) , ( L θ e ¯ α ) β = − ( ω ,αβ − ω β,α ) = 0 , ( L θ e α ) ¯ β = − ( ω , ¯ α ¯ β − ω ¯ β, ¯ α ) = 0 , ( L θ e ¯ α ) ¯ β = − ( ω ,α ¯ β − ω ¯ β,α ) , (2.131)where we have used the eq. (2.119). Furthermore, time-independence of g and h mn implythat ω α, β = ω ,α ¯ β . (2.132)Also, as in the null case analysis above, study the remaining Lie derivatives w.r.t. curvedcoordinates. Among others, one obtains the following results dg = − g ( ω , α e α + ω , α e ¯ α ) , (2.133) g dw = ω ,αβ e α ∧ e β + ω ,α ¯ β e α ∧ e ¯ β + ω , ¯ αβ e ¯ α ∧ e β + ω , ¯ α ¯ β e ¯ α ∧ e ¯ β . (2.134)Notice that, by construction, eq. (2.133) and the choice of g = f is consistent with eq. (2.125).Construct now three complex structures J i ( i = 1 , ,
3) on B , which give rise to thefollowing antiself-dual K¨ahler 2-forms, K = ˜ e ∧ ˜ e + ˜ e ¯1 ∧ ˜ e ¯2 , (2.135) K = − i (cid:16) ˜ e ∧ ˜ e ¯1 + ˜ e ∧ ˜ e ¯2 (cid:17) , (2.136) K = − i (cid:16) ˜ e ∧ ˜ e − ˜ e ¯1 ∧ ˜ e ¯2 (cid:17) . (2.137)These J i fulfill the algebra of imaginary unit quaternions, and the important thing to noticeis that, by means of eqs. (2.123)-(2.125), one can show that dK i = 0 . (2.138) W.r.t. the base-space B , where we have adopted the convention that (cid:15) = 1. CHAPTER 2. CLASSIFICATION OF SUGRA SOLUTIONS
As commented in appendix B, this implies that B is a hyper-K¨ahler manifold.We are left with the analysis of the field equations. Due to the KSIs, we only need todemand the fulfilling of the Maxwell equations, as the rest are automatically satisfied. Inparticular, the field strength is given by F = √ ω , α e ∧ e α + √ ω , α e ∧ e ¯ α + √ ω ,αβ e α ∧ e β + √ ω , ¯ α ¯ β e ¯ α ∧ e ¯ β + 12 √ (cid:0) ω α, β + ω µ, µ δ α ¯ β (cid:1) e α ∧ e ¯ β + 12 √ (cid:0) ω ¯ α, β + ω ¯ µ, µ δ ¯ αβ (cid:1) e ¯ α ∧ e β . (2.139)We now use eqs. (2.132), (2.133) and (2.134) to simplify this expression into F = − √
32 ( dt + w ) ∧ dg + √ g dw − √ G + , (2.140)where we have defined a B -self-dual 2-form G + (cf. eq. (2.48)) as G + = ω ,α ¯ β e α ∧ e ¯ β + ω , ¯ αβ e ¯ α ∧ e β + ω ,µµ e α ∧ e ¯ α , (2.141) i.e. G + α ¯ β = ω ,α ¯ β + 12 (cid:15) α ¯ βσ ¯ τ ω ,σ ¯ τ . (2.142)This is the same field strength we found in section 2.1.2, and thus the same field equationsapply.Having detailed the two approaches to classifying solutions to Killing Spinor equations,and how to use them in order to obtain supersymmetric solutions to supergravity theories,we are ready to consider fakeSupergravity. hapter 3 N = 2 d = 4 gauged fakeSUGRAcoupled to non-Abelian vectors This chapter is a recapitulation of [79], where we presented the classification of solutionsto N = 2 d = 4 fakeSupergravity coupled to non-Abelian vector multiplets, otherwise alsoreferred to as Wick-rotated N = 2 d = 4 Supergravity, having allowed for gaugings of theisometries of the scalar manifold . This is because, as explained in section 1.2, this theory canbe obtained from gauged N = 2 d = 4 SUGRA coupled to non-Abelian vector multiplets bymeans of a Wick rotation. In this sense, since we are allowing for non-Abelian couplings, itis not the coupling constant that is rotated, but rather the Fayet-Iliopoulos term responsiblefor gauging the R -symmetry.The outline of this chapter is the following: in section 3.1 we set up the fake-Killing spinorsequations that we are going to solve. We see that, as we are Wick rotating the FI term, therelations between the equations of motion one can derive from the integrability equation aresimilar to the ones obtained in the usual supersymmetric case, and hence the implications asfar as the checking of equations of motion are identical. This was to be expected since weare not changing the characteristics of the Killing spinors. Some information about SpecialGeometry and the gauging of isometries in special geometries, needed to understand theset-up, is given in appendix (B.2).As in the classification of supersymmetric solutions, we split the problem into two differentcases, depending on the norm of the vector one constructs as a bilinear of the fake-Killingspinors; the timelike case, i.e. when the norm does not vanish, shall be treated in section 3.2.In section 3.3 we will have a go at the null case, i.e. when the norm of the vector vanishesidentically. In that section, we will be ignoring the possible non-Abelian couplings and hencewe shall not obtain a complete characterisation; instead we will find that the solutions haveholonomy group contained in Sim (2), and we shall discuss the general features of such asolution. This will be illustrated by two examples, namely the Nariai cosmos in the minimaltheory in section 3.3.1, and a general class of solutions with holomorphic scalars that canbe seen as a back-reacted intersection of a cosmic string with a Nariai/ Robinson-Bertottisolution, in section 3.3.2.The reader might feel that the generic theories that can be treated in our setting are rathercryptic, as their connection with supergravity theories or Einstein-Yang-Mills-Λ theories canbe considered weak. However, it is well known that there are choices for the FI terms in gauged For gauging in N = 2 SUGRA, see e.g. appendix B.2, or [29] for a complete review. CHAPTER 3. N = 2 D = 4 GAUGED FAKESUGRA N = 2 d = 4 SUGRA for which this equals the bosonic part of an ungauged supergravity[80]. This means that for such choices, our fake-supersymmetric solutions are nothing morethan non-BPS solutions to an ordinary ungauged supergravity. The easiest model in whichone can see this happening is the model which can be obtained by dimensionally-reducingminimal N = 1 d = 5 Supergravity, and we shall discuss some simple solutions to thismodel in section 3.4, as well as their uplift to five dimensions. Finally, in section 3.5, weshall give our conclusions and a small outlook for related work in higher dimensions. Thetensorial conventions are presented in appendix A.1.1, and the spinorial ones in appendixA.2.1. Furthermore, the interested reader will find information about the normalisation ofthe bilinears and the curvatures for the null case in appendices A.3.1 and C.1, respectively. N = 2 Einstein-Yang-Mills
The bosonic field content of N = 2 d = 4 Supergravity coupled to n v vector multiplets consistsof the graviton, g µν , ¯ n = n v + 1 vector fields A Λ (where Λ = 0 , . . . , n v ) and n v complex scalarfields Z i (for i = 1 , . . . , n v ). The self-interaction of the scalars and their interaction withthe vector fields can be derived from a geometric structure called Special Geometry, of whichmore is given in appendix B.2.As commented above, the scenario we want to consider can be obtained from ordinary N = 2 d = 4 gauged SUGRA coupled to vector multiples (but not to hyper-multiplets) byWick rotating the Fayet-Iliopoulos term . In other words, we Wick rotate the constant tri-holomorphic map P x Λ → i C Λ δ x , where C Λ are real and constant. In usual supersymmetricstudies the FI term gauges a U (1) in the hyper-multiplets’ SU (2), and the effect of the Wickrotation is that we are gauging instead an R -symmetry through the effective connection C Λ A Λ [28].The presence of a FI term is compatible with the gauging of non-Abelian isometries of thescalar manifold, so long as the action of the gauge group commutes with the FI term (see e.g. [29]). Taking the gauge algebra to have structure constants f ΛΣΓ , this then implies that wemust impose the constraint f ΛΣΩ C Ω = 0. One result of the introduction of the C Λ constantsis that the dimension of the possible non-Abelian gauge algebra is not ¯ n = n v + 1, but rather n v , as 1 vector field is already used as the connection for the R -symmetry.The gauging of isometries implies that the field strengths of the physical fields are givenby D Z i ≡ dZ i + gA Λ K i Λ , F Λ ≡ dA Λ + g f ΣΓΛ A Σ ∧ A Γ , (3.1)where K i Λ is the holomorphic part of the Killing vector K Λ (see appendix B.2.2 for the minimalinformation needed, or [29, 81] for a fuller account). One implication of the above definitionis that C Λ F Λ = d (cid:2) C Λ A Λ (cid:3) , so that the linear combination C Λ A Λ is indeed an Abelian vectorfield.As mentioned, we are introducing an R -connection on top of the existent K¨ahler/ U (1)-symmetry due to the vector coupling. This means that we should define the covariant deriva- See appendix (B.3.2) for more information about FI terms and their Wick rotation. .1. FAKE N = 2 EINSTEIN-YANG-MILLS D a (cid:15) I = ∇ a (cid:15) I + i Q a (cid:15) I + ig A Λ a [ P Λ + i C Λ ] (cid:15) I ≡ D a (cid:15) I − g C Λ A Λ a (cid:15) I , (3.2)where P Λ is the momentum map corresponding to an isometry K Λ of the special geometry.Using the above definitions we can write the fake Killing spinor equations as D a (cid:15) I = − ε IJ T + ab γ b (cid:15) J − ig C Λ L Λ γ a ε IJ (cid:15) J , (3.3) D a (cid:15) I = ε IJ T + ab γ b (cid:15) J − ig C Λ L Λ γ a ε IJ (cid:15) J , (3.4) i/ D Z i (cid:15) I = − ε IJ /G i + (cid:15) J − W i ε IJ (cid:15) J , (3.5) i/ D Z ¯ ı (cid:15) I = − ε IJ /G ¯ ı − (cid:15) J − W ¯ ı ε IJ (cid:15) J , (3.6)where for clarity we have given also the rules for D a (cid:15) I and / D Z ¯ ı (cid:15) I even though they can beobtained by complex conjugation from the other two rules. Furthermore, we have introducedthe abbreviation W i = − ig ¯ f i Λ [ P Λ + i C Λ ] , W ¯ ı = W i , (3.7)and we have used the standard N = 2 d = 4 SUGRA definitions [29] T + ≡ i L Λ F Λ+ , G i + ≡ − ¯ f i Λ F Λ + . (3.8)The integrability conditions for the above system of equations can easily be calculatedand give rise to E ab γ b (cid:15) I = − i L Λ (cid:104) / B Λ − N ΛΣ / B Σ (cid:105) ε IJ γ a (cid:15) J , (3.9)where we defined not only the Bianchi identity as (cid:63) B Λ = D F Λ (= 0), but also the followingequations, where E ab = 0 is the Einstein equation, B Λ = 0 the Maxwell equation and V thepotential of the theory E ab = R ab + 2 G i ¯ D ( a Z i D b ) Z ¯ + 4Im ( N ) ΛΣ (cid:2) F Λ ac F Σ b c − η ab F Λ cd F Σ cd (cid:3) − η ab V , (3.10) (cid:63) B Λ = D (cid:2) N ΛΣ F Σ − + N ΛΣ F Σ+ (cid:3) − g Re (cid:16) K Λ¯ ı (cid:63) D Z ¯ ı (cid:17) ≡ D F Λ − g Re (cid:16) K Λ¯ ı (cid:63) D Z ¯ ı (cid:17) , (3.11) V = g (cid:104) C Λ C Σ L Λ L Σ + f Λ i ¯ f i Σ ( P + i C ) Λ ( P + i C ) Σ (cid:105) . (3.12)This potential is not real, and imposing it to be so implies that we must satisfy the constraint0 = Im ( N ) − | ΛΣ P Λ C Σ , (3.13)which is a gauge-invariant statement. However, for our choice of non-Abelian gaugings com-patible with the FI term, this constraint is satisfied identically; one can see that indeed In the notation that we will follow throughout this chapter, D will be the total connection, whereas wewill reserve D for the connection without the R -part and D for the K¨ahler-connection, i.e. the connectionappearing in ungauged supergravity. CHAPTER 3. N = 2 D = 4 GAUGED FAKESUGRA contracting the last equation in eq. (B.102) with f Σ i and using identities (B.75) and (B.97)one obtains the identity Im ( N ) − | ΛΣ P Σ = 4 i L Σ L Ω f ΣΩΛ , (3.14)which upon contracting with C Λ gives the desired result. Therefore the potential V reads V = g (cid:104) (cid:12)(cid:12) C Λ L Λ (cid:12)(cid:12) + f Λ i ¯ f i Σ ( P Λ P Σ − C Λ C Σ ) (cid:105) , = g (cid:104) (cid:12)(cid:12) C Λ L Λ (cid:12)(cid:12) + Im ( N ) − | ΛΣ ( C Λ C Σ − P Λ P Σ ) (cid:105) , (3.15)which is similar to the supersymmetric result in [29], upon Wick rotating the Fayet-Iliopoulosterm. Likewise, the above equations of motion can be obtained from the action (cid:90) √ g (cid:104) R + 2 G i ¯ D a Z i D a Z ¯ + 2Im ( N ) ΛΣ F Λ ab F Σ ab − N ) ΛΣ F Λ ab (cid:63) F Σ ab − V (cid:105) , (3.16)which as stated in the introduction has correctly normalised kinetic terms, and Im( N ) is anegative definite matrix.In SUGRA the integrability condition for the scalars relates the scalar equation of motion(EOM) with the Maxwell EOM, and the same happens here. A straightforward calculationresults in B i (cid:15) I = − i ¯ f i Λ (cid:104) / B Λ − N ΛΣ / B Σ (cid:105) ε IJ (cid:15) J , (3.17)where we have introduced the equation of motion for the scalars Z i as B i = (cid:3) Z i − i∂ i N ΛΣ F Λ+ ab F Σ+ ab + i∂ i N ΛΣ F Λ − ab F Σ − ab + ∂ i V . (3.18)It is thus clear that the integrability conditions for equations (3.3 - 3.6) give relations betweenthe EOMs. These, modulo the changes in the form of the tensors, are exactly the same asfound in regular Supersymmetry, which was to be expected. The implication of relations (3.9)and (3.17) is then also the same as in SUSY [1, 54], namely that the independent number ofequations of motion one has to check in order to be sure that a given solution to eqs. (3.3 - 3.6)is also a solution to the equations of motion is greatly reduced . The minimal set of equationsof motion one has to check depends on the norm of the vector bilinear V a = i(cid:15) I γ a (cid:15) I .In the timelike case we only need to solve the timelike direction of the Bianchi identity, i.e. ı V (cid:63) B Λ = 0, and the Maxwell (Yang-Mills) equations. This case will be considered insection 3.2. If the norm of the bilinear is null, i.e. V a V a = 0, then a convenient set of EOMsis given by N a N b E ab = 0 , N a B Λ a = 0 and N a B Λ a = 0 , where N is a vector normalised by V a N a = 1 . This case will be considered in section 3.3. In this section we consider the timelike case and the strategy is the usual one employedin the characterisation of supersymmetric solutions. We analyse the differential constraints(coming from the fKSEs) on the bilinears constructed from the spinors (cid:15) I , which are defined As we are using the same conventions as [82], we can copy the arguments there as they stand. .2. ANALYSIS OF THE TIMELIKE CASE D X = g C Λ L Λ V + i ı V T + , (3.19) D a V b = g | X | C Λ R Λ η ab + 4Im (cid:0) X T + ab (cid:1) , (3.20) D V x = g C Λ R Λ V ∧ V x + g C Λ I Λ (cid:63) [ V ∧ V x ] , (3.21)where, following [82], we have introduced the real symplectic sections of K¨ahler weight zero, R = Re ( V /X ) , I = Im ( V /X ) −→ | X | = (cid:104)R | I(cid:105) , (3.22)where V and the symplectic inner product (cid:104)−|−(cid:105) are explained in appendix B.2. As in theungauged theory, the 2¯ n real functions I play a fundamental rˆole in the construction of BPSsolutions and the 2¯ n real functions R depend on I . Given a Special Geometric model, findingthe explicit I -dependence of R is known as the stabilisation equation , and solutions are knownfor different models.A difference of this analysis w.r.t. the usual supersymmetric case lies in the character ofthe bilinear V . In such a case it is always a Killing vector, whereas this is not true here, as canbe seen from eq. (3.20). We can of course still use it to introduce a timelike coordinate τ bychoosing an adapted coordinate system through V = V a ∂ a = √ ∂ τ , but now the componentsof the metric will depend explicitly on τ , as was to be expected for instance from [22].As the V x contain information about the metric on the base-space, it is important todeduce its behaviour under translations along V ; we calculate £ V V x = ı V dV x + d ( ı V V x ) = g C Λ ı V A Λ V x + 2 g | X | C Λ R Λ V x . (3.23)This implies that by choosing the gauge-fixing ı V A Λ = − | X | R Λ , (3.24)we find that £ V V x = 0. As a matter of fact, the above gauge-fixing is the actual result oneobtains when considering timelike supersymmetric solutions in N = 2 d = 4 Supergravitytheories [82, 81].The above result has some interesting implications, the first of which is derived by con-tracting eq. (3.20) with V a V b , namely (cid:104)∇ V R | I(cid:105) + (cid:104)R | ∇ V I(cid:105) = ∇ V | X | = g C Λ R Λ . (3.25)0 CHAPTER 3. N = 2 D = 4 GAUGED FAKESUGRA
We can rewrite the above equation to a simpler form by observing that (cid:104)V /X | d ( V /X ) (cid:105) = X − (cid:104)V | D V(cid:105) − X − D X (cid:104)V | V(cid:105) = 0= (cid:104)R | d R(cid:105) − (cid:104)I | d I(cid:105) + i (cid:104)R | d I(cid:105) + i (cid:104)I | d R(cid:105) , (3.26)which implies that (cid:104) d R | I(cid:105) = (cid:104)R | d I(cid:105) , (3.27) (cid:104)R | d R(cid:105) = (cid:104)I | d I(cid:105) . (3.28)If we then introduce the real symplectic section C T = (0 , C Λ ), we can rewrite eq. (3.25) in thesimple and suggestive form 0 = (cid:104)R | ∇ V I + g C (cid:105) . (3.29)The above equation could also have been obtained from the contraction of eq. (3.19) with V , i.e. X D V X = − g (cid:104)R| C (cid:105) + ig (cid:104)I| C (cid:105) , (3.30)and taking its real part. By taking the imaginary part and using the identityIm (cid:18) X D X (cid:19) = − (cid:104)I | D I(cid:105) , (3.31)we also find that we must have 0 = (cid:104)I | ∇ V I + g C (cid:105) . (3.32)These equations suggest that the derivative of the symplectic section I in the direction V is constant. We can confirm this by considering eqs. (3.5) and (3.6). The contraction ofeq. (3.5) with ¯ (cid:15) K γ a ε KI reads 2 X D Z i = 4 ı V G i + − W i V , (3.33)which upon contraction with V leads to D V Z i = − X W i . (3.34)Using the gauge-fixing condition (3.24), the identity ¯ f Λ i P Λ = i L Λ K i Λ and the fact that for ourchoice of possible non-Abelian gauge groups we have L Λ K i Λ = 0 (cf. eq. (B.101)), the aboveequation gets converted to ∇ V Z i = − g X ¯ f Λ i C Λ . (3.35)Using the Special Geometry identity (cid:104)U i | U ¯ (cid:105) = i G i ¯ , we can rewrite the above equation to (cid:104)∇ V I + g C | U ¯ (cid:105) = i (cid:104)∇ V R | U ¯ (cid:105) , (3.36)which can be manipulated by using Special Geometry properties and a renewed call toeq. (3.34), giving (cid:104)∇ V I + g C | U ¯ (cid:105) = 0 . (3.37) These expressions were derived in [83] starting from a prepotential and using the homogeneity of thesymplectic section R . The derivation presented here is far less involved, and also holds in situations where noprepotential exists. .2. ANALYSIS OF THE TIMELIKE CASE ∇ V I = − g C , (3.38)which implies that the τ -dependence of the functions I is actually at most linear; in fact itis so only for the I Λ , as the I Λ are τ -independent.At this point it is necessary to introduce a complete coordinate system ( τ, y m ), which weshall take to be adapted to V and compatible with the Fierz identities in appendix A.3.1 V a ∂ a = √ ∂ τ , V = 2 √ | X | ( dτ + ω ) ,V xa ∂ a = − √ | X | V xm ( ∂ m − ω m ∂ τ ) , V x = √ V xm dy m , (3.39)where ω = ω m dy m is a (possibly τ -dependent) 1-form and we have introduced V xm by V xm V ym = δ xy ; as the V xm act as a Dreibein on a Riemannian space, the x -indices can beraised and lowered with δ xy , so that we shall not be distinguishing between co- and con-travariant x -indices.Putting the Vierbein together with the Fierz identity (A.60), we find that the metric takeson the conforma-stationary form ds = 2 | X | ( dτ + ω ) − | X | h mn dy m dy n , (3.40)where h mn = V xm V xn is the metric on the three-dimensional base-space.W.r.t. our choice of coordinates we have that £ V V x = 0 implies ∂ τ V xm = 0. The V x areof course also constrained by eq. (3.21), which in the chosen coordinate system and using thedecomposition A Λ = − R Λ V + ˜ A Λ m dy m −→ F Λ = − D (cid:2) R Λ V (cid:3) + ˜ F Λ , (3.41)reads dV x = g C Λ ˜ A Λ ∧ V x + g C Λ I Λ ε xyz V y ∧ V z . (3.42)For consistency we must have C Λ ∂ τ ˜ A Λ = 0. Furthermore, we can use the residual gaugefreedom C Λ ˜ A Λ → C Λ ˜ A Λ + dφ ( y ) , V x → e gφ V x to take C Λ I Λ to be constant, a possibility wewill however not use. At last, the integrability condition d V x = 0 implies0 = g (cid:104) ε xyz C Λ ˜ F Λ yz + √ V mx ˜ D m C Λ I Λ (cid:105) , (3.43)where we have introduced ˜ F Λ xy ≡ V mx V ny ˜ F Λ mn and˜ D m I = ∂ m I + g C Λ ˜ A Λ m I + g ˜ A Λ m S Λ I , ˜ D x ≡ V mx ˜ D m . (3.44)The system leading to (3.42) was analysed by Gauduchon and Tod in [84], in the study offour-dimensional hyper-hermitian Riemannian metrics admitting a tri-holomorphic conformalKilling vector. They observed that the geometry of the base-space belongs to a subclass ofthree-dimensional Einstein-Weyl spaces, called hyper-CR or (subsequently) Gauduchon-Todspaces . The additional constraint demanded on the EW spaces is nothing more than the See appendix D.1 for some technical information about these geometries. CHAPTER 3. N = 2 D = 4 GAUGED FAKESUGRA integrability condition (3.43), which is called the generalised Abelian monopole equation . Weshall see below that the equation determining the seed function I Λ will be a generalisednon-Abelian monopole equation, i.e. the straightforward generalisation of the standard Bo-gomol’nyi equation on R to GT spaces.In [28], Behrndt and Cvetiˇc realised that their five-dimensional cosmological solutionscould be dimensionally reduced to four-dimensional ones, which raises the question of whether/which of the solutions found by Grover et al. in [30] can be reduced to solutions that we areclassifying. In such a case, we would be deadling with a map between the five-dimensionaltimelike case and the four-dimensional timelike case, and thus the dimensional reduction hasto be over the four-dimensional base-space, which was found to be hyperK¨ahler-torsion [30].The key to identifying the subclass of five-dimensional solutions that can be reduced to oursthen lies in a further result of Gauduchon and Tod (see [84, remark 2]), which states thatsolutions to eqs. (3.42) and (3.43) are obtained by reduction of a conformal hyper-K¨ahlerspace along a tri-holomorphic Killing vector. As discussed in [30, sec. 3.2], this kind of spacesare actually particular instances of HKT spaces, which thus allows for the reduction. In fact,this inheritance of geometrical structures also ocurrs in ordinary supergravity theories in six,five and four dimensions [85], and it is reasonable to suppose that this also holds for fake(Wick-rotated) supergravities. As a final comment, let us mention that the three-dimensionalKilling spinor equation on a GT manifold allows non-trivial solutions [86, 87].Before turning to the equations of motion, we deduce the following equation for ω fromthe antisymmetrised version of eq. (3.20) and the explicit coordinate expression in (3.39). Ascan be seen, this calculation needs the explicit form for the 2-form T + , which can be obtainedfrom eq. (3.19) and the rule that a general imaginary self-dual 2-form B + is determined byits contraction with V by means of (see [88] for more detail) B + = 14 | X | (cid:0) V ∧ ı V B + + i (cid:63) (cid:2) V ∧ ı V B + (cid:3)(cid:1) . (3.45)The result thus reads dω + g C Λ ˜ A Λ ∧ ( dτ + ω ) = √ (cid:63) [ V ∧ (cid:104)I | D I(cid:105) ] . (3.46)Contracting the above equation with V we find that £ V ω = g √ C Λ ˜ A Λ −→ ω = g C Λ ˜ A Λ τ + (cid:36) , (3.47)where (cid:36) = (cid:36) m dy m is τ -independent. Substituting the above result into eq. (3.46) andevaluating its R.H.S. we obtain d(cid:36) + g C Λ ˜ A Λ ∧ (cid:36) + g C Λ ˜ F Λ τ = (cid:104)I | ˜ D m I − ω m ∂ τ I(cid:105) V xm ε xyz V y ∧ V z . (3.48)If now take the τ -derivative of this equation, we shall find again eq. (3.43). The equationdetermining (cid:36) is then found by splitting up the τ -dependent part; it reads˜ D (cid:36) = ε xyz (cid:104) ˜ I | ˜ D x ˜ I − (cid:36) x ∂ τ I(cid:105) V y ∧ V z , (3.49)where we have introduced ˜ I = I (at τ = 0). .2. ANALYSIS OF THE TIMELIKE CASE F T = ( F Λ , F Λ ) can then easily be deduced to give thestandard supersymmetric result F = − D ( R V ) − (cid:63) [ V ∧ D I ]= − D ( R V ) − √ ε xyz V mx (cid:104) ˜ D m I − ω m ∂ τ I (cid:105) V y ∧ V z , (3.50)which agrees completely with the imposed gauge-fixing (3.24).At this point we shall treat the Bianchi identity D F Λ = 0 as in [81], namely as leadingto a Bogomol’nyi equation determining the pair ( ˜ A Λ , I Λ ). Since we were given the gaugepotential in eq. (3.41), the Bianchi identity is solved identically, and thus does not implyfurther contraints. Nevertheless one needs to make sure that such potential leads to a fieldstrength with the form prescribed by fakeSupersymmetry in eq. (3.50). If we impose that, weobtain ˜ F Λ xy = − √ ε xyz ˜ D z I Λ , (3.51)which due to eq. (3.38) it is manifestly τ -independent. This is the equation which we re-ferred to above as the generalisation of the standard Bogomol’nyi equation on R to a three-dimensional Gauduchon-Tod space. One can see that, upon contraction with C Λ , the aboveequation implies the constraint (3.43).We would now like to show that the timelike solutions (to the fKSEs) we have just char-acterised are indeed solutions to the EOMs. For this we need to impose the Yang-Millsequations, eq. (3.11). This equation consists of two parts, namely one in the time-direction, i.e. B t Λ , and one in the space-like directions, B x Λ . A tedious but straightforward calcula-tion shows that B t Λ = 0 identically, in full concordance with the discussion in section 3.1.The EOMs in the x -direction, however, do not vanish identically. Instead, they impose thecondition (cid:16) ˜ D x − ω x ∂ τ (cid:17) I Λ = g f Λ(ΩΓ f ∆)ΓΣ I Ω I ∆ I Σ − g f ΛΩΣ I Ω I Σ C Γ I Γ , (3.52)which in the limit C → i.e. fSUSY → SUSY) coincides with the result obtained in [81]. Asimplification on the above equation can be performed by employing eqs. (3.44) and (3.47),which implies that ∂ τ (cid:16) ˜ D m I Λ − ω m ∂ τ I Λ (cid:17) = ∂ τ ∂ m I Λ = 0 . (3.53)Using this and the fact that I Λ is linear in τ , we can thus rewrite eq. (3.52) as˜ D x ˜ I Λ − (cid:16) ˜ D x (cid:36) x (cid:17) ∂ τ I Λ = g f Λ(ΩΓ f ∆)ΓΣ I Ω I ∆ ˜ I Σ − g f ΛΩΣ I Ω ˜ I Σ C Γ I Γ , (3.54)which is a τ -independent equation. Before making some general comments on the behaviour of the solutions, we describe howto construct solutions using the results obtained in the foregoing section. The first step is todecide which model to consider, i.e. one has to specify what special geometric manifold toconsider, what non-Abelian groups to gauge, and furthermore the constants C Λ . Given this4 CHAPTER 3. N = 2 D = 4 GAUGED FAKESUGRA model, one must then decide what GT space will be describing the geometry of the three-dimensional base-space; this is equivalent to finding the triplet ( V x , C Λ ˜ A Λ , C Λ I Λ ) that solveseq. (3.42). This allows in principle to solve the Bogomol’nyi equation (3.51), giving ( ˜ A Λ , I Λ ).The next step is to determine the τ -independent part of the seed functions I Λ 7 , usingequation (3.54). As this equation contains not only the I Λ but also (cid:36) , one is forced todetermine both objects, and to make sure that eq. (3.49) is satisfied. After this, the onlysteps left include to determine the field strengths by means of eq. (3.50), and writing downthe physical scalars Z i = L i / L and the metric by determining the stationarity 1-form ω through eq. (3.47) and the metrical factor | X | through eq. (3.22). As usual, the explicitconstruction of the last fields goes through solving the stabilisation equation , which determinesthe symplectic section R in terms of the seed functions I . For many models, solutions to thisare known.However, the Bogomol’nyi equation eq. (3.51) is not easy to solve. Since we only knowexplicit solutions to the Bogomol’nyi equation on R , this means that for the momentthe only non-trivial non-Abelian solution backgrounds we can build are the ones that followfrom the supersymmetric ones, satisfying C Λ I Λ = 0, by using the substitution rule I Λ →I Λ − g C Λ τ / (2 √ C Λ ˜ A Λ is gauge trivial, and thus the base-space is still R . This being so, the equations determining the τ -independent part of the I , i.e. eqs. (3.51)and (3.54), reduce to the ones for N = 2 E-YM deduced in [81]; indeed the only differencelies in the divergence of (cid:36) occurring in eq. (3.54), and in the R -case there is no obstructionto choosing it to vanish from the onset.The construction of fake-supersymmetric solutions then boils down to the substitutionprinciple put forward by Behrndt and Cvetiˇc in [28]: given a supersymmetric solution to N = 2 d = 4 E-YM Supergravity, Abelian [47] or non-Abelian [81, 91], substitute I Λ →I Λ − g C Λ τ / (2 √
2) and impose the restriction C Λ I Λ = 0. As explained before, when dealingwith non-Abelian gauge groups not all choices for C Λ are possible, and one must respect theconstraint f ΛΣΓ C Γ = 0.The first observation is that generically the asymptotic form of the solution is not DeSitter but rather Kasner, i.e. the τ -expansion of the base-space is power-like, making thedefinition of asymptotic mass even more cumbersome than in the De Sitter case . The secondobservation is that the metric has a curvature singularity at those events/ points for which | X | − = 0, which may be located outside our chosen coordinate system. This raises thequestion of horizons, or, in other words, how to decide in a practical manner when does oursolution describe a black hole. Observe that in the original solution for one single black holeof Kastor and Traschen, this question is readily resolved by changing coordinates to obtainthe time-independent spherically symmetric extreme RNdS black hole, for which the criteriato have an horizon is known: the existence of a black hole in the original coordinate systemcan be expressed in terms of the existence of a Killing horizon for a timelike Killing vector,which covers the singularity. It is interesting to note that generically there is no such timelikeKilling vector in this case.To see it, consider for instance the CP -model. This model only has one complex scalar The τ -dependence is fixed by eq. (3.38). Observe that this is purely a non-Abelian restriction, as GT metrics (solutions to the generalised Abelianmonopole equation) are known, see e.g. [89, 90]. As a side note, note that the resulting Kasner spaces have a timelike conformal isometry of the kind usedin [92] to define a conformal energy. .3. ANALYSIS OF THE NULL CASE Z living on the coset space SL (2; R ) /SO (2) and an associated K¨ahler potential e K =1 − | Z | , so that we have the constraint 0 ≤ | Z | <
1. Making the choice C Λ = ( − , V = 2 g (cid:2) e K (cid:3) , (3.55)which is manifestly positive. Imposing I = 0 in order to have R as the base-space, and I = 0 in order to have a static solution, i.e. ω = 0, the EOMs imply that a simple solutionis given by I = gτ √ , I = √ gλ −→ | X | = g (cid:2) τ − λ (cid:3) , (3.56)where λ is a real constant. If λ = 0 the above solution leads to dS , whereas if λ (cid:54) = 0 we canintroduce a new coordinate t through τ = λ cosh ( gt ), such that the solution is given by ds = dt − sinh ( gt ) d(cid:126)x , (3.57) Z = − i cosh − ( gt ) . (3.58)This says that at late times the metric is dS , but it is singular when t = 0; at that pointin time also the scalar becomes problematic, as | Z ( t = 0) | = 1, violating the bound, whichin its turn implies that the contribution of the scalars to the energy-momentum tensor blowsup. It is paramount that in this case no timelike Killing vector exists. Had we on the otherhand taken I = √ gpr − , for which a timelike Killing vector does exist, the metric can betransformed to the static form ds = p + R − g R R dt − R ( R + p )( p + R − g R / dR − R dS . (3.59)This metric has one Killing horizon, identified with the cosmological horizon for R >
0, andhas therefore a naked singularity located at R = 0. In the static coordinates the scalar fieldreads Z = − ip ( p + R ) − / , which explicitly breaks the bound 0 ≤ | Z | < R = 0,showing again the link between the regularity of the metric and that of the scalars.In view of all this, it would be very desirable to have at hand a manageable prescriptionfor deciding when a solution describes a black hole. In this respect, we would like to mentionthe isolated horizon formalism (see e.g. [93]), which attempts to give a local definition ofhorizons, without a reference to the existence of timelike Killing vectors. This formalism wasapplied in the context of SUGRA in [94, 95], and similar work for fake SUGRAs might proveof interest. In this section we characterise the fake-supersymmetric solutions in the null case, i.e. when V = 0. For simplicity we shall restrict ourselves to theories with no YM-type couplings;a full analysis is possible [81], but likely to not be very rewarding. As in the timelike case,the difference with the supersymmetric case lies in the fact that the vector-bilinear is not aKilling vector. Furthermore, introducing an adapted coordinate v through L = L a ∂ a = ∂ v ,one can see that the metric will be explicitly v -dependent, unlike in the supersymmetric case.The aim of this section is then to determine this v -dependence, and to give two minimal andsimple (albeit generic) solutions, that illustrate the changes generated by the R -gauging.6 CHAPTER 3. N = 2 D = 4 GAUGED FAKESUGRA
In the null case the norm of the vector V vanishes, whence X = 0. This means that thetwo spinors (cid:15) I are parallel, and following [82, 81] we shall put (cid:15) I = φ I (cid:15) , for some functions φ I and the independent spinor (cid:15) . The decomposition of (cid:15) I follows from its definition as (cid:15) I = ( (cid:15) I ) ∗ , which then implies that (cid:15) I = φ I (cid:15) ∗ , where we have defined φ I = φ I . Furthermore,we can without loss of generality normalise the functions φ such that φ I φ I = 1. Having takeninto account this normalisation, one can write down the following completeness relation forthe I -indices ∆ I J = φ I φ J + ε IK Φ K ε JL Φ L , (3.60)which is such that ∆ I J φ J = φ I , ∆ I J ε JK φ K = ε IK φ K . Moreover, one can see that ∆ I J =∆ J I .Projecting the fKSEs (3.3)-(3.6) onto the functions we obtain0 = D a (cid:15) + φ I ∇ a φ I (cid:15) , (3.61)0 = (cid:16) T + ab + ig C Λ L Λ η ab (cid:17) γ b (cid:15) ∗ − ε IJ φ I ∇ a φ J (cid:15) , (3.62)0 = i /∂Z i (cid:15) ∗ , (3.63)0 = (cid:104) /G i + + W i (cid:105) (cid:15) . (3.64)We shall now introduce an auxiliary spinor η , normalised by (cid:15)η = √ = − η(cid:15) . This spinorialfield allows us to introduce four new null vectors L a = i(cid:15)γ a (cid:15) ∗ , N a = iηγ a η ∗ ,M a = iηγ a (cid:15) ∗ , M a = i(cid:15)γ a η ∗ , (3.65)where L and N are real vectors and by construction M ∗ = M , whence the notation. Observethat eq. (A.58) implies that the vector L is nothing but V , but where it has been denotedby L (ightlike), as we are now in the null case. Given the above definitions, it is a tedious yetstraightforward calculation to show that they form an ordinary normalised null tetrad, i.e. the only non-vanishing contractions are L a N a = 1 = − M a M a , (3.66)which implies that η ab = 2 L ( a N b ) − M ( a M b ) . (3.67)Apart from these vectors, one can also define imaginary self-dual 2-forms analogous to theones defined in eq. (A.59), byΦ ab ≡ (cid:15)γ ab (cid:15) , Φ = √ L ∧ M , Φ ab ≡ (cid:15)γ ab η , Φ = √ (cid:2) L ∧ N + M ∧ M (cid:3) , Φ ab ≡ ηγ ab η , Φ = −√ N ∧ M , (3.68)where the identification on the RHS follows from the Fierz identities.The introduction of the above auxiliary spinorial field is not unique, and one still has thefreedom to rotate (cid:15) and η by (cid:15) → e iθ (cid:15) and η → e − iθ η . This does not affect L nor N , butit does produce M → e − iθ M and M → e iθ M , which we shall use it to get rid of a phasefactor when introducing a coordinate expression for the tetrad. A second freedom is the shift .3. ANALYSIS OF THE NULL CASE η → η + δ (cid:15) , with δ a complex function. This shift does not affect the normalisation condition,but on the vectors it generates L → L , M → M + δ L , N → N + | δ | L + δ M + ¯ δ M , (3.69)and can also be used to restrict the coordinate expressions of the tetrad.The first step is introducing a coordinate v through L (cid:91) ≡ L a ∂ a = ∂ v , and using eq. (3.61)to derive ∇ a L b = g C Λ A Λ a L b . (3.70)This equation says that L is a recurrent null vector. Having such a vector field is the defin-ing property of a space with holonomy Sim( d −
2) (see [96] for more information) and thecombination g C Λ A Λ is called the recurrence 1-form. Antisymmetrising this expression we seethat dL = g C Λ A Λ ∧ L , which implies not only C Λ F Λ ∧ L = 0 , but also L ∧ dL = 0. This lastresult states that the vector L is hyper-surface orthogonal, which implies the local existenceof functions Y and u such that L = Y du . Seeing that L is charged under the R -symmetry,we can always gauge-transform the function Y away, arriving at L = du , (3.71)whence also that C Λ A Λ = Υ L , for some function Υ. We can write eq. (3.70) as ∇ a L b = g Υ L a L b , which immediately implies ∇ L L = 0 , (3.72)so that L is a geodesic null vector. Given this information and the normalisation of the tetrad,we can choose coordinates u , v , z and ¯ z such that L = du , L (cid:91) = ∂ v ,N = dv + Hdu + (cid:36)dz + (cid:36)d ¯ z , N (cid:91) = ∂ u − H∂ v ,M = e U dz , M (cid:91) = − e − U ( ∂ ¯ z − (cid:36)∂ v ) ,M = e U d ¯ z , M (cid:91) = − e − U ( ∂ z − (cid:36)∂ v ) , (3.73)where we have used the U (1) rotation M → e − iθ M to get rid of a possible phase in theexpression of M and M . The spin connection and the curvatures for this tetrad are given inappendix C.1. A last implication of the Fierz identities is that ε (4) ≡ ε abcd e a ∧ e b ∧ e c ∧ e d = i L ∧ N ∧ M ∧ M = i e + ∧ e − ∧ e • ∧ e ¯ • , (3.74)whence ε + −• ¯ • = i . Furthermore, one finds from (3.72) ∂ v H = g Υ , and 0 = ∂ v U = ∂ v (cid:36) = ∂ v (cid:36) , (3.75)whence the only v -dependence of the metric resides in H . The resulting form of the metric ds = 2 du ( dv + Hdu + (cid:36)dz + (cid:36)d ¯ z ) − e U dzd ¯ z (3.76)is a Kundt wave, that is, it admits a non-expanding, shear- and twist-free geodesic null vector(cf. appendix C.3 for more details). To see this take e.g. L (cid:91) = ∂ v . Moreover, it is in theWalker form, which implies that the space has holonomy contained in Sim (2) [97].8
CHAPTER 3. N = 2 D = 4 GAUGED FAKESUGRA
To determine Υ we shall be using the identity C Λ F Λ = d (cid:0) C Λ A Λ (cid:1) = d Υ ∧ L , whichpresupposes knowing F Λ . The generic form of F Λ can be derived from the fKSEs (3.62) and(3.64). Contraction of the first with i(cid:15) and iη leads to ı L T + = ig C Λ L Λ L , (3.77) ı M T + = ig C Λ L Λ M + i √ φ I ε IJ dφ J . (3.78)Using these and the fact that, as T + is an imaginary-self-dual 2-form it must be expressiblein terms of the Φs defined in eq. (3.68), one obtains T + = ℵ L ∧ M − ig C Λ L Λ (cid:2) L ∧ N + M ∧ M (cid:3) , (3.79)with √ ℵ = iφ I ε IJ ∇ N φ J , and moreover √ φ I ε IJ ∇ M φ J = g C Λ L Λ , φ I ε IJ ∇ L φ J = φ I ε IJ ∇ M φ J . (3.80)Giving eq. (3.64) a similar treatment leads to G i + = ℵ i L ∧ M − W i (cid:2) L ∧ N + M ∧ M (cid:3) , (3.81)where ℵ i are still undetermined functions. Using the rule F Λ+ = i L Λ T + + 2 f Λ i G i + we findthat F Λ+ = ϕ Λ L ∧ M + V Λ (cid:2) L ∧ N + M ∧ M (cid:3) , (3.82)where we have introduced V Λ = g (cid:16) L Λ L Σ + Im( N ) − | ΛΣ (cid:17) C Σ (3.83)and ℵ = 2 i L Λ ϕ Λ , ℵ i = − ¯ f i Λ ϕ Λ ←→ ϕ Λ = i ℵ L Λ + 2 ℵ i f Λ i . (3.84)Using F Λ = F Λ+ + F Λ − = 2Re (cid:0) F Λ+ (cid:1) and using d Υ ∧ L = C Λ F Λ , we obtain ∇ L Υ = − C Λ (cid:2) V + V (cid:3) Λ , (3.85) ∇ M Υ = C Λ ϕ Λ , (3.86) ∇ M Υ = C Λ ϕ Λ . (3.87)Eq. (3.85) is the key to the possible v -dependence. We want to integrate it to obtain H through eq. (3.75); for this we need to know the coordinate dependence of the scalars Z . Thiscan be obtained by contracting eq. (3.63) with the i(cid:15) and iη . The result is that0 = ∇ L Z i = ∂ v Z i , ∇ M Z i = e − U ∂ ¯ z Z i , (3.88)so that the Z i depend only on u and z . Likewise, the Z ¯ ı depend only on u and ¯ z .Using the fact that the scalars are v -independent, integration of eq. (3.85) is straightfor-ward and leads toΥ = − g (cid:104) (cid:12)(cid:12) C Λ L Λ (cid:12)(cid:12) + Im ( N ) − | ΛΣ C Λ C Σ (cid:105) v + Υ ( u, z, ¯ z ) , (3.89) H = − g (cid:104) (cid:12)(cid:12) C Λ L Λ (cid:12)(cid:12) + Im ( N ) − | ΛΣ C Λ C Σ (cid:105) v + Υ v + Υ ( u, z, ¯ z ) . (3.90) .3. ANALYSIS OF THE NULL CASE = 0 by doing a coordinate transformation v → v + f ( u, z, ¯ z ), but we shallbe ignoring this possibility for the moment. H can be written in terms of the potential V ineq. (3.15) with P Λ = 0, since we are ignoring possible non-Abelian couplings, as H = (cid:104) g (cid:12)(cid:12) C Λ L Λ (cid:12)(cid:12) − V (cid:105) v + Υ v + Υ , (3.91)which is calculationally advantageous when V is known.At this point we have nearly completely specified the v -dependence of the solution, theonly field missing being A Λ ; with this in mind it is worthwhile to impose the gauge-fixing ı L A Λ = 0, which is always possible and furthermore it is consistent with the earlier result C Λ A Λ = Υ L . As a result of this gauge fixing we have that ∂ v A Λ = £ L A Λ = d (cid:0) ı L F Λ (cid:1) = − (cid:0) V + V (cid:1) Λ L , (3.92)so that A Λ = − (cid:0) V + V (cid:1) Λ v L + ˜ A Λ = g F − | ΛΣ C Σ v L + ˜ A Λ , (3.93)where ˜ A Λ is a v -independent 1-form satisfying ı L ˜ A Λ = 0, and F is the imaginary part ofthe prepotential’s Hessian, cf. eq. (B.83). Given this expression, the Bianchi identity isautomatically satisfied, but just as in the timelike case this does not (necessarily) mean thatany ˜ A Λ leads to a field strength of the desired form. Calculating the comparison one findsthat d ˜ A Λ = (cid:0) V − V (cid:1) Λ M ∧ M + (cid:16) φ Λ + θ M ( v (cid:0) V + V (cid:1) Λ ) (cid:17) L ∧ M + (cid:16) φ Λ + θ M ( v (cid:0) V + V (cid:1) Λ ) (cid:17) L ∧ M . (3.94)Let us at this point return to the fKSEs, and evaluate eq. (3.5) using eqs. (3.81) and(3.88). This results in iθ + Z i γ + (cid:15) I + iθ • Z i γ • (cid:15) I = − ε IJ (cid:2) W i γ − − α i γ ¯ • (cid:3) γ + (cid:15) J . (3.95)The above equation is readily seen to be solved by observing that the constraint γ + (cid:15) J = 0 (3.96)leads to γ + (cid:15) I = 0 under complex conjugation, as well as to γ ¯ • (cid:15) I = 0 and γ • (cid:15) I = 0, due tohaving chiral spinors and the normalisation prescribed in eq. (3.74). A similar analysis onthe fKSE (3.3) in the v -direction shows that the spinor (cid:15) I is v -independent, and thus also (cid:15) I .The other equations become D ¯ • (cid:15) I = 0 , (3.97) D • (cid:15) I = ig C Λ ∗ ε IJ γ ¯ • (cid:15) J , (3.98) D + (cid:15) I = −ℵ ε IJ γ ¯ • (cid:15) J . (3.99)Using the definition (3.2) and the spin connection in eq. (C.3), we can expand eqs. (3.97)and (3.98) as 0 = θ ¯ • (cid:15) I − θ ¯ • (cid:0) U + K (cid:1) (cid:15) I , (3.100)0 = θ • (cid:15) I + θ • (cid:0) U + K (cid:1) (cid:15) I − ig C Λ L Λ ε IJ γ ¯ • (cid:15) J , (3.101)0 CHAPTER 3. N = 2 D = 4 GAUGED FAKESUGRA
The first is easily integrated by putting (cid:15) I = exp (cid:0) S (cid:1) χ I ( u, z ) , with S ≡ U + K , (3.102)which upon substitution into eq. (3.101) leads to ∂ z χ I + ( ∂ z S ) χ I = ig C Λ X Λ ε IJ γ ¯ • e S χ J . (3.103)This last equation is potentially dangerous, as it has a residual ¯ z -dependence (even though η and X Λ are ¯ z -independent). We use avoiding this inconsistency as a mean to fix S ; derivingeq. (3.103) w.r.t. ¯ z and using the complex conjugated version of eq. (3.103) to get rid of η I in the resulting equations, the result is that S has to satisfy ∂ z ∂ ¯ z S = − g e S (cid:12)(cid:12) C Λ X Λ (cid:12)(cid:12) −→ e − S = g (cid:12)(cid:12) C Λ X Λ (cid:12)(cid:12) (cid:0) | z | (cid:1) . (3.104)This unique choice for S is a necessary condition for eqs. (3.100) and (3.101) admitting asolution, but it may not be sufficient. In the next subsection we shall discuss the simplestnull case solution to the minimal theory, and show that the system can indeed be solvedcompletely. The lesson to be learnt is that, once we introduce S , the system (3.100, 3.101)corresponds to an equation determining spinors on a 2-sphere, and it has solutions. In fact,the C Λ X Λ factor can be absorbed by redefining χ I = (cid:112) C Λ X Λ η I , which converts eq. (3.104)into ∂ z η I + (cid:16) ∂ z ˜ S (cid:17) η I = ig ε IJ γ ¯ • e ˜ S η J , with e − ˜ S = g √ (cid:0) | z | (cid:1) , (3.105)which is just the spinor equation on S in stereographic coordinates. The minimal theory is obtained by selecting V T = (1 , − i/ N = − i/ N ) = 0. If we further fix C = 2, the minimal De Sitter theoryaction is given by (cid:90) √ g (cid:0) R − F − g (cid:1) . (3.106)Using the general results obtained earlier, we can write down the following solution ds = 2 du (cid:0) dv − g v du (cid:1) − dzd ¯ zg (1 + | z | ) ,A = − gv du . (3.107)This metric is nothing more than dS × S , albeit in a non-standard coordinate system, andthe solutions is known to the literature as the electrically-charged Nariai solution [98, 99].Note that the local holonomy of the Nariai solution is not the full sim (2), but rather so (1 , ⊕ so (2) ⊂ sim (2) [96].We now proceed to discuss the preserved fake-supersymmetries. For this, it is easier towrite the metric as ds = 2 du (cid:0) dv − g v du (cid:1) − g (cid:0) dθ + sin ( θ ) dϕ (cid:1) , (3.108) .3. ANALYSIS OF THE NULL CASE (cid:15) , ∇ a (cid:15) − gA a (cid:15) = − /F γ a σ (cid:15) − g γ a σ (cid:15) . (3.109)The solution to the above equation is seen to be (cid:15) = exp (cid:0) θ γ σ (cid:1) exp (cid:0) − ϕ γ (cid:1) (cid:15) , with γ + (cid:15) = 0 , (3.110)where (cid:15) is a 2-vector of constant spinors.We finish this subsection with a couple of comments. In SUGRA one can associate a Liesuperalgebra to a given supersymmetric solution [100, 101], and for the AdS × S maximallysupersymmetric solutions in minimal N = 2 d = 4, this algebra is su (1 , | u - and v -independent. Since thespinors are gauge-dependent objects, one should consider an R -covariant Lie derivative onthem [102, 103]. This derivative is defined for Killing vectors X and Y as L X (cid:15) = ∇ X (cid:15) + ( ∂ a X b ) γ ab (cid:15) − gξ X (cid:15) , with (cid:26) dξ X = £ X A ,ξ [ X,Y ] = £ X ξ Y − £ Y ξ X . (3.111)Using this Lie derivative, one can see that L X (cid:15) = 0 for any X ∈ Isom( dS ). In the usual supersymmetric case there are two generic classes of null solutions whose Su-persymmetry is straightforward to see: the first are the pp-waves which are characterised bythe fact that the scalars depend only on u , and the cosmic strings which are characterisedby vanishing vector potentials A Λ , vanishing Sagnac connection (cid:36) = 0, and a holomorphicspacetime dependence of the scalars, i.e. Z i = Z i ( z ) [45, 82]. In this subsection we shallbe considering the fSUGRA analogue of the latter case, and impose (cid:36) = 0 and that Z i is afunction of z only. However, because of eq. (3.93) the vector potentials cannot vanish, and weshall be looking for the minimal expression of ˜ A Λ for which the Bianchi identity, eq. (3.94),is solved: minimality implies that φ Λ = ve − U ∂ ¯ z (cid:16) V Λ + V Λ (cid:17) and the Bianchi identity reducesto d ˜ A Λ = 2 i Im (cid:18) X Λ g C Σ X Σ (cid:19) dz ∧ d ¯ z (1 + | z | ) , (3.112)a solution to which exists locally, and determines ˜ A u = 0 and ˜ A Λ z and ˜ A Λ¯ z as functions of z and ¯ z .Given the above identifications, we can use eq. (3.82) to calculate the constraints imposedby the Maxwell equations, i.e. B Λ = 0 in eq. (3.11), which leads to N ΛΣ ∂ z (cid:0) V + V (cid:1) Σ = ∂ z (cid:104) N ΛΣ V Σ + N ΛΣ V Σ (cid:105) , (3.113) N ΛΣ ∂ ¯ z (cid:0) V + V (cid:1) Σ = ∂ ¯ z (cid:104) N ΛΣ V Σ + N ΛΣ V Σ (cid:105) , (3.114) ∂ z (cid:104) N ΛΣ ∂ ¯ z (cid:0) V + V (cid:1) Σ (cid:105) = ∂ z (cid:104) N ΛΣ ∂ z (cid:0) V + V (cid:1) Σ (cid:105) , (3.115)2 CHAPTER 3. N = 2 D = 4 GAUGED FAKESUGRA where the contribution due to ˜ A Λ has been dropped out identically. As eq. (3.114) is thecomplex conjugated version of (3.113), and eq. (3.115) is the integrability condition for (3.113)and (3.114), we only need to demand eq. (3.113) to hold. Using the holomorphicity of thescalars to write ∂ z = ∂ z Z i ∂ i , one can express it as an equation in Special Geometry, namely ∂ i N ΛΣ V Σ + ∂ i N ΛΣ V Σ = 2 i Im ( N ) ΛΣ ∂ i V Σ = gi L Λ C Γ f Γ i − gi ∂ i Im ( N ) ΛΣ Im ( N ) − | ΣΓ C Γ . (3.116)Some straightforward algebra using the expressions (B.77) and (B.78) shows that the aboveequation holds, and thus the Maxwell equations are solved for arbitrary scalar functions Z i ( z ).Had we obtained the most general solution above, we would be certain that the fieldssolve the fKSEs, and thus the KSIs would have told us that we only need to further verify E ++ = 0 to make sure that the proposed configuration solves all the equations of motion. Wedid however check that all of the EOMs are indeed satisfied, which, as was to be expectedfrom the discussion above, they all reduced to Special Geometry calculations.In conclusion, given an expression for Z i = Z i ( z ), we need to find the local expression for˜ A Λ from eq. (3.112), so the solution is given by ds = 2 du (cid:0) dv − H v du (cid:1) − g | C Λ L Λ | dzd ¯ z (1 + | z | ) , (3.117) A Λ = g F − | ΛΣ C Σ v du + ˜ A Λ , (3.118)where H = V − g (cid:12)(cid:12) C Λ L Λ (cid:12)(cid:12) . (3.119)Furthermore, one can obtain deformations of the Nariai cosmos by taking the scalars Z i tobe constants, in which case the z ¯ z -part of the metric describes a 2-sphere of radius g | C Λ L| .Depending on H , the uv -part of the metric describes dS ( H > R , ( H = 0) or AdS ( H < sim (2). The solution forgeneric Z i ( z ) however has proper sim (2) holonomy. N = 2 SUGRA
As is well known, there are models in N = 2 d = 4 SUGRA coupled to vector multiplets forwhich one can choose the Fayet-Iliopoulos terms such that the hyper-multiplet contributionto the potential vanishes (see e.g. [80] or [29, sec. 9] for a discussion of this point). Sincethe construction we are considering in this chapter is a Wick-rotated version of the generalsupersymmetric set-up, with no hyperscalars, this implies that there are fake-supersymmetricmodels in which the only contribution to the potential comes from the gauging of the isome-tries, as the FI contributions to the latter vanish. In that case, the bosonic action (3.16)coincides with that of an ordinary YMH-type of supergravity theory, and for those specificmodels the solutions we obtained are in fact non-BPS solutions of a regular supergravitytheory. Let us illustrate this with the dimensional reduction of minimal d = 5 SUGRA.The dimensional reduction of minimal five-dimensional SUGRA leads to a specific N = 2 d = 4 SUGRA, namely minimal SUGRA coupled to one vector multiplet, with a prepotentialgiven by F ( X ) = − (cid:0) X (cid:1) X . (3.120) .4. NON-BPS SOLUTIONS TO N = 2 SUGRA Z = X / X , one finds that the scalar manifold is SL(2; R ) / U(1) withthe corresponding K¨ahler potential e K = Im ( Z ) (note that this implies the constraintIm ( Z ) > P = 0, we can calculate the potential in eq. (3.15) to find V = g (cid:2) C Im − ( Z ) + 6 C C Re( Z ) Im − ( Z ) (cid:3) . (3.121)There are two interesting subclasses we can consider. The first one is C Λ = (0 , C ), for whichthe potential is of the correct form to correspond to the dimensionally-reduced version of thetheory considered in [30].The second case is C Λ = ( C , I = 0 , so thatthe base-space is R , and I = 0 as to ensure staticity, i.e. ω = 0 . The regularity of thesolution to the stabilisation equations, or equivalently the consistency of the metrical factor | X | , imposes the constraint I (cid:0) I (cid:1) <
0. With this information, the solution is determinedby 12 | X | = (cid:114) (cid:12)(cid:12)(cid:12) I ( I ) (cid:12)(cid:12)(cid:12) , Z = 2 i (cid:115)(cid:12)(cid:12)(cid:12)(cid:12) I I (cid:12)(cid:12)(cid:12)(cid:12) , (3.122)so that the solution is asymptotically Kasner. As the effective radius of the compactified fifthdirection is proportional to Im( Z ), which grows linear in τ , this solution is asymptoticallydecompactifying. The resulting five-dimensional metric is found to be (shifting I → √ H ) ds = 2 H − dy (cid:16) dτ − √ |I | dy (cid:17) − H d(cid:126)x . (3.123)which can be transformed to a Walker metric for a space of holonomy Sim(3) [97]. Observethat the relation between ( d + 1)-dimensional spaces of holonomy in Sim( d −
1) and time-dependent black holes, of which the foregoing is one example, was first introduced in [96].The generic solution in subsection 3.3.2 can readily be adapted to the model at hand, andreads ds = 2 du (cid:0) dv + λ v Z − du (cid:1) − λ Z dzd ¯ z (1 + | z | ) , (3.124)where we have introduced the abbreviations √ λ = g C and Z = Im( Z ). The vector fieldsare given by the expression (3.93) with ˜ A = 0. ˜ A needs to satisfy d ˜ A = √ iλ Z dz ∧ d ¯ z (1 + | z | ) , (3.125)which presupposes knowing the explicit dependence of Z on z . Lifting this solution up to fivedimensions we obtain, after the coordinate transformations v → e √ λy w where y is the fifthdirection, the following solution ds = 2 Z − e √ λy du dw − Z (cid:20) dy + 2 λ dzd ¯ z (1 + | z | ) (cid:21) , (3.126)ˆ A = √ Z ) (cid:104) dy + 2 √ λ Z − vdu (cid:105) − √ A , (3.127)4 CHAPTER 3. N = 2 D = 4 GAUGED FAKESUGRA where ˜ A is determined by the condition (3.125). This solution is a deformation of themaximally-supersymmetric AdS × S solution, and deformations of the other maximally-supersymmetric five-dimensional solutions can be obtained by using the Sp (2; R )-dualitytransformations before oxidation, in a similar way to how the four- and five-dimensionalvacua are related (see e.g. [104]).Let us end this section by pointing out that there are more models for which the FIcontribution to the potential vanishes [29]. One of them is the ST [2 , m ] model, which inungauged supergravity allows for the embedding of monopoles and the construction of non-Abelian black holes [81]. A convenient parameterisation is given by the symplectic section V = (cid:18) L Λ η ΛΣ S L Σ (cid:19) , with η = diag([+] , [ − ] m ) , η ΛΣ L Λ L Σ . (3.128)The FI part of the potential is easily calculated to give [80, 29] V F I = − g Im − (S) C Λ η ΛΣ C Σ , (3.129)so that V F I = 0 whenever C is a null vector w.r.t. η .Taking ST [2 ,
4] as the model to work with and C to be a null vector, we can gauge an SU (2) group, and by further taking C Λ I Λ = 0 (which implies that the base-space is R ) wecan generalise the solutions found in [91] to cosmological solutions. For that, take the indicesΛ to run over (0 , + , − , i ) and let C + be the only non-vanishing element in C . We find astatic solution, i.e. ω = 0, by selecting I ± = I = I i = 0 ; this allows for the embeddingof a ’t Hooft-Polyakov monopole in the I i . If we then further take ˜ I + = 0, where ˜ I is thepart of the function I independent of τ , and normalise the metric on constant- τ slices tobe asymptotically R (which is equivalent to taking ˜ I − and I to be suitable constants) themetric is determined through eq. (3.40) and12 | X | = √ τ (cid:114) µ g (cid:104) − H (cid:105) , (3.130)where H is a completely regular function of r ∈ R coming from the ’t Hooft-Polyakovmonopole, which reads H = coth ( µr ) − µr , (3.131)and is monotonic with H ( r = 0) = 0 and asymptoting to H ( r → ∞ ) = 1. This means thatthe constant- τ slices are geodesically-complete. The full metric however suffers from an initialsingularity at τ = 0, and also from Kasner expansion (the base-space has a time-dependencewhich behaves power-like).More general backgrounds can be constructed by considering the hairy or coloured (generalAbelian) solutions of [47], and (non-Abelian) of [91, 81]. Comments made in section (3.2.1)apply to these solutions. Where 0 is a timelike direction, ± are null directions and i = 1 , , .5. SUMMARY OF THE CHAPTER In this chapter we have studied the fake-supersymmetric solutions that can be obtainedfrom N = 2 d = 4 gauged Supergravity coupled to (non-Abelian) vector multiplets, by Wickrotating the FI term needed to obtain a gauged supergravity. As is usual in the classification ofsupersymmetric backgrounds, the solutions are divided into two classes, denoted the timelikeand the null case, which are distinguished by the norm of the vector built out of the preservedKilling spinor.In the timelike case we have found that the metric is of the standard conforma-stationaryform, also appearing naturally in the supersymmetric timelike solutions, with the differencethat the metric is to have a specific time dependence. This is such that there is a natural sub-stitution principle, as first pointed out by Behrndt and Cvetiˇc [28], for creating solutions fromthe known supersymmetric backgrounds to N = 2 d = 4 SUGRA coupled to (non-Abelian)vector multiplets. Apart from this time-dependence, we find that the base-space must be asubclass of three-dimensional Einstein-Weyl spaces known as hyper-CR or Gauduchon-Todspaces [84], and that half of the seed functions, namely the I Λ , must obey the Bogomol’nyiequation generalised to GT spaces.In the null case we have found that the solutions must have a holonomy contained inSim(2), which arguably can be considered to be a minor detail. It was however shown in [105]that the purely gravitational solutions of this kind have rather special properties with respectto quantum corrections, and it is not unconceivable that this holds for the more general classof solutions with Sim(2) holonomy in supergravity theories, such as the one presented insection 3.4.We did not develop a full-fledged characterisation of the solutions in the null case, butinstead focussed on the new characteristics induced by the interplay between Sim(2) holonomyand Special Geometry. The general solution to the null case is the Nariai solution in eq. (3.107)with the substitution g uu = − g v → − g v + 2Υ ( z, ¯ z ), where ∂ z ∂ ¯ z Υ = 0. This can beseen from [106], that gives the characterisation for the minimal case. The end result iswhat can be considered to be a back-reacted solution describing the intersection of a Nariai/Robinson-Bertotti space with a generic (stringy) cosmic string [82].The fact that the holonomy is contained in Sim(2) is due to having gauged an R -symmetry,whence one can deduce that the null vector (constructed as a bilinear of the preserved Killingspinor) is a gauge-covariantly constant null vector. In other words, it is a recurrent null vector,and thus why the four-dimensional space has holonomy Sim (2) [96]. As the Wick rotationneeded to create fake supergravities from ordinary gauged supergravities will always introducean R -gauging, one might be inclined to think that null fake-supersymmetric solutions willalways have infinitesimal holonomy in sim ( d − et al. [30]. In that case one can see that therecurrency condition (3.70) still holds but with the Levi-Civit`a connection replaced with ametric compatible, torsionful connection, where the torsion is completely antisymmetric andproportional to the Hodge dual of the graviphoton field strength. As the connection is metric,the link between the recurrency relation and Sim holonomy going through mutatis mutandis ,we see that in fake N = 1 d = 5 gauged Supergravity theories, there is a Sim(3) holonomyeven though in general this is not associated to the Levi-Civit`a connection.Furthermore, as was shown in [96] and illustrated in section 3.4, time-dependent solutionsof the kind found in the timelike case can be obtained by dimensional reduction of spaces with Sim holonomy. Moreover, they can also be obtained from solutions in the five-dimensional6
CHAPTER 3. N = 2 D = 4 GAUGED FAKESUGRA timelike case. This strongly suggests that the ordinary hierarchy of supersymmetric solutionsand the geometric structures appearing in them, between theories in six, five and four dimen-sions with eight supercharges [85], has a fake analogue. A graphical description of what thiswould be like is given in fig. (3.1).As a matter of fact, in chapter 4 we will investigate the solutions to five-dimensionalminimal fakeSupergravity, and we shall see that those precise relations hold between the five-dimensional null case and the structures obtained in this chapter. In fact, the maps giverelations between Einstein-Weyl spaces, since a HKT space is nothing more than a four-dimensional Weyl geometry admitting covariantly constant spinors. In view of this, we thenpostulate that the relations in fig. (3.1) also hold for a N = (1 , d = 6 fakeSUGRA, whichto the best of our knowledge has not yet been constructed, due to the inherent ungaugeabilityof the regular minimal SUGRA theory. .5. SUMMARY OF THE CHAPTER N = (1 , d = 6 g = 2 e + e − − HKT (4)
Jones-Tod [107] N = 1 d = 5 g = (cid:0) e (cid:1) − HKT (4) g = 2 e + e − − GT (3) Jones-Tod N = 2 d = 4 g = deform. (2 e + e − − S ) g = (cid:0) e (cid:1) − GT (3) Figure 3.1: A graphical depiction of the relations between the fSUGRA theories and theirfake-supersymmetric solutions, based on dimensional reduction over a spacelike circle. Ver-tical full lines indicate dimensional reduction over a S in the basespace, which changes thecharacteristics of the basespace. Dotted lines represent dimensional reduction over a spacelikecircle in the null-cone, a reduction which does not change the characteristics of the basespace.In the fakeSupergravities considered in this thesis the geometry of the base-spaces are allEinstein-Weyl manifolds.8 CHAPTER 3. N = 2 D = 4 GAUGED FAKESUGRA hapter 4 N = 1 d = 5 minimal fakeSUGRA This chapter studies the classification of solutions to N = 1 d = 5 minimal fakeSUGRA, usingthe bilinear method. In particular, it gives details corresponding to the null case of such atheory, since the timelike case was studied in [30]. For an equivalent analysis using spinorialgeometry methods one can consult [77].The outline of the chapter is the following: in section 4.1 we introduce a fake Killing spinorequation (fKSE) and use a subset of its integrability equations to see that they give rise torelations between the equations of motion. The introduction of bilinears constructed out of theKilling spinors allows us to introduce a five-dimensional frame, w.r.t. which the implications ofthe integrability conditions are discussed. In section 4.2 we analyse the differential constraintson the bilinears and obtain a necessary set of demands on the five-dimensional metric andthe gauge field for the existence of a Killing spinor; in 4.2.1 we show that said constraintsare also sufficient, furthermore showing that we are dealing with solutions that are half fake-BPS. In section 4.3 we discuss the constraints imposed by the equations of motion on theconfigurations, to which we give some simple solutions in sec. 4.3.1. Section 4.4 considersthe dimensional reduction of the theory to N = 2 d = 4 fSUGRA, and studies the relationbetween the general null solution in five dimensions and the general timelike solution in four.At last, section 4.5 contains a summary of the chapter. The conventions used are given inappendix A.1.1, and A.3.2 contains the bilinears which will be used extensively in section 4.2.The interested reader will find technical information on the solution geometry in appendix(C.2), and a small introduction to Gauduchon-Tod spaces in appendix D.1. The bosonic field content of minimal N = 1 d = 5 Supergravity comprises only of the metric g µν and one vector field A µ . It is this simplicity of the field content which allows for aclear derivation of the relevant geometrical structures, which rely on the form of the fakegravitino equation. In chapter 5, when we also consider the coupling to matter multiplets,these structures will be somewhat obfuscated.Since fSUGRA gauges an R -symmetry, we define the following connection on the spinors D a (cid:15) i = ∇ a (cid:15) i − ξ A a (cid:15) i . (4.1)Using the above definition for the gauge-covariant derivative, we can write the fake super-590 CHAPTER 4. N = 1 D = 5 MINIMAL FAKESUGRA symmetry rule as D a (cid:15) i = √ (cid:2) a /F + 2 /F a (cid:3) (cid:15) i + √ ξ γ a (cid:15) i , (4.2)where a /F ≡ γ abc F bc and /F a ≡ − F ab γ b . The integrability condition for this fKSE can beseen to be [108] ˜ E ab γ b (cid:15) i = − √ M a (cid:15) i , (4.3)where we have defined˜ E ab = E ab − g ab E cc , (4.4) E ab = R ab − ξ η ab − (cid:0) F ac F bc − η ab F (cid:1) , (4.5) (cid:63) M = d (cid:63) F − √ F ∧ F . (4.6)As before, the above integrability condition leads to relations between components of theEOMs, and in the (null) case at hand they can be found by using the bilineals of [108]. Ingeneral, there are three types of bilinears that can be constructed out of the spinors (cid:15) i : ascalar f = i(cid:15) i (cid:15) i , a vector V a = i(cid:15) i γ a (cid:15) i and three 2-forms Φ xab = ( σ x ) ij (cid:15) j γ ab (cid:15) i ( x = 1 , , f (cid:54) = 0 was already treated in [30], we shall restrictour attention to the null case and put f = 0. Furthermore, we shall again rename V as L (ightlike).The Fierz identities imply various relations between the bilinears, which in this case read[108] ı L ˆ L = 0 , (4.7) ı L Φ x = 0 , (4.8)ˆ L ∧ Φ x = 0 , (4.9) δ xy L a L b = Φ xac Φ ycb , (4.10)Φ x ∧ Φ y = 0 . (4.11)Eq. (4.7) of course implies that L is a null vector, and eqs. (4.8) and (4.9) imply thatΦ x = ˆ L ∧ E x with ı L E x = 0 , (4.12)for some 1-forms E x , which automatically satisfy (4.11). A minor calculation shows that(4.10) implies that the three 1-forms are actually orthonormal, in the sense that g − ( E x , E y ) = − δ xy , (4.13)which implies that the E x can actually be used to build up a F¨unfbein. This we do byintroducing the missing linearly independent 1-form N , normalised such that ı L N = 1 and ı N (cid:91) E x = 0. The five-dimensional metric is then ds = ˆ L ⊗ N + N ⊗ ˆ L − E x ⊗ E x . (4.14)Using the above base one can now express the implications of the integrability condition(4.3) as 0 = M ∧ ˆ L , (4.15)0 = ˜ E ab L b , (4.16)0 = ˜ E ab Φ xcb −→ E ab E xb . (4.17) .2. DIFFERENTIAL CONSTRAINTS E ++ ≡ N a N b E ab = 0 , M + ≡ N a M a = 0 , (4.18)to have a bona fide fake-supersymmetric solution.In this section we have analysed the non-differential constraints on the bilinears. We nowproceed to discuss the differential ones. Using the fermionic rule (4.2), one can derive the following constraints ı L F = − ξ ˆ L , (4.19) D a L b = − √ ( (cid:63)F ) ab c L c , (4.20) D a Φ xbc = − √ ξ ( (cid:63) Φ x ) abc − √ F de ( (cid:63) Φ x ) de [ b η c ] a + √ F ad ( (cid:63) Φ x ) bcd + √ ( (cid:63) Φ x ) ad [ b F c ] d . (4.21)Eq. (4.20) can actually be rewritten in a more suggestive form D a L b ≡ ∇ a L b − S abc L c = 2 ξ A a L b , with S abc = − √ ( (cid:63)F ) abc . (4.22)As described in chapter 3, from this one can see that if the S -contribution were absent, thespace-time would have local holonomy contained in sim (3) [96]. Moreover, S can be seen asa totally antisymmetric torsion which is metric-compatible , and thus Hol( D ) ⊆ Sim (3). Wewill actually see later that if the field strength satisfies the radiation condition F ∧ ˆ L = 0, thespace has again local holonomy contained in sim (3). This is another way of stating that the generalised holonomy of the fKSE is contained in Sim (3), which is precisely what is neededfor the existence of Killing spinors.Using the base given above we can express the fieldstrength F in terms of it, and theconstraint (4.19) prescribes F = 2 ξ ˆ L ∧ N + (cid:37) x ˆ L ∧ E x + f xy E x ∧ E y , (4.23)for some (still) undetermined entities (cid:37) x and f xy . Using that ı L (cid:63) F = (cid:63) (cid:16) ˆ L ∧ F (cid:17) one can thenwrite the antisymmetrised version of eq. (4.20) as D ˆ L = − √ (cid:63) (cid:16) ˆ L ∧ F (cid:17) . (4.24)Since L is a null vector we know that ε xyz f yz ≡ √ ξ ℵ x −→ (cid:63) (cid:16) ˆ L ∧ F (cid:17) ≡ − √ ξ ˆ L ∧ ℵ , (4.25) A torsionful connection is said to be metric if the torsion tensor S ( X, Y, Z ) ≡ g ( S X Y, Z ) is antisymmetricin the last two entries, i.e. S ( X, Y, Z ) = − S ( X, Z, Y ), in fact implying that S is a 3-form. Furthermore, thetorsionful connection is said to be strong if S is closed, i.e. dS = 0. In the case at hand we have that S ∼ (cid:63)F ,whence dS ∼ d (cid:63) F ∼ αF ∧ F which generally non-vanishing, so that our torsion-structure is not strong. Where we have chosen the orientation ε + − xyz = ε xyz , for ε xyz the three-dimensional Riemannian totallyantisymmetric tensor. CHAPTER 4. N = 1 D = 5 MINIMAL FAKESUGRA where we have decomposed ℵ = ℵ x E x as follows from (4.23). This implies that ˆ L ishypersurface-orthogonal, i.e. ˆ L ∧ d ˆ L = 0, which in its turn implies the existence of twofunctions Y and u such that ˆ L = Y du ; but since ˆ L has R -weight 2, we can gauge-fix Y = 1,and thus ˆ L = du . Plugging all of this into eq. (4.24) then implies that A = Υ ˆ L + ℵ , (4.26)for some function Υ. This form of the vector potential has ı L A = 0, which after contractingeq. (4.20) with L a implies that L is a geodesic vector, i.e. ∇ L L = 0.Let us further introduce a coordinate v adapted to the vector L by L = ∂ v , and threemore coordinates y m ( m = 1 , ,
3) such that the F¨unfbein can be taken to be E + = ˆ L = du , θ + = N (cid:91) = ∂ u − H∂ v ,E − = N = dv + Hdu + ω m dy m , θ − = L = ∂ v ,E x = E xm dy m , θ x = E xm ( ∂ m − ω m ∂ v ) , (4.27)where we have defined E A ( θ B ) = δ AB and also the Dreibein E xm , which leads to the three-dimensional metric h mn ≡ E xm E xn . The resulting line element then takes on the Walker-form[97] ds = 2 du ( dv + Hdu + ω ) − h mn dy m dy n . (4.28)Given this base and the result (4.26), we see that the symmetrised version of eq. (4.20) impliesthat ∂ v H = 2 ξ Υ , (4.29) ∂ v ω m = 2 ξ ℵ m , (4.30) ∂ v h mn = 0 . (4.31)In order to fix the v -dependence of the metric we need the v -dependence of the vectorpotential A . This can be obtained by calculating £ L A = ı L F = − ξ ˆ L = ∂ v Υ ˆ L + £ L ℵ = ∂ v Υ ˆ L + ( ∂ v ℵ m ) dy m , (4.32)which implies that ℵ is v -independent, and also thatΥ = − ξ v + Υ ( u, y ) , (4.33) H = − ξ v + 2 ξ Υ v + Υ ( u, y ) , (4.34) ω m = 2 ξ v ℵ m + (cid:36) m ( u, y ) . (4.35)As discussed in [96], Υ can be made to vanish by means of a suitable coordinate transfor-mation v → v + U ( u, y ), and thus from now on we shall be taking Υ = 0.Having now the explicit coordinate dependence, we can proceed to find the expressionsfor (cid:37) x and f xy in the field strength. (cid:37) x = E xm (cid:16) ˙ ℵ m − ξ ω m (cid:17) . (4.36)It is clear from eq. (4.26) that f mn = 2 ∂ [ m ℵ n ] , or equivalently f = ð ℵ , if we introducethe three-dimensional exterior derivative ð = dy m ∂ m . This implies that the definition of .2. DIFFERENTIAL CONSTRAINTS ℵ in eq. (4.25) is actually a constraint on ℵ . In fact this type of constraint is part of thedefinition of our previously considered Gauduchon-Tod spaces [84], which appear naturallyas the geometric structure of the three-dimensional base-space in N = 2 fakeSupergravities,as discussed in chapter 3. To confirm it, we investigate the totally antisymmetric form of theconstraint (4.21). In form-notation this reads D Φ x = − √ ξ (cid:63) Φ x . (4.37)Using the definition (4.12) we can rewrite the LHS as D Φ x = − ˆ L ∧ ( dE x − ξ ℵ ∧ E x ) , (4.38)so that eq. (4.37) can be recast as0 = ˆ L ∧ (cid:16) dE x − ξ ℵ ∧ E x + √ ξ ε xyz E y ∧ E z (cid:17) . (4.39)Ignoring the possible u -dependence, we can recast the above equation in terms of purelythree-dimensional objects as ð E x = 2 ξ ℵ ∧ E x − √ ξ ε xyz E y ∧ E z , (4.40)which offers a way to define Gauduchon-Tod spaces [84]. The conclusion then is that thegeometry of the transverse space is a Gauduchon-Tod manifold, albeit with a possible u -dependence.Furthermore, eq. (4.21) provides us with an additional constraint. Let us calculateˆ L ∧ ∇ N E x = D N Φ x − D N ˆ L ∧ E x , (4.41)and use D N ˆ L = ξ ℵ , (4.42)and eq. (4.21) to arrive at0 = ˆ L ∧ (cid:104) ∇ N E x − ξ ℵ x N + √ ε xyz (cid:37) y E z (cid:105) = E + ∧ (cid:104) ∇ + E x − ξ ℵ x E − + √ ε xyz (cid:37) y E z (cid:105) . (4.43)As the E s form a frame, one can use the spin connection in ∇ + E x = Ω + ,xa E a to rewrite theabove as ( Ω + , − x − ξ ℵ x ) E + ∧ E − = (cid:34) Ω + ,xy + √ ε xyz (cid:37) z (cid:35) E + ∧ E x . (4.44)The fSUSY thus imposes the constraintsΩ + , − x = ξ ℵ x = θ − ω x , (4.45)Ω + ,xy = − √ ε xyz (cid:37) z , (4.46) This can be obtained from eq. (4.20). CHAPTER 4. N = 1 D = 5 MINIMAL FAKESUGRA where eq. (4.30) has been used in the second equality on (4.45). In fact, comparing it witheq. (C.14), we see that they agree automatically, and thus this gives us no further information.The situation w.r.t. (4.46) is different, however, as it imposes a constraint whose consequenceswe now investigate. By using the explicit forms in eqs. (4.36) and (C.15), we see that the v -dependent part drops out and one finds2 √ ξ (cid:36) x − ε xyz B yz = √ E xm ˙ ℵ m + ε xyz E my ˙ E zm , (4.47)where we have defined B ≡ ð (cid:36) + 2 ξ ℵ ∧ (cid:36) . (4.48)As we now see, the constraints derived thus far are enough to have have a fake-supersymmetricconfiguration. So far we have derived necessary constraints on a configuration to be fake-supersymmetric.In this subsection we shall show that they are actually sufficient; we do this by plugging theminto the fKSE (4.2), and confirming that no further constraints arise.Imposing γ + (cid:15) i = 0 implies that the (cid:15) i are v -independent. Using this fact and the chosenorientation one can then see that γ xyz (cid:15) i = ε xyz (cid:15) i , γ xy (cid:15) i = − ε xyz γ z (cid:15) i , γ x (cid:15) i = − ε xyz γ yz (cid:15) i . (4.49)We can use these relations to massage the fKSE in the + direction into the form θ + (cid:15) i = − (cid:0) √ (cid:37) x + ε xyz Ω + ,yz (cid:1) γ x (cid:15) i = 0 , (4.50)which vanishes due to eq. (4.46). Hence, the fake Killing spinor is both u - and v -independent.Doing a similar thing to the fKSE in the x -directions gives0 = ∇ ( λ ) x (cid:15) i + ξ γ xy ℵ y (cid:15) i + √ ξ γ x (cid:15) i , (4.51)where ∇ ( λ ) is the three-dimensional covariant derivative for the spin-connection λ (cf. ap-pendix C.2). In [86] equations like these are called Killing equations in Weyl geometry ,and one can see that eq. (4.51) corresponds to a Killing equation for a three-dimensionalweightless spinor. The existence theorem for solutions to the above equation is found in [87],where it is shown that the above equation has solutions if and only if the Weyl geometry isGauduchon-Tod.The amount of preserved fSUSY remains to be studied. For this we can use eq. (C.16) toderive ∇ ( λ ) x (cid:15) i ≡ ∂ x (cid:15) i − Ω x,yz γ yz (cid:15) i = ∂ x (cid:15) i − ξ γ xy ℵ y (cid:15) i − √ ξ γ x (cid:15) i , (4.52)where we made use of the relations in (4.49). Comparing this equation with (4.51), one seesthat ∂ x (cid:15) i = 0, so that the Killing spinor is constant. As the only restriction imposed has been γ + (cid:15) i = 0, the configuration thus breaks half of the fakeSupersymmetry. Note that the Fierz identity [108, eq. (A.27)] instructs us to do so. The identification is obvious from [86, Th. 1.2] once one sees that [86, eq. (1)] implies that (cid:8) γ i , γ j (cid:9) = − δ ij ,and furthermore that the 1-form used differs from the one here by θ [86] = − ξ ℵ . .3. SOLVING THE EOMS In order to study the Maxwell equation of motion we first calculate (cid:63) F = 2 √ ξ E + ∧ E − ∧ ℵ − ε xyz (cid:37) x E y ∧ E z ∧ E + − ξ ε xyz E x ∧ E y ∧ E z . (4.53)With this expression, the Maxwell EOM can be seen to give ∇ ( λ ) x (cid:37) x = 4 ξ(cid:37) x ℵ x + 24 ξ v ℵ x ℵ x + √ ξ ℵ x ε xyz B yz + ω x ∂ v (cid:37) x + 2 ξ E xm θ + E xm . (4.54)We shall have a look at some particular subcases.Consider first the u -independent case with (cid:36) = 0. The above equation implies that ∇ ( λ ) x ℵ x = 0, whence the three-dimensional metric h is Gauduchon, which was already impliedby eq. (4.25). If we relax the condition that (cid:36) = 0, we find that ∇ ( λ ) x (cid:36) x = −√ ξ (cid:16) (cid:63) (3) B x − √ ξ (cid:36) x (cid:17) ℵ x . (4.55)Combining this with eq. (4.47) for the u -independent case, we see that one arrives at ∇ ( λ ) x (cid:36) x = 0 . (4.56)As indicated by fSUSY, the only component of the Einstein equations that needs to bechecked explicitly is E ++ . In the u -independent case this is easily calculated to give0 = ∇ ( λ ) x ( ∂ x + 2 ξ ℵ x ) Υ . (4.57)This means that in the u -independent case the full solution is given by a Gauduchon-Tod spacedetermining the pair ( E x , ℵ ) through eq. (4.40), an Υ which is determined by eq. (4.57) anda (cid:36) determined through eq. (4.47), which in this case reads2 √ ξ (cid:36) x = ε xyz ( ð (cid:36) + 2 ξ ℵ ∧ (cid:36) ) yz . (4.58)Knowing ( E x , ℵ , Υ , (cid:36) ), the full solution can be written down using eqs. (4.33) - (4.35) and(4.26), (4.27). An example is given next. A first explicit solution is given by ds = 2 du (cid:0) dv − ξ v du (cid:1) − ξ dS ,A = − ξ v du , (4.59)where dS is the standard metric on the 3-sphere. The above background is an electrically-charged five-dimensional Nariai cosmos [109]. Observe that as this solution satisfies F ∧ ˆ L = 0,it has Hol( ∇ ) ⊆ Sim (3); in fact it has Hol( ∇ ) = so (1 , ⊕ so (3) ⊂ Sim (3).6
CHAPTER 4. N = 1 D = 5 MINIMAL FAKESUGRA
We can of course also use the Berger sphere as a GT structure, i.e. employ eqs. (D.12) toconstruct a different solution, which we shall call the squashed
Nariai cosmos. Explicitly, thesolution reads ds = 2 du (cid:0) dv − ξ v du + sin( µ ) v [ dχ − cos( φ ) dϕ ] (cid:1) − cos ( µ )12 ξ dB χ,φ,ϕ ] , (4.60) A = − ξ v du + sin( µ )2 ξ ( dχ − cos( φ ) dϕ ) , (4.61)where dB χ,φ,ϕ ] is the metric of the Berger sphere.Note that in the squashed case ℵ (cid:28)
0, so that we are dealing with a background where therelevant holonomy group is w.r.t. the connection D . Also observe that the Berger sphere is theonly compact GT space that is not an Einstein space and has non-vanishing Weyl-scalar [84,prop. 6]. Thus the only Einstein space that can be used to construct a solution backgroundis the 3-sphere, which leads to the Nariai solution in eq. (4.59); this of course can also beobtained from the squashed Nariai presented here with the choice µ = 0. d = 4 fakeSUGRA As is well known, minimal N = 1 d = 5 Supergravity can be dimensionally reduced to fourdimensions, where it can be identified with N = 2 d = 4 Supergravity coupled to one vector-multiplets. The field content is a metric g , two vector fields A Λ (Λ = 0 , Z . The resulting Special Geometry is governed by the cubic prepotential F ( X ) = − ( X ) X . (4.62)The explicit KK-Ansatz necessary for obtaining this identification reads ds = k − ds − k (cid:0) dy + A (cid:1) , (4.63)ˆ A = −√ (cid:2) A − B (cid:0) dy + A (cid:1)(cid:3) , (4.64) Z = B + ik . (4.65)Along the same lines, the theory studied in this chapter can also be dimensionally reducedto four dimensions, and using the above KK-Ansatz the resulting four-dimensional theory fallsinto the class studied in chapter 3, with a potential V that is given by V = 16 ξ k − = 16 ξ Im − ( Z ) . (4.66)Comparing it to the general potential given in eq. 3.121 of chapter 3, we see that C = 0 , ( C ) = 24 ξ , (4.67)where the coupling constant g appearing in eq. (3.121) has been absorbed into C to avoidconfusion. This is because the Weyl scalar is constrained to be W = − ξ , which is non-vanishing. .5. SUMMARY OF THE CHAPTER u -direction, which implies that one must take the solution to be u -independent. Wecan thus reduce the general solution to ds = k − ( dv + ω ) − k h mn dy m dy n , (4.68) A = − k − ( dv + ω ) , (4.69) A = − − / (cid:0) ℵ + 2 ξ v A (cid:1) , (4.70)where the scalars are given by k = 4 ξ v − , B = − ξ √ v . (4.71)These solutions have a stationarity vector ω that is at most linear in τ , and the geometry ofthe transverse space is a Gauduchon-Tod space. As expected, they thus resemble those foundin the timelike class in four dimensions (cf. section 3.2), where v plays the rˆole of the timecoordinate τ . This chapter has analysed the characterisation of solutions to five-dimensional fSUGRA withone sympletic-Majorana spinor, also commonly referred to as N = 1 d = 5 De Sitter SUGRA.Our main result was that all solutions to the theory admitting fake-Killing spinors, from whicha null vector field can be constructed, fall into the following family of backgrounds ds = 2 du (cid:18) dv + (cid:0) Υ − ξ v (cid:1) du + 2 ξv ℵ + (cid:36) (cid:19) − ds GT , (4.72) F = χ du ∧ dv + 12 d ℵ , (4.73)where 8 ξ is the cosmological constant, GT is a u -dependent Gauduchon-Tod space [84], andΥ , ℵ and (cid:36) are, respectively, a function and two 1-forms on GT which may also depend on u (but not on v ).Gauduchon-Tod spaces were initially discussed in the context of hyper-hermitian spacesadmitting a tri-holomorphic Killing vector field 84. They are special types of Einstein-Weyl 3-spaces, obeying the constraint (4.40), and we have seen that they play a rˆole in the solutionsto both four- and five-dimensional fSUGRA. In this sense, they were used e.g. in [30] toconstruct examples of timelike solutions of d = 5 minimal fSUGRA for which the base-spaceis not conformally hyper-K¨ahler. In d = 4, it was shown in chapter 3 that the timelikesolutions are defined by a base-space which is GT. But whereas the Ricci curvature of theWeyl connection is always non-flat in the solutions we have described in this chapter, thefour-dimensional timelike solutions also allow flat GT spaces.As for the null supersymmetric solutions of minimal five-dimensional ungauged and gaugedSUGRA theories, the family of backgrounds (4.72) admits a geodesic, expansion-free, twist-free and shear-free null vector field N . To see this, consider the null vector field N = ∂/∂v .The congruence of integral curves affinely parametrised by v fulfills ∇ N N = 0 (geodesic). N CHAPTER 4. N = 1 D = 5 MINIMAL FAKESUGRA is hypersurface orthogonal, i.e. e − ∧ d e − = 0, which means that the congruence is twist-free.It is also non-expanding, ∇ µ N µ = 0, and shear-free, ∇ ( µ N ν ) ∇ µ N ν = 0. As we saw in section3.3, such geometries are dubbed Kundt metrics in four-dimensional General Relativity [110].It is a special case of the higher-dimensional metrics considered in [111 - 114].But N has distinct properties in the De Sitter theory, as compared with the Minkowskior AdS theories. In these latter, the null vector is always Killing and for some special cases itbecomes covariantly constant. Then the Kundt geometries become plane-fronted waves withparallel rays ( pp-waves ). This is not the case for the De Sitter theory. For the special casewith ℵ = 0, however, the null vector acquires an interesting property; it becomes recurrent ,that is, it obeys ∇ µ N ν = C µ N ν , (4.74)for some non-trivial, recurrence one form C µ . This means that the geometries (4.72) have spe-cial holonomy Sim (3), which is the maximal proper subgroup of the Lorentz group SO (4 , Sim group, holonomy and recurrency can be found inappendix E.The four parameter Similitude group,
Sim (2), became a focus of interest due to theproposal of
Very Special Relativity (VSR) [115]. Cohen and Glashow asked the question if theexact symmetry group of Nature could be isomorphic to a proper subgroup of the Poincar´egroup, rather than the Poincar´e group itself. The proper subgroup they considered was
ISim (2), obtained by adjoining the maximal proper subgroup of the Lorentz group,
Sim (2),with spacetime translations. The theory based on this symmetry group, VSR, actually impliesSpecial Relativity if a discrete symmetry, namely CP, is also added. But since the latter isbroken in nature, VSR is necessarily distinct from Special Relativity. Additionally, therehas been other subsequent developments on theories employing the Sim and ISim groups,including the
General Very Special Relativity of [116]. In this sense, studies of d -dimensionalLorentzian geometries with Sim ( d −
2) holonomy have been carried out recently [96]. Theresulting geometries have interesting properties, such as the possibility of vanishing quantumcorrections [105]. Possible connections to Supersymmetry were also hinted in [112]. In thischapter, as well as in chapter 3, we have shown how indeed these geometries emerge in anexplicit supersymmetric computation.In the next chapter we shall extend the investigations to include the coupling to N = 1 d = 5 Abelian matter. hapter 5 N = 1 d = 5 fakeSUGRA coupled toAbelian vectors This chapter studies the classification of the null class solutions to N = 1 d = 5 fSUGRAcoupled to Abelian vector multiplets. Its content is essentially that of [117], on which it ismodelled. It is a generalisation of the analysis of chapter 4, and one can indeed see thatthose solutions (where only the gravity supermultiplet was present) are a limiting case ofthe ones presented here. To obtain this theory we analytically continue the supersymmetrytransformations of the gravitino, as well as those for the gauginos, of the regular SUGRAtheory. The vanishing of these transformations produces fake Killing spinor equations, andwe consider fake-supersymmetric solutions which admit (non-trivial) spinors satisfying theequations.The outline of this chapter is the following: section 5.1 gives a summary of the basicequations of the theory. In section 5.2 we analyse them, focusing on the case where the 1-formspinor bilinear is null. The conditions obtained from the gravitino equation are derived fromthe analysis of the minimal fake-supersymmetric solutions in chapter 4. We also introducelocal coordinates, and show that, not too surprisingly, the solutions are given in terms of aone-parameter family of three-dimensional Gauduchon-Tod (GT) spaces. Imposing fSUSYtogether with the Bianchi identity and the gauge field equations is sufficient to ensure that allthe remaining equations, with the exception of one component of the Einstein equations, holdautomatically. Section 5.3 gives some simple examples of our solutions, including some near-horizon geometries and an explicit model with one gauge multiplet, which under vanishingof the potential provides non-BPS solutions to a SUGRA theory. In section 5.4 we providesolutions where the GT space is the Berger sphere. Section 5.5 considers the conditions forwhich the null Killing spinor 1-form is recurrent, and investigate the properties of variousscalar curvature invariants. Finally, section 5.6 has the summary of the chapter. N = 1 d = 5 fSUGRA and Killing spinors The theory we start from is N = 1, d = 5 gauged Supergravity coupled to Abelian vectormultiplets [118]. Apart from the fields already present in its minimal version, this theoryalso has n v vector fields and n v real scalar fields. As it is customary, the ¯ n = n v + 1 vectorfields are denoted jointly by A I ( I = 0 , . . . , n ). Similarly to four-dimensional supergravity,the scalar self-interactions and their interaction with the vector fields can be derived from a690 CHAPTER 5. N = 1 D = 5 FAKESUGRA COUPLED TO ABELIAN VECTORS geometrical structure, which in this case is known as
Real Special geometry . The bosonicaction is given by S = 116 πG (cid:90) (cid:0) − R + 2 g V (cid:1) ∗ − Q IJ (cid:0) − dX I ∧ (cid:63)dX J + F I ∧ ∗ F J (cid:1) − C IJK F I ∧ F J ∧ A K , (5.1)where I, J, K take values 1 , . . . , n and F I = dA I are 2-forms representing gauge field strengths(one of the gauge fields corresponds to the graviphoton). The constants C IJK are symmetricin { I, J, K } , and we shall be assuming that Q IJ (the gauge coupling matrix) is positive-definiteand invertible, with inverse Q IJ . The X I are scalar fields subject to the constraint16 C IJK X I X J X K = X I X I = 1 . (5.2)The fields X I can thus be regarded as being functions of ( n −
1) unconstrained scalars φ r .We list some useful relations associated with N = 2, d = 5 gauged Supergravity Q IJ = 92 X I X J − C IJK X K ,Q IJ X J = 32 X I , Q IJ dX J = − dX I , V = 9 V I V J ( X I X J − Q IJ ) , (5.3)where V I are constants. The De Sitter supergravity theory is obtained by sending g to − g in eq. (5.1).Fake-supersymmetric De Sitter solutions admit a Dirac spinor (cid:15) satisfying a gravitino andthe gauginos fake Killing spinor equations. The gravitino fKSE reads (cid:20) ∇ M + 18 γ M H N N γ N N − H M N γ N − g ( 12 Xγ M − A M ) (cid:21) (cid:15) = 0 , (5.4)where we have defined V I X I = X , V I A I M = A M , X I F I MN = H MN . (5.5)The gaugino fKSE is given by (cid:18) ( − F IMN + X I H MN ) γ MN + 2 ∇ M X I γ M − gV J ( X I X J − Q IJ ) (cid:19) (cid:15) = 0 . (5.6)We adopt a mostly minus signature for the metric, which is written in a null frame as ds = 2 e + e − − δ ij e i e j , (5.7)for i, j, k = 1 , , The modifier
Real is used to stress the fact that the scalar fields are now real-valued. A small introductionto this structure is given in appendix B.4. Even though the notation used there is different from the one usedin this section, the essence of the structure remains the same. .2. ANALYSIS OF GRAVITINO KILLING SPINOR EQUATION We proceed with the analysis of the fake Killing spinor equations, focusing on the case forwhich the 1-form spinor bilinear generated from the Killing spinor is null; our basis is chosensuch that this bilinear 1-form is given by e − .The analysis of the gravitino fKSE has already been completed in the case of minimal d = 5 De Sitter Supergravity in chapter 4. But for the present purposes, in which we areemploying spinorial geometry techniques (cf. section 2.2), a more suitable analysis is given in[77]. The conditions on the geometry and the fluxes obtained from eq. (5.4) can thus be readoff from the results in [77, sec. 3], which are listed in equations (3.1)-(3.20). We shall not beincorporating any of the conditions obtained from the Bianchi identity in [77] here, becausethe 2-form flux H which appears in eq. (5.4) is not the exterior derivative of A , in contrastto the minimal theory. Also, to establish the correspondence between the fKSE solved in [77]and (5.4) one makes the following replacements F → √ H , χ → − √ gX , χA → − gA , (5.8)where the quantities on the LHS of these expressions are the field strength, the cosmologicalconstant and the gauge potential of the minimal theory, using the conventions of [77].Following the reasoning given there, one can without loss of generality work in a gaugefor which the conditions on the geometry are d e − = 0 , (5.9) d e + = − g e + ∧ A − ω − , − i e − ∧ e i − ω [ i, |−| j ] e i ∧ e j , (5.10) d e i = 2 ω [ − ,j ] i e − ∧ e j + B ∧ e i + 3 gX (cid:63) e i , (5.11) L N e i = 0 , (5.12)where N is the vector field dual to e − , and ω is the five-dimensional spin connection. B is a1-form given in terms of the spin connection by B = − ω i, + − e i . (5.13)In addition, the 1-form A satisfies − gA = − ω − , + − e − + B , (5.14)and the 2-form flux H satisfies H = (cid:63) B + gX e + ∧ e − + 13 e − ∧ (cid:63) ( ω − ,ij e i ∧ e j ) . (5.15)Here (cid:63) denotes the Hodge dual taken w.r.t. the 1-parameter family of 3-manifolds E equippedwith metric ds E = δ ij e i e j , (5.16)whose volume form (cid:15) (3) satisfies γ ijk (cid:15) = (cid:15) (3) ijk (cid:15) . (5.17)2 CHAPTER 5. N = 1 D = 5 FAKESUGRA COUPLED TO ABELIAN VECTORS
Furthermore, in this gauge, the spinor (cid:15) can be taken to be a constant, which satisfies γ + (cid:15) = 0 , (5.18)or equivalently γ + − (cid:15) = (cid:15) . (5.19) The gaugino fake Killing spinor equation (5.6) can be rewritten as (cid:18) F I + − − F I + i γ − γ i − F Iij (cid:15) ijk γ k − X I H + − + 2 X I H + i γ − γ i + X I H ij (cid:15) ijk γ k , +2 ∇ + X I γ − + 2 ∇ i X I γ i − gXX I + 6 gQ IJ V J (cid:19) (cid:15) = 0 , (5.20)where we have made use of the identities γ ij (cid:15) = (cid:15) (3) ijk γ k (cid:15) (5.21)and γ i = − δ ij γ j . Acting on eq. (5.20) with the projectors (1 ± γ + − ) one obtains twoequations of the form ( α + β i γ i ) (cid:15) = 0 , (5.22)for real α , β i coefficients. As (cid:15) is non-zero, the only solution of such an equation is α = 0 , β i = 0 , and on evaluating the resulting conditions on the fluxes and scalars obtained fromthese equations, one finds that L N X I = 0 (5.23)and F I + − = 3 g ( XX I − Q IJ V J ) , (5.24) F Iij = X I ( (cid:63) B ) ij + (cid:15) ijk ∇ k X I , (5.25) F I + i = F I − i = 0 , (5.26)where we have adopted the convention that ( (cid:15) ) ijk = ( (cid:15) ) ijk , i.e. indices on the volume formare raised with the metric of signature (+ , + , +). As e − is a closed form, one can introduce local coordinates u, v, y α for α = 1 , , e − = du , N = ∂∂v , e i = e iα dy α . (5.27)Observe that possible du terms in e i can be removed without loss of generality by using agauge transformation which leaves the spinor (cid:15) invariant, as described in [77]. The three-dimensional Dreibein e iα does not depend on v , but in general it depends on y α and u , as dothe scalars X I . .2. ANALYSIS OF GRAVITINO KILLING SPINOR EQUATION v -dependence of e + . Note that eq. (5.10) implies that L N e + = − gA , (5.28)and eqs. (5.14) and (5.24) that L N A = 3 g ( X − Q IJ V I V J ) e − . (5.29)Moreover, eq. (5.29) together with eq. (5.14) implies that one can take A = 3 g ( X − Q IJ V I V J ) vdu − g B . (5.30)Note also that if ˜ d denotes the exterior derivative restricted to hypersurfaces of constant u, v ,eq. (5.11) then implies that ˜ d e i = B ∧ e i + 3 gX (cid:63) e i . (5.31)This is again the Gauduchon-Tod structure found in the study of a special class of Einstein-Weyl spaces [84, 89] (cf. appendix D).Furthermore, eqs. (5.12) and (5.23) imply that L N B = 0 , (5.32)and likewise eq. (5.31) implies that˜ d B + 3 g (cid:63) ( X B + ˜ dX ) = 0 . (5.33)It is then straightforward to integrate eq. (5.28) up to find e + = dv − g ( X − Q IJ V I V J ) v du + W du + v B + φ i e i , (5.34)where W is a v -independent function, and φ = φ i e i is a v -independent 1-form.Next we wish to determine the v -dependence of the field strengths F I . Note first that L N F I = d ( i N F I ) = ˜ d (3 g ( XX I − Q IJ V J )) ∧ du , (5.35)on making use of the Bianchi identity dF I = 0. From eq. (5.24) one also finds that F I = 3 g ( XX I − Q IJ V J ) e + ∧ e − + (cid:63) ( X I B + ˜ dX I ) + e − ∧ S I , (5.36)where S I = S I j e j . (5.37)One thus takes the Lie derivative of this expression and compares with eq. (5.35), to find that L N S I = 3 g ( XX I − Q IJ V J ) B − g ˜ d ( XX I − Q IJ V J ) , (5.38)and whence F I = 3 g ( XX I − Q IJ V J )( v B + dv + φ ) ∧ du + (cid:63) ( X I B + ˜ dX I ) , + du ∧ (cid:18) gv (cid:0) ( XX I − Q IJ V J ) B − ˜ d ( XX I − Q IJ V J ) (cid:1) + T I (cid:19) , (5.39)4 CHAPTER 5. N = 1 D = 5 FAKESUGRA COUPLED TO ABELIAN VECTORS where T I = T I j e j (5.40)are v -independent 1-forms on E .Having this expression for the flux, we continue by imposing two consistency conditions.The first is V I F I = dA , where F I is given by eq. (5.39) and A is obtained from eq. (5.30). Itgives the following condition on the 1-forms T I , V I T I = − g ˙ B + 3 g ( X − Q IJ V I V J ) φ , (5.41)where ˙ B = L ∂∂u B . We also impose X I F I = H , where H is given in eq. (5.15), and we notethat ω − ,ij = −
12 ( v ˜ d B + ˜ dφ − φ ∧ B ) ij + 12 ( ˙ e i ) j −
12 ( ˙ e j ) i . (5.42)This gives an additional condition on T I X I T I = − (cid:63) (cid:18) ˜ dφ − φ ∧ B + δ ij ˙ e i ∧ e j (cid:19) . (5.43)Finally, we impose the Bianchi identities dF I = 0, which give the conditions˜ dT I = L ∂∂u (cid:63) ( X I B + ˜ dX I ) + 3 g ˜ d ( XX I − Q IJ V J ) ∧ φ + 3 g ( XX I − Q IJ V J ) ˜ dφ , (5.44)0 = ˜ d (cid:63) (cid:0) X I B + ˜ dX I (cid:1) . (5.45)This exhausts the content of the fake Killing spinor equations. In addition to conditions imposed by fSUSY, we require that our configurations solve the fieldequations. We start by evaluating the gauge field equations d (cid:63) ( Q IJ F J ) + 14 C IJK F J ∧ F K = 0 , (5.46)where the five-dimensional volume form (cid:15) is related to the three-dimensional volume form (cid:15) by (cid:15) = e + ∧ e − ∧ (cid:15) . (5.47)One obtains the following condition˜ d (cid:63) ( Q IJ T J ) = − g L ∂∂u (cid:0) ( 32 XX I − V I ) (cid:15) (cid:1) + 32 φ ∧ B ∧ ˜ dX I + 32 ˜ dφ ∧ ( B X I − ˜ dX I )+3 gφ ∧ (cid:63) (cid:0) ( 32 XX I − V I ) B − X I ˜ dX + 32 X ˜ dX I + Q IJ ˜ dQ JN V N (cid:1) + 12 C IJK T J ∧ (cid:63) ( X K B + ˜ dX K ) . (5.48) .2. ANALYSIS OF GRAVITINO KILLING SPINOR EQUATION i.e. having only thegravity supermultiplet), it becomes satisfied automatically. This is the rather peculiar featureof the null case of minimal fSUGRA theories, which was noticed before [77, 106].Next, consider the Einstein field equations. It is straightforward to show that the inte-grability conditions of the fKSEs imply that all components of the Einstein equations holdautomatically, with the exception of the “ −− ” component. The field equations are0 = R αβ + Q IJ F Iαµ F Jβ µ − Q IJ ∇ α X I ∇ β X J + g αβ (cid:18) − Q IJ F Iβ β F Jβ β − g ( 12 Q IJ − X I X J ) V I V J (cid:19) , (5.49)and hence the “ −− ” component is R −− − Q IJ F I − i F J − j δ ij − Q IJ ∇ − X I ∇ − X J = 0 . (5.50)This equation imposes the additional condition0 = ˜ ∇ W + ˜ ∇ i ( W B i ) − ˜ ∇ i ˙ φ i − gφ i V I ( T I ) i − (¨ e i ) i −
3( ˙ e j ) i X I ( (cid:63) T I ) ij + 12 C IJK X K (cid:0) ( T I ) i ( T J ) i + ˙ X I ˙ X J (cid:1) . (5.51)We also require that the solution satisfies the scalar field equations. However, the integrabilityconditions of the fake Killing spinor equations, together with the gauge field equations, implythat the scalar field equations hold with no additional conditions. In order to construct a supersymmetric solution in the null class, we introduce local coor-dinates u, v, y α , together with a family of Gauduchon-Tod 3-manifolds GT, and write themetric as ds = 2 e + e − − ds , ds = δ ij e i e j , (5.52)with e + = dv − g v ( X − Q IJ V I V J ) du + W du + v B + φ i e i , e − = du , e i = e αi dy α , (5.53)where the basis elements e i do not depend on v , but can depend on u , W is a v -independentfunction and B = B i e i , φ = φ i e i are v -independent 1-forms. The metric (5.52) is a Kundtwave, whose recurrency properties we shall analyse in section 5.5. Because the base-space isGauduchon-Tod, the basis elements e i have to satisfy˜ d e i = B ∧ e i + 3 gX (cid:63) e i , (5.54)where ˜ d is the exterior derivative restricted to hypersurfaces of constant v, u and (cid:63) is theHodge dual on the GT. The scalar fields satisfy˜ d (cid:63) (cid:0) X I B + ˜ dX I (cid:1) = 0 , (5.55)6 CHAPTER 5. N = 1 D = 5 FAKESUGRA COUPLED TO ABELIAN VECTORS and ˜ d B + 3 g (cid:63) ( X B + ˜ dX ) = 0 . (5.56)The field strengths are given by F I = 3 g ( XX I − Q IJ V J )( dv + φ ) ∧ du + (cid:63) ( X I B + ˜ dX I ) , + du ∧ (cid:16) − gv ˜ d ( XX I − Q IJ V J ) + T I (cid:17) , (5.57)where T I = T I j e j (5.58)are v -independent 1-forms on the GT. The 1-forms T I must further satisfy˜ d (cid:63) ( Q IJ T J ) = − g L ∂∂u (cid:0) ( XX I − V I )dvol GT (cid:1) + φ ∧ B ∧ ˜ dX I + ˜ dφ ∧ ( B X I − ˜ dX I ) + C IJK T J ∧ (cid:63) ( X K B + ˜ dX K )+3 gφ ∧ (cid:63) (cid:0) ( XX I − V I ) B − X I ˜ dX + X ˜ dX I + Q IJ ˜ dQ JN V N (cid:1) , (5.59)as well as V I T I = − g ˙ B + 3 g ( X − Q IJ V I V J ) φ (5.60)and X I T I = − (cid:63) (cid:18) ˜ dφ − φ ∧ B + δ ij ˙ e i ∧ e j (cid:19) (5.61)and ˜ dT I = L ∂∂u (cid:63) ( X I B + ˜ dX I ) + 3 g ˜ d (cid:18) ( XX I − Q IJ V J ) φ (cid:19) . (5.62)Finally, the function W is found by solving˜ ∇ W + ˜ ∇ i ( W B i ) − ˜ ∇ i ˙ φ i − gφ i V I ( T I ) i − (¨ e i ) i −
3( ˙ e j ) i X I ( (cid:63) T I ) ij + 12 C IJK X K (cid:0) ( T I ) i ( T J ) i + ˙ X I ˙ X J (cid:1) = 0 . (5.63)We remark that eqs. (5.59), (5.62) and (5.63) always admit solutions, however it is notapparent a priori that eqs. (5.60) and (5.61) can always be solved. We now present some simple solutions to the system prescribed above. .3. SOME SIMPLE EXAMPLES The near-horizon geometries found in [119] are in fact all examples of fake-supersymmetricsolutions in the null class. We take ∂/∂u as a symmetry of the full solution, and set the X I to be constant, as well as W = 0, φ = 0, T I = 0. The remaining conditions on the geometrythus simplify considerably, and one finds e − = du , (5.64) e + = dv − g ( X − Q IJ V I V J ) v du + v B , (5.65) e i = e αi dy α , (5.66)where ˜ d e i = B ∧ e i + 3 gX (cid:63) e i , (5.67)and the gauge field strengths are F I = 3 g ( XX I − Q IJ V J ) dv ∧ du + X I (cid:63) B , (5.68)with the 1-form B satisfied ˜ d B + 3 gX (cid:63) B = 0 . (5.69)This solution can be interpreted as the (fake-supersymmetric) near-horizon geometry of a(possibly non-fake-supersymmetric) black hole. The case for which V I X I = 0 , B (cid:54) = 0 is ofparticular interest, as the spacetime geometry is M × S , where M is a U (1) vibration over AdS related to the near-horizon extremal Kerr solution, and the spatial cross-sections of theevent horizon are S × S . A particularly simple class of solutions can be constructed with only one vector multiplet.Since the five-dimensional gravity multiplet does not contain scalars, there is only one physicalscalar field ϕ , and we choose the only non-vanishing value for the symmetric constant C IJK to be given by C = 1. With this choice, X I = (cid:18) ϕ − , √ ϕ (cid:19) , X I = 13 (cid:18) ϕ , √ ϕ − (cid:19) ,Q IJ = diag ( ϕ , ϕ − ) , Q IJ = 2diag ( ϕ − , ϕ ) . (5.70)The equations resulting from the classification of the theory, summarised in subsection 5.2.4,must also be satisfied, however we shall not present such analysis here.This model is interesting as it provides a simple setting for non-supersymmetric solutionsto a supersymmetric theory. The potential is given by V = 9 V ( V ϕ + 2 √ V ϕ − ) , (5.71)and one can immediately see that if V = 0 the theory is supersymmetric ( i.e. it was thedefinite-positiveness of V that granted us a De Sitter-like fSUGRA structure). This solution,8 CHAPTER 5. N = 1 D = 5 FAKESUGRA COUPLED TO ABELIAN VECTORS however, is non-BPS, since the presence of V means its KSE is not that of standard five-dimensional SUGRA. This kind of behaviour was already present in the classification offour-dimensional fSUGRA (see section 3.4), and thus one can also make use here of theoxidation/ dimensional reduction relations between supergravity theories (see e.g. [104] or[120, sec. (5.3)]) to obtain solutions to minimal N = (2 , d = 6 Supergravity.To achieve this, we make use of the results developed in [104], which provide the five-dimensional action (obtained through a KK compactification over an S ) and compare it tothe action for our model. The fields in these actions, however, do not promptly correspond,and they have to be appropiately identified. The dimensionally-reduced action is given by S = (cid:90) d x (cid:112) | g | k (cid:32) − R − k F ( A ) − k − F ( B ) + (cid:15) µνρστ (cid:112) | g | k − F ( A ) µν F ( B ) ρσ B τ (cid:33) . (5.72)The kinetic term for the graviton in this action does not have a canonical form, so we proceedto rescale the metric by a scalar k , and hence unveil the kinetic term for the scalars hiddenin the Einstein-Hilbert term g µν → k − g µν . (5.73)The action hence becomes S = (cid:90) d x (cid:112) | g | (cid:32) − R + 43 k − ( ∂k ) − k F ( A ) − k − F ( B ) + (cid:15) (cid:112) | g | F ( A ) F ( B ) B (cid:33) . (5.74)We then compare this action with that of our model S = (cid:90) d vol (cid:18) − R + 3 ϕ − ( ∂ϕ ) − ϕ ( F ) − ϕ − ( F ) , − (cid:15) (cid:112) | g | F F A − (cid:15) (cid:112) | g | F F A (cid:33) , (5.75)where the topological term is integrated by parts to identify the gauge fields. Upon inspection, k = aϕ , where a is just a real constant of integration, and A = − a − A , (5.76) B = ± a A . (5.77)To identify the remaining elements we consider the supersymmetric variation of the gaugino, i.e. eq. (5.6), and that obtained from the dimensional reduction of the gravitino KSE of thesix-dimensional theory (see e.g. [104, eq. (1.15)]). We obtain that A µ = − A µ , B µ = − A µ and k = − ϕ .This gives the identification of fields, and one can use the equations of the reduction overa circle to obtain the six-dimensional ones g (6) µν = g (5) µν − ϕ A µ A ν , g (6) µ(cid:93) = − ϕ A µ , g (6) (cid:93)(cid:93) = − ϕ ,H − ab(cid:93) = ϕ − F ab , H − abc = − ϕ − (cid:0) (cid:63) (6) ( e (cid:93) ∧ F ) (cid:1) abc . (5.78) We have adapted the conventions of [104] to those of here. In particular, the Riemmann tensor has theopposite defining sign. In flat indices, the six-dimensional space is labelled by a = { , , , , } and (cid:93) , where a spans the five-dimensional space. .4. THE BERGER SPHERE PROVIDES A SOLUTION N = (2 , d = 6SUGRA with (the bosonic part of the) action given by (cid:90) d x (cid:112) | g | (cid:18) R + 124 ( H − ) (cid:19) , (5.79)where as usual one considers the antiself-duality of H − as an additional constraint on thetheory, rather than on the actual action. Consider now the Gauduchon-Tod space presented in [90] , embedded into an fSUGRAbackground as ds = 2 du ( dv − g v ( X − Q IJ V I V J ) du + W du + v B + φ i e i ) − dx − | x + h | ds S (5.80) B = 2 x + h + ¯ h | x + h | dx , X I = − i ( h − ¯ h ) C I g | x + h | , (5.81)where C I are constants such that V I C I = 1, and h = h ( z ) is a holomorphic and monotonicfunction. This wave fulfills eq. (5.55). However, the non-constancy of the X I spoils theidentities in (5.2), and hence this GT space (without rescaling) is not a good cross-section fora solution to the fSUGRA theory.Alternatively, one might consider scaling the metric so that eq. (5.31) is compatible witheq. (D.9). This is ds = 2 du ( dv − g v ( X − Q IJ V I V J ) du + W du + v B + φ i e i )+ ( h − ¯ h ) | x + h | g X (cid:0) dx + | x + h | ds S (cid:1) . (5.82)This inmediately says that the choice h = ¯ h , which was prohibited in the minimal case [77],also cannot be used here. Furthermore X (cid:54) = 0, and it is constrained by eqs. (5.55) and (5.56). In this section we concentrate on the case for which the GT space is compact and withoutboundary for all u , and such that X I , T I , φ, W are smooth. The near-horizon geometries ofthe previous section are special examples of these solutions.Consider eq. (5.55) and contract it with X I . One finds that˜ ∇ i B i + 23 Q IJ ˜ ∇ i X I ˜ ∇ i X J = 0 . (5.83)On integrating over GT one finds the constraint (cid:90) GT Q IJ ˜ ∇ i X I ˜ ∇ i X J = 0 , (5.84) See also the end of appendix D.1. CHAPTER 5. N = 1 D = 5 FAKESUGRA COUPLED TO ABELIAN VECTORS which, assuming that Q IJ is positive-definite, implies that X I = X I ( u ) and X = X ( u ).We shall consider eq. (5.54) with X (cid:54) = 0 and take GT to be the Berger sphere [121, 84].One can then write ds = cos ( µ )9 g X (cid:18) cos µ ( σ L ) + ( σ L ) + ( σ L ) (cid:19) , (5.85) B = sin( µ ) cos( µ ) σ L , (5.86)where µ = µ ( u ), and σ iL are the left-invariant 1-forms on SU (2) satisfying˜ dσ iL = − (cid:15) ijk σ jL ∧ σ kL . (5.87)Note that eqs. (5.60) and (5.62) can be solved by making use of eq. (5.56), to give T I = −L ∂∂u (cid:18) X I gX B (cid:19) + 3 g ( XX I − Q IJ V J ) φ + Θ I , (5.88)where Θ I are 1-forms on GT satisfying V I Θ I = 0 (5.89)and ˜ d Θ I = 0 . (5.90)Next, simplify eq. (5.59) using eqs. (5.61) and (5.62), using the identity˜ d (cid:18) δ ij ˙ e i ∧ e j (cid:19) = 2 B ∧ (cid:18) δ ij ˙ e i ∧ e j (cid:19) + 9 g ˙ X dvol GT + 3 gX L ∂∂u dvol GT . (5.91)After some manipulation, one finds that (5.59) can be rewritten as˜ d (cid:18) X I δ ij ˙ e i ∧ e j − gX Q IJ T J ∧ B − X V I φ ∧ B (cid:19) + 92 g (cid:0)
12 ˙ XX I − X ˙ X I (cid:1) dvol GT + (cid:0) gV I − gXX I (cid:1) L ∂∂u dvol GT + 12 gX (cid:18) − ˙ X I (cid:63) B ∧ B + X I ( L ∂∂u (cid:63) B ) ∧ B (cid:19) = 0 . (5.92)For the Berger sphere the second and third lines of this expression can be written in the form Q I ( u )dvol GT , and hence on integrating over the GT one obtains two separate conditions˜ d (cid:18) X I δ ij ˙ e i ∧ e j − gX Q IJ T J ∧ B − X V I φ ∧ B (cid:19) = 0 (5.93)and 92 g (cid:0)
12 ˙ XX I − X ˙ X I (cid:1) dvol GT + (cid:0) gV I − gXX I (cid:1) L ∂∂u dvol GT + 12 gX (cid:18) − ˙ X I (cid:63) B ∧ B + X I ( L ∂∂u (cid:63) B ) ∧ B (cid:19) = 0 . (5.94) If X = 0 then the GT is either S × S or T , according whether B (cid:54) = 0 or B = 0, respectively. We do notconsider those cases here. .4. THE BERGER SPHERE PROVIDES A SOLUTION d (cid:18) (cid:63) Θ I + 13 gX Θ I ∧ B + 3 g ( XX I − Q IJ V J ) (cid:63) φ + 12 X I δ ij ˙ e i ∧ e j (cid:19) = 0 (5.95)and − L ∂∂u (cid:18) gX B (cid:19) + gXφ + X I Θ I = − (cid:63) (cid:18) ˜ dφ − φ ∧ B + δ ij ˙ e i ∧ e j (cid:19) . (5.96)On contracting eq. (5.95) with X I , and using eq. (5.96), one finds L ∂∂u dvol GT = − XX dvol GT , (5.97)which for the Berger sphere implies that the squashing of the S is u -independent, i.e. µ isconstant in eq. (5.85), and the u -dependence of the metric on the GT is inside the overallconformal factor of X − . It follows that L ∂∂u (cid:63) B = − ˙ XX (cid:63) B , (5.98)and hence eq. (5.94) can be simplified to give˙ XX I − X ˙ X I − ˙ XX V I + 118 g X (cid:0) − X ˙ X I − ˙ XX I (cid:1) B = 0 . (5.99)On contracting this expression with X I and Q IJ V J one finds˙ X B = 0 , ( X − Q IJ V I V J ) ˙ X = 0 . (5.100)Suppose first that X − Q IJ V I V J (cid:54) = 0. Then ˙ X = 0, and eq. (5.99) further implies that˙ X I = 0 (5.101)Furthermore, contracting eq. (5.95) with V I gives˜ d (cid:63) φ = 0 , (5.102)so eq. (5.95) now reads ˜ d (cid:63) Θ I + Θ I ∧ (cid:63) B = 0 . (5.103)As ˜ d Θ I = 0, and the Berger sphere is simply connected, this equation implies that the Θ I areexact Θ I = ˜ dH I , (5.104)so that ˜ ∇ H I + B i ˜ ∇ i H I = 0 . (5.105)2 CHAPTER 5. N = 1 D = 5 FAKESUGRA COUPLED TO ABELIAN VECTORS
It follows that ˜ dH I = 0, and so is Θ I = 0. Also, eq. (5.63) implies that W = W ( u ). Itremains to consider the condition 5.96 gXφ = − (cid:63) (cid:0) ˜ dφ − φ ∧ B (cid:1) . (5.106)After some manipulation, it can be shown that it implies˜ ∇ φ + B i ˜ ∇ i φ = 2 ˜ ∇ ( i φ j ) ˜ ∇ ( i φ j ) + 3 (cid:18) B φ − ( B φ ) (cid:19) . (5.107)As the RHS of this expression is a sum of two non-negative terms, it follows from the maximumprinciple that φ is constant, and˜ ∇ ( i φ j ) = 0 , B φ − ( B φ ) = 0 . (5.108)These conditions imply that one can take, without loss of generality, φ = k σ L (5.109)for a constant k , irrespectively of whether B vanishes or not. Also note that by making acoordinate transformation of the formˆ u = f ( u ) , v = h ( u )ˆ v + g ( u ) , ψ = ˆ ψ + (cid:96) ( u ) , (5.110)where we have taken the vector field dual to σ L to be ∂/∂ψ , one can choose the functions f, h, g, (cid:96) such that the form of the metric and gauge field strengths is preserved, and in thenew coordinates W = 0.To summarise, if X − Q IJ V I V J (cid:54) = 0, then the Berger sphere squashing-parameter µ andthe X I are constant, and the background is given by ds = 2 du (cid:18) dv − g v ( X − Q IJ V I V J ) du + ( v sin µ cos µ + k ) σ L (cid:19) − cos µ g X (cid:18) cos µ ( σ L ) + ( σ L ) + ( σ L ) (cid:19) , (5.111) F I = 3 g ( XX I − Q IJ V J ) dv ∧ du + X I gX sin µ cos µ σ L ∧ σ L , (5.112)where k is a constant.In the special case X − Q IJ V I V J = 0 there are two possibilities. If B (cid:54) = 0 then eq. (5.99)implies that the X I are again constant, whereas if B = 0 then it implies that X I = 23 X − V I + X Z I , (5.113)for constant Z I . Also, eqs. (5.95) and (5.96) imply H I − X ( XX I − Q IJ V J ) X N H N = L I (5.114)and φ = − gX (cid:18) ˜ d ( X I H I ) + X I H I B (cid:19) + k ( u ) σ L , (5.115)where Θ I = ˜ dH I , H I are functions, and L I = L I ( u ) satisfy X I L I = 0. Of course a simplifiedeq. (5.63) still needs to be solved, determining the function W . .5. SOLUTIONS WITH A RECURRENT VECTOR FIELD The geometry we have found, eq. (5.52), is again a five-dimensional Kundt wave [110] (seealso appendix C.3). Thus it admits a null vector generating a geodesic null congruence that ishypersurface orthogonal, non-expanding and shear-free. As in the case of minimal De Sitter d = 5 Supergravity, the null vector field N is not Killing. We shall consider the necessaryand sufficient conditions for N to be recurrent, which places additional restrictions on theholonomy of the Levi-Civit`a connection. Recurrency with respect to this connection is definedas ∇ µ N ν = C µ N ν , (5.116)where C is the recurrent one-form [96]. d -dimensional geometries that allow recurrent vectorfields have holonomy group contained in the Similitude group Sim ( d − d − d + 4) / SO ( d − , E ( d −
2) augmented by homotheties. It is also the maximal propersubgroup of the Lorentz group, and hence connections admitting
Sim ( d −
2) have a minimal(non-trivial) holonomy reduction. See appendix E for some technical information about thisgroup.As commented in chapter 4, theories with the Similitude group have received some atten-tion in the past few years, as they have been shown to hold interesting physical features. Theyare linked to theories with vanishing quantum corrections [105] and to the recently proposedtheories of
Very Special Relativity [115] and
General Very Special Relativity [116]. For oursolutions, note that ∇ − N j = 12 B j , (5.117)and so a necessary condition for the N to be recurrent is B = 0. In fact, this is also sufficient,and one finds that if B = 0 then ∇ µ N ν = − g ( X − Q IJ V I V J ) vN µ N ν . (5.118)For the remainder of this section we take B = 0, and investigate the resulting conditionsimposed on the geometry.Consider first eq. (5.54). If X (cid:54) = 0 then the GT space is S , whereas if X = 0 it is flat.Next consider eq. (5.55), on contracting with X I this condition is equivalent to Q IJ ˜ ∇ i X I ˜ ∇ i X J = 0 , (5.119)and since Q IJ is positive-definite this implies that X I = X I ( u ). Eqs. (5.60) and (5.62) thenfurther imply that T I = 3 g ( XX I − Q IJ V J ) φ + K I , (5.120)where K I are 1-forms on the GT satisfying˜ dK I = 0 , V I K I = 0 , (5.121)and eq. (5.61) simplifies to gXφ + X I K I = − (cid:63) ˜ dφ . (5.122)4 CHAPTER 5. N = 1 D = 5 FAKESUGRA COUPLED TO ABELIAN VECTORS
The conditions obtained from eq. (5.59) are˙ X = 0 , (5.123)and ˜ d (cid:63) K I = − g ( XX I − Q IJ V J ) ˜ d (cid:63) φ + 3 gX ˙ X I dvol GT . (5.124)In particular, note that if X − Q IJ V I V J (cid:54) = 0, then on contracting this condition with V I onefinds that ˜ d (cid:63) φ = 0 , ˜ d (cid:63) K I = 3 gX ˙ X I dvol GT . (5.125)The function W must satisfy˜ ∇ W = ˜ ∇ i ˙ φ i + 9 g ( X − Q IJ V I V J ) φ · φ − C IJK X K (cid:0) ( T I ) i ( T J ) i + ˙ X I ˙ X J (cid:1) , (5.126)as a consequence of eq. (5.63).Thus given K I , φ, X I , W satisfying these conditions, the metric and field strengths are ds = 2 du (cid:18) dv − g v ( X − Q IJ V I V J ) du + W du + φ i e i (cid:19) − ds , (5.127) F I = 3 g ( XX I − Q IJ V J ) dv ∧ du − K I ∧ du . (5.128)As a simple example, take X (cid:54) = 0, X I constant and K I = 0. Then the Gauduchon-Todspace is S . All of the conditions are satisfied if one takes φ = ξ i σ iL , where ξ i = ξ i ( u ).With this choice of φ , eq. (5.126) implies that W is a ( u -dependent) harmonic function on S . This solution describes gravitational waves propagating through a generalized squashedNariai universe. Generically, these waves will be plane-fronted waves, as N is not a Killingvector. However, if X − Q IJ V I V J = 0 they are pp-waves.Alternatively, taking again X I constant with X (cid:54) = 0, one can instead set φ = 0 and K I = K Ii ( u ) σ iL . (5.129)Then the conditions which must be satisfied are V I K I = X I K I = 0 , ˜ ∇ W = Q IJ ( K I ) i ( K J ) i . (5.130)If K I (cid:54) = 0, one must thus have non-vanishing W .In [77] it was shown that, for recurrent solutions in the minimal theory, all scalar curva-ture invariants constructed purely algebraically from the Riemann tensor are constant. Bycomputing the Ricci scalar for the recurrent solutions constructed here, one can see that anecessary and sufficient condition for the Ricci scalar to be constant is that Q IJ V I V J is con-stant. This condition is also sufficient to ensure that all the other algebraic scalar curvatureinvariants are also constant. To see this, define ψ µνλ = 23 ∇ µ ∇ [ ν φ λ ] + 13 ∇ ν ∇ [ µ φ λ ] − ∇ λ ∇ [ µ φ ν ] , (5.131) θ µν = ∇ µ ∇ ν W + 9 g ( X − Q IJ V I V J ) v ∇ ( µ φ ν ) + 14 ( dφ ) µλ ( dφ ) ν λ , (5.132) .6. SUMMARY OF THE CHAPTER N µ ψ µνλ = 0 , N µ θ µν = 0 . (5.133)The Riemann tensor satisfies R µνλτ = ( R ) µνλτ + 4 N [ µ θ ν ][ λ N τ ] + N µ ψ νλτ − N ν ψ µλτ + N λ ψ τµν − N τ ψ λµν , (5.134)where R is the Riemann tensor for the metric g (obtained from the metric in eq. (5.127)by setting W = 0, φ = 0, where ( R ) µνλτ ≡ g µκ ( R ) κνλτ ). g is the metric on AdS × GT, R , × GT or dS × GT according to whether X − Q IJ V I V J is negative, zero or positive.It then follows, from exactly the same reasoning as set out for the minimal case, that allalgebraic scalar curvature invariants constructed from the metric g and the Riemann tensor R are identical to the same invariants constructed from g and R , and hence they areconstant. The status of scalar curvature invariants constructed from covariant derivatives ofthe Riemann tensor remains to be determined.For B (cid:54) = 0, one can also provide the construction with a
Sim -holonomy structure. This isrelevant e.g. for the embedding of the Berger sphere considered in section 5.4. By consideringthe gravitino fake Killing spinor equation, one obtains ∇ µ N ν = − gA µ N ν + 12 X I [ (cid:63) ( N ∧ F I )] µν . (5.135)So define a new covariant derivative D by D µ N ν ≡ ∇ µ N ν − S µν ρ N ρ = − gA µ N ν , where S = 12 (cid:63) ( X I F I ) (5.136)can be interpreted as a totally antisymmetric torsion 3-form, and thus the new connectionis metric-compatible. It is clear that that N is then recurrent w.r.t. the connection D , andconsequently its holonomy is a subgroup of Sim (3) [96].
In this chapter we have analysed the geometric structure of the null case solutions of N = 1 d = 5 fSUGRA coupled to Abelian vector multiplets. The general fake-supersymmetricsolution is given by the Kundt wave ds = 2 du ( dv − g v ( X − Q IJ V I V J ) du + W du + v B + φ i e i ) − h mn dy m dy n , (5.137)where h mn = e im e in is the metric on the Gauduchon-Tod 3-space, W is a v -independentfunction and B = B i e i , φ = φ i e i are v -independent 1-forms. The field strengths are given by F I = 3 g ( XX I − Q IJ V J )( dv + φ ) ∧ du + (cid:63) ( X I B + ˜ dX I )+ du ∧ (cid:16) − gv ˜ d ( XX I − Q IJ V J ) + T I (cid:17) , (5.138)where T I = T I j e j are v -independent 1-forms on the GT.6 CHAPTER 5. N = 1 D = 5 FAKESUGRA COUPLED TO ABELIAN VECTORS
Furthermore, we have studied the conditions for which the 1-form bilinear is recurrent, sothe holonomy of the Levi-Civit`a connection is inside
Sim (3), and investigated the propertiesof various scalar curvature invariants. We have found that recurrency is obtained by setting B = 0, and that depending on whether the norm of X is vanishing or not, the GT spaceis either R or S , respectively. In addition, we find that our general recurrent solutionsinclude plane-fronted waves propagating through a generalised squashed Nariai cosmos. For X − Q IJ V I V J = 0 these actually become pp-waves. Moreover, having X − Q IJ V I V J equalto a constant guarantees that all scalar curvature invariants constructed algebraically fromthe metric and the Riemann tensor are constant, and thus the ideas of [105] also apply in thisset-up. hapter 6 Classification of ‘Supersymmetric’Einstein-Weyl spaces
The work in this chapter is similar in construction to that of previous ones, where we alsoemploy techniques inherited from the classification of supersymmetric solutions to SUGRAtheories to attack a problem of a different nature. We consider a ‘novel’ KSE (in the sense thatsuch KSE is not related a priori to any supersymmetric setting), whose relevance becomesapparent once we analyse its integrability condition. This is the same as in previous chapters,but our motivation now is different from that of characterisation of solutions to fSUGRAtheories. We are interested in classifying Lorentzian Einstein-Weyl spaces of arbitrary di-mension, and the KSE is chosen in such a way that the integrability condition matches thegeometric constraint for a manifold to be of Einstein-Weyl type. The chapter follows [122],which contains the original work.As said, the tools use here are the same ones employed in the programme of classificationof solutions to supergravity theories, and we thus split the problem at hand according towhether they employ a timelike or null vector field. The characterisation we give is that ofthose EW spaces that arise from the existence of a
Killing spinor , i.e. a spinor that fulfills theKSE we propose, and it is in this sense that we refer to them as supersymmetric geometries.The outline of the chapter is the following: section 6.1 introduces the spinorial rule, itsintegrability condition (which resembles the geometric constraint for Einstein-Weyl spaces)and a short manipulation on a vector bilinear valid for all dimensions and cases. Section 6.2analyses all possible timelike cases, showing their triviality. Section 6.3 describes the nullsolutions for the N = 1, d = 4 case, while section 6.4 treats the d = 6 null case, andsection 6.5 includes the remaining ones. Section 6.6 has a summary of the chapter. For theinterested reader, appendix D gives some information on Weyl geometry and Einstein-Weylspaces and appendix A.2 presents the spinorial notation we employ. A little description onthe geometry of Kundt waves, which show up as solutions, is given in appendix C.3. Consider the following rule for the covariant derivative of some spinor (cid:15) , which we shall taketo be Dirac In order to construct bilinears, it is useful to impose a bit more structure on the spinor. This naturallyleads to question the compatibility of eq. (6.1) with the conditions for the existence of different kinds of spinors
CHAPTER 6. ‘SUPERSYMMETRIC’ EINSTEIN-WEYL SPACES ∇ a (cid:15) = − d A a (cid:15) + γ ab A b (cid:15) , (6.1)where d is the number of spacetime dimensions and A is just some real 1-form, which at thispoint is completely unconstrained. This equation is related to the fKSE (4.51) in chapter 4,in that the latter can be obtained from the former (for d = 4) by applying the Jones-Todreduccion mechanism [107]. We shall call the solutions (cid:15) to eq. (6.1) Killing spinors, and thecorresponding metric and 1-form a supersymmetric field configuration. Observe that withour choice of Dirac conjugate, the above rule implies ∇ a (cid:15) = − d A a (cid:15) − A b (cid:15)γ ab . (6.2)A straightforward calculation of the integrability condition leads to γ a /F (cid:15) = W ( ab ) γ b (cid:15) , (6.3)where F ≡ dA is called the Faraday tensor andW ( ab ) = R ( g ) ab − ( d − ∇ ( a A b ) − ( d − A a A b − g ab ( ∇ c A c − ( d − A c A c ) , (6.4)which is readily identified with (the symmetric part of) the Ricci tensor in Weyl geometry(see appendix D.1 for a small introduction). Contracting the above integrability conditionwith γ a one finds that d /F (cid:15) = W (cid:15) , (6.5)which when combined with eq. (6.3) leads to (cid:0) W ( ab ) − d η ab W (cid:1) γ b (cid:15) = 0 . (6.6)In the Riemannian setting the above is enough to conclude that if we find a spinor (cid:15) satisfyingeq. (6.1), then the underlying geometry is Einstein-Weyl. In the non-Riemannian settingthis conclusion is however not true; as in the classification of supersymmetric solutions tosupergravity theories, there are two quite different cases to be considered, namely the timelikeor the null case. The minimal set of equations of motion that need to be imposed in orderto guarantee that all EOMs are satisfied is different in each case: in the timelike case asupersymmetric field configuration automatically satisfies the EW condition, whereas in thenull case the minimal set consists of only one component of the EW condition, namely theone lying in the double direction of the null vector bilinear.Seeing the similarity of the integrability condition of the spinorial rule with the geometricconstraint for EW spaces, it should not come as a surprise that eq. (6.1) is invariant underthe following Weyl transformations g = e w ˜ g , e a = e w ˜ e a ,A = ˜ A + dw , θ a = e − w ˜ θ a ,(cid:15) = e αw ˜ (cid:15) , α = − d . (6.7)This symmetry can in fact be used to obtain the RHS of eq. (6.1), which would otherwisehave to be wild-guessed: since by definition the structure of all EW spaces will contain it, the (Weyl, Majorana, Majorana-Weyl, etc.). Indeed, such additional constrains are in fact compatible with thegiven rule of parallelity of (cid:15) . .2. TIMELIKE SOLUTIONS D a = ∇ a − γ ab A b . Furthermore, an additional αA a -term serves to preserve thespinor transformation rule, and the constant α is chosen by demanding that the integrabilitycondition of the resulting equation includes the criterion for EW spaces. This thus allows usto formulate the parallelity equation as D a (cid:15) = − d A a (cid:15) . In other words, we have the structureof a weighted Killing spinor in Weyl geometry.The next step is to define the bilinear ˆ L = L µ dx µ = (cid:15)γ µ (cid:15) dx µ , which (as shown inappendix A.2) is a real 1-form and for a Lorentzian spacetime is either timelike, i.e. g ( L, L ) > g ( L, L ) = 0. In any case, one can always derive from the spinorialequation (6.1) the following differential rule for the bilinear ∇ a L b = 4 − d A a L b − L a A b + ı L A g ab , (6.8)whose totally antisymmetric part reads d ˆ L = 6 − d A ∧ ˆ L , (6.9)singling out the d = 6 case as special, as ˆ L is then closed.We start the analysis by considering the timelike case. Suppose that L is timelike and define f ≡ g ( L, L ). We can straightforwardly use eq. (6.8) tofind df = (4 − d ) A f . (6.10)This implies that, as long as d (cid:54) = 4, the Weyl structure is exact and any supersymmetricEW space is equivalent to a metrical space allowing for a parallel spinor (w.r.t. the Levi-Civit`a connection). Bryant classified all the pseudo-Riemannian spaces admitting covariantlyconstant spinors for a different number of dimensions [123]. Thus, this prescribes the timelike Einstein-Weyl metrics with Lorentzian signature in dimensions three (flat), five and six ( g = R ,d − × ˜ g , where ˜ g is a four-dimensional Ricci-flat K¨ahler manifold). A general study forthe remaining dimensions is still an open problem, as far as we know. However, Galaev andLeistner provide a partial answer by giving a blueprint for the geometry of simply-connected,complete Lorentzian spin manifolds that admit a Killing spinor [124, Th. 1.3].For the d = 4 case, we use the same building blocks as in chapter 3 to set up the wholecalculus of spinor bilinears. We deal with the spinor structure of N = 2 d = 4 supersymmetry,which allows us to decompose a Dirac spinor as a sum of two Majorana spinors, which we canthen project onto its anti-chiral part, denoted (cid:15) I ( I = 1 , (cid:15) I . Note thathere the position of the I -index indicates exclusively the chirality, and these are interchangedby complex conjugation, i.e. ( (cid:15) I ) ∗ = (cid:15) I , so the theory has two independent spinorial fields.Doing this decomposition, the rule (6.1) can then be written as ∇ a (cid:15) I = γ ab A b (cid:15) I and ∇ a (cid:15) I = γ ab A b (cid:15) I . (6.11)Using the spinors one can then construct (cf. chapter 3) a complex scalar X ≡ ε IJ ¯ (cid:15) I (cid:15) J ,three complex 2-forms Φ x ( x = 1 , ,
3) that will not play any rˆole in what follows, and four0
CHAPTER 6. ‘SUPERSYMMETRIC’ EINSTEIN-WEYL SPACES real 1-forms V a = i ¯ (cid:15) I γ a (cid:15) I . These latter ones form a linearly independent base and can beused to write the metric g as 4 | X | g = η ab V a ⊗ V b , (6.12)whence V ∼ L . Given the definitions of the bilinears we can calculate dX = 0 , (6.13) dV a = A ∧ V a , (6.14)meaning that X is just a complex constant. The integrability condition of eq. (6.14) is F ∧ V a = 0 which, due to the linear-independency of the V a , implies that F = 0. Locally, then,we can transform A to zero and introduce coordinates x a such that V a = 4 | X | dx a , resultingin a Minkowski metric. Thus, a timelike supersymmetric four-dimensional Lorentzian EWspace is locally conformal to Minkowski space.The conclusion then w.r.t. the timelike solutions to the rule (6.1) is that they are trivialin the sense that they are always related by a Weyl transformation to a Lorentzian spaceadmitting parallel spinors, i.e. spinors satisfying the rule ∇ a (cid:15) = 0.The analysis of the null cases is more involved, mainly due to a lack of systematics inthe bilinears , but also because the bilinear approach to the classification of supersymmetricsolutions becomes unwieldy for d >
6. Instead of attempting to do a complete analysis in allthe cases where the bilinear approach can be applied, we shall analyse the cases d = 4 and d = 6 explicitly, and then give some generic comments about the rest in section (6.5). N = 1 d = 4 solutions In view of the explicit case treated in the foregoing section, the natural starting point for thisanalysis would be the null case in N = 2 d = 4. Prior experience with this case in Supergravity,however, shows that this is related to the simpler case of N = 1 d = 4 Supergravity [125],a theory for which the vector bilinear L is automatically a null vector, and the spinor is ofWeyl type. As commented above, the KSE (6.1) is in fact compatible with the truncation of (cid:15) to a chiral spinor, and thus for the rest of this section we shall take (cid:15) to be a Weyl spinor.The first rule one can derive for the bilinear is ∇ a L b = − L a A b + ı L A g ab , (6.15)which is enough to see that L (cid:91) is a geodesic null vector. Its antisymmetric and symmetricparts read d ˆ L = A ∧ ˆ L , (6.16) ∇ ( a L b ) = − A ( a L b ) + ∇ · L g ab . (6.17)Another bilinear that can be constructed is a 2-form defined as Φ ab = (cid:15)γ ab (cid:15) [125]. By usingthe propagation rule one can deduce ∇ a Φ bc = 2Φ a [ b A c ] − g a [ b Φ c ] d A d , (6.18)which through antisymmetrisation gives rise to d Φ = 2 A ∧ Φ . (6.19) The exception is the vector bilinear L , as one can see from eq. (6.8). .3. NULL N = 1 D = 4 SOLUTIONS L ∧ d ˆ L = 0, whence ˆ L is hypersurface orthogonal, and one can usethe Frobenius theorem to introduce two real functions u and P such that ˆ L = e P du . Sinceby eq. (6.16) ˆ L has gauge charge 1 under A , one can perform a Weyl-gauge transformationto take P = 0 and thus obtain ˆ L = du . This further implies that A = Υ ˆ L , where Υ isa real function whose coordinate dependence needs to be deduced, and also that ı L A = 0.Furthermore, we see that d † ˆ L = 0 and ∇ L L = 0, i.e. L is the tangent vector to an affinelyparametrised null geodesic.Observe that one can apply the same reasoning for eq. (6.9) in dimensions different fromsix: as long as d (cid:54) = 6 we can always use a Weyl transformation to fix ˆ L = du and write A = Υ ˆ L .The fact that in d = 6 the 1-form ˆ L is automatically closed has profound implications, as willbe shown in section (6.4).Having fixed the Weyl symmetry, we can now introduce a normalised null tetrad [126] anda corresponding coordinate representation byˆ L = du , L = ∂ v , ˆ N = dv + Hdu + (cid:36)dz + ¯ (cid:36)d ¯ z , N = ∂ u − H∂ v , ˆ M = U dz , M = − ¯ U − ( ∂ ¯ z − ¯ (cid:36)∂ v ) , ˆ M = ¯ U d ¯ z , M = − U − ( ∂ z − (cid:36)∂ v ) , (6.20)for which the metric reads g = ˆ L ⊗ ˆ N + ˆ N ⊗ ˆ L − ˆ M ⊗ ˆ M − ˆ M ⊗ ˆ M = 2 du ( dv + Hdu + (cid:36)dz + ¯ (cid:36)d ¯ z ) − | U | dzd ¯ z . (6.21)A straightforward calculation shows that the constraint (6.17) implies thatΥ = − ∂ v H , ∂ v (cid:36) = 0 , ∂ v ¯ (cid:36) = 0 , ∂ v | U | = 0 , (6.22)so that the only v -dependence resides in the function H , and we have thus determined thegauge field A in terms of it. Moreover, in N = 1 d = 4 theory one can see that Φ = ˆ L ∧ ˆ M (see e.g. eq. (3.68) in chapter 3). Combining this with eq. (6.19) one has0 = ˆ L ∧ d ˆ M = d ¯ U ∧ d ¯ z ∧ du , whence ¯ U = ¯ U ( u, ¯ z ) . (6.23)This result means that we can take U = 1 by a suitable coordinate transformation Z = Z ( u, z )with ∂ z Z = U , which leaves the chosen form of the metric invariant.To finish the analysis we shall investigate eq. (6.18). As A ∼ ˆ L we have that ı A Φ ∼ ı L Φ = 0 , (6.24)and we find that ∇ a Φ bc = 2Υ Φ a [ b L c ] . Combining this with Φ ab = 2 L [ a M b ] we have0 = L [ b | ∇ a M | c ] , (6.25)which can be evaluated on the chosen coordinate basis to give0 = ∂ ¯ z (cid:36) − ∂ z ¯ (cid:36) , (6.26)2 CHAPTER 6. ‘SUPERSYMMETRIC’ EINSTEIN-WEYL SPACES which implies (cid:36) = ∂ z B , ¯ (cid:36) = ∂ ¯ z B , (6.27)for B a real function. As is well known, one can then get rid of (cid:36) altogether by a suitableshift of the coordinate v → v − B .The end result is that, given a Weyl spinor (cid:15) , any solution to the equation (6.1) is relatedby a Weyl transformation to ds = 2 du ( dv + Hdu ) − dzd ¯ z , (6.28) A = − ∂ v H du . (6.29)As a matter of fact, this metric is a special case of a more general metric referred to as aKundt metric in the Physics literature (see appendix C.3 for more information). This kindof metric appears naturally in the null case of not only Supergravity solutions [112], but alsofakeSupergravity ones, as was seen for example in chapters 3 and 4, or in [106].We now return to the topic of pseudo-Riemannian signatures and certain EOMs (the EWconditions in this case) being automatically satisfied. Since we are trying to give a prescriptionfor EW spaces, we are obviously bound to satisfy eq. (D.8). An explicit calculation showsthat the integrability conditions (6.6) are automatically satisfied, with the only non-trivialcomponent being W( N, N ) L c γ c (cid:15) . Adapting the Fierz identities to the null case scenario oneobtains the constraint L c γ c (cid:15) = 0 (see e.g. [127, eq. (5.12)]), satisfying this way all of (6.6) .However, one still needs to ensure that the local geometry (6.28) indeed solves all EW con-ditions (D.8), and we must therefore impose by hand that W( N, N ) = 0. A small calculationshows that this implies that H must satisfy the following differential equation ∂ u ∂ v H − H∂ v H = ∂ ¯ ∂H . (6.30)One can easily see that the above equation is invariant under the substitutions u → u , v → λ v , z → λz , ¯ z → λ ¯ z and H → λ H , which implies that for cases for which H is aweighted-homogeneous function, i.e. H (cid:0) u, λ v, λz, λ ¯ z (cid:1) = λ H ( u, v, z, ¯ z ) , (6.31)there exists a homothety KK = 2 v∂ v + z∂ + ¯ z ¯ ∂ −→ £ K g = 2 g . (6.32)It should be noted that it is not the case that every solution to eq. (6.30) is homogeneous, asexemplified by the following three solutions. i ) H = uv + | z | , ii ) H = − v | z | , iii ) H = v∂F + v ¯ ∂ ¯ F + ¯ z∂ u F + z∂ u ¯ F , where F = F ( u, z ) . (6.33) By solution we refer to a geometry that arises from the existence of a spinor that fulfills eq. (6.1). Moreover, it is in the Walker form [97]. Aditionally, one reaches the same conclusion by analysing the sufficiency of the derived constraints arisingfrom the KSE. .4. NULL N = (1 , D = 6 SOLUTIONS F = zf ( u ), for any function f ( u ).Moreover, one can see that the field A in example i ) is pure gauge, namely A = d ( − u / g → e ω ˜ g and A = ˜ A + dω such that ˜ A = 0, i.e. we take ω = − u /
2. The coordinate transformations v = e ω t , z = e ω y followed by t = s − ue ω | y | ,then takes the metric ˜ g to a Minkowski metric. Case i ) is thus conformally Minkowski. Thisis however only a particular example, and one cannot generically say that all solutions thatexhibit an exact Weyl structure ( i.e. trivial solutions) are conformally related to flat space. Forillustrative purposes note that for A = 0 the wave profile is of the type H = f ( u, z ) + ¯ f ( u, ¯ z );if one then calculates the Riemannian curvature one obtains R + ◦ + ◦ = ∂ f and R + • + • = ¯ ∂ ¯ f ,hence it is not generically a flat space. In the general case A = dw , the wave profile isprescribed to be H = − vf ( u ) + K ( u, z, ¯ z ). The requirement for being an EW space dictatesthat ∂ u f ( u ) = − ∂ ¯ ∂K , and the non-vanishing components of the Weyl tensor are C + ◦ + ◦ = ∂ K , C + • + • = ¯ ∂ K , (6.34)whence only for cases where ∂ K = 0 = ¯ ∂ K is our space conformal to Minkowski space.Case i ) falls under these conditions.Furthermore, example iii ) above is a Weyl-scalar-flat background. It is a four-dimensionalgeneralisation of the three-dimensional Weyl-flat EW geometries obtained in [128]. It givesrise to a non-trivial EW space as long as ∂ F (cid:54) = 0. More about this correspondence will besaid in section (6.5). N = (1 , d = 6 solutions As in the foregoing section, we shall be considering a chiral spinor (cid:15) . The Fierz identitiesimply that the vector bilinear is null. Moreover, one can also make use of [120], where thesupersymmetric solutions of ungauged chiral supergravity in six dimensions, i.e. minimal N = (1 , d = 6 Supergravity [129], are prescribed. This theory is in itself quite curious, andso are the spinor bilinears: there is only a null vector L and a triplet of self-dual 3-forms Φ r (3) ( r = 1 , , L a ≡ − ε IJ (cid:15) cI γ a (cid:15) J , (cid:15) cI γ a (cid:15) J = − ε IJ L a , Φ rabc ≡ i [ σ r ] IJ (cid:15) cI γ abc (cid:15) J , (cid:15) cI γ abc (cid:15) J = i [ σ r ] IJ Φ rabc , (6.35)where (cid:15) c = (cid:15) T C denotes the Majorana conjugate. These bilinears satisfy the following Fierzrelations L a L a = 0 , (6.36) ı L Φ r (3) = 0 −→ ˆ L ∧ Φ r (3) = 0 , (6.37)Φ r fab Φ sfcd = 4 δ rs L [ a L [ c η b ] d ] − ε rst L [ a | Φ t | b ] cd + ε rst L [ c Φ t abd ] . (6.38)With means of eqs. (6.37) and (6.38) one finds that Φ r (3) = ˆ L ∧ K r (2) , with ı L K r (2) = 0.We use the rule (6.1) to calculate the effect of parallel-transporting the bilinears. Theresult is that for an arbitrary vector field X we have ∇ X ˆ L = − ı X A ˆ L − ı X ˆ L A + ı L A ˆ X , (6.39) ∇ X Φ r = − ı X A Φ r + ˆ X ∧ ı A (cid:91) Φ r − A ∧ ı X Φ r . (6.40)4 CHAPTER 6. ‘SUPERSYMMETRIC’ EINSTEIN-WEYL SPACES
From eq. (6.39) it is clear that L is a null geodesic, i.e. ∇ L L = 0, and, as we already knewfrom eq. (6.9), that d ˆ L = 0.Moreover, we introduce a Vielbein, adapted to the null nature of L , in terms of the naturalcoordinates u , v and y m ( m = 1 , . . . , E + = du , θ + = ∂ u − H∂ v ,E − = dv + Hdu + S m dy m , θ − = ∂ v ,E i = e mi dy m , θ i = e im [ ∂ m − S m ∂ v ] , (6.41)where ˆ L ≡ E + and L ≡ θ − . As usual we can then define the metric on the base-space by h mn ≡ e mi e ni , and write the full six-dimensional Kundt metric as ds = 2 du (cid:16) dv + Hdu + ˆ S (cid:17) − h mn dy m dy n . (6.42)We expand the 2-forms as 2 K r ≡ K rij E i ∧ E j w.r.t. the above Vielbein, and by choosing thelight-cone directions such that ε + − = 1 = ε we see that (cid:63) (4) K r = − K r . Defining the(1 , J r through h ( J r X, Y ) ≡ K r ( X, Y ), we can see that eq. (6.38) implies J r J s = − δ rs + ε rst J t , (6.43)so that the four-dimensional base-space is an almost quaternionic manifold .At this point we shall fix part of the Weyl gauge symmetry by imposing the gauge-fixingcondition ı L A = 0; consequently, we can expand the gauge field as A = Υ ˆ L + A m dy m . (6.44)Using this expansion and the explicit form of the Vielbein in terms of the coordinates, we cananalyse eq. (6.39), resulting in Υ = − ∂ v H , (6.45) ∂ v ˆ S = − A , (6.46)0 = ∂ v h mn . (6.47)Contrary to what is usually the case in (fake)supergravities, we do not know the full v -dependence of H , and therefore we cannot completely fix the v -dependence of the unknowns.The above results comprise all the information contained in eq. (6.39).In order to analyse the content of eq. (6.40) we first take X = L , to find that ∇ L Φ r = 0 . (6.48)When this is evaluated in the chosen coordinate system it implies that ∂ v K rmn = 0. This ap-parently innocuous result fixes however the v -dependence of A : from the totally antisymmetricpart of eq. (6.40) one obtains d Φ r = 2 A ∧ Φ r −→ L ∧ ( dK r − A ∧ K r ) , (6.49)where we have introduced the exterior derivative on the base-space d ≡ dy m ∂ m . As the K r are v -independent and ˆ L = du , we see that the consistency of the above equation also requires A to be v -independent. Thus, we also obtain from eq. (6.46) thatˆ S = − v A + (cid:36) , (6.50) .4. NULL N = (1 , D = 6 SOLUTIONS (cid:36) a 1-form living on the base-space such that ∂ v (cid:36) m = 0. It should be clear fromeq. (6.49) that the y -dependence of the K r is given by the equation dK r = 2 A ∧ K r , (6.51)whose integrability condition reads F ∧ K r = 0 , (6.52)where we have defined F = dA . Actually, this last equation implies that F is self-dual, i.e. (cid:63) (4) F = F , whence A is a self-dual connection or, in physics-speak, an R -instanton.The analysis of eq. (6.40) in the direction X = θ + is straightforward, and leads to thefollowing constraints on the spin connection ω + − k K rkj = − A k K rkj , (6.53)0 = ω + ik K rkj + ω + jk K rik . (6.54)By using the results in appendix C.3, we see that eq. (6.53) is automatically satisfied. Asmall investigation of eq. (6.54) shows that it implies the base-space 2-form ω + ij E i ∧ E j isself-dual. Coupling this observation with eq. (C.34), and taking into account F ’s self-duality,we see that the base-space 2-form 2Ω = Ω ij E i ∧ E j , whose components are defined byΩ ij ≡ D [ i (cid:36) j ] + 2 e [ im ∂ u e j ] m , where D (cid:36) ≡ d (cid:36) − A ∧ (cid:36) , (6.55)has to be self-dual, i.e. (cid:63) (4) Ω = Ω.In order to completely drain eq. (6.40) we need to consider X lying on the base-space.Let X be such a vector. Then, we find that ∇ ( λ ) X K r = X (cid:93) ∧ (cid:63) (4) [ A ∧ K r ] − A ∧ ı X K r , (6.56)where ∇ ( λ ) is the ordinary spin connection on the base-space using the λ in eq. (C.34).Following [30] we can then introduce a torsionful connection ∇ X Y ≡ ∇ ( λ ) X Y − S X Y , with thetorsion being totally antisymmetric and proportional to the Hodge dual of the R -gauge field, i.e. h ( S X Y , Z ) ≡ − (cid:2) (cid:63) (4) A (cid:3) ( X , Y , Z ) , (6.57)such that eq. (6.56) can be written compactly as ∇ K r = 0. Almost quaternionic manifoldsadmitting a torsionful connection parallelising the almost quaternionic structure are called hyper-K¨ahler Torsion manifolds (HKT), a name that first appeared in [130] to describe thegeometry of supersymmetric sigma model manifolds with torsion [131].As pointed out in [30], one can make use of the residual Weyl symmetry in eq. (6.7) with w = w ( y ), i.e. a Weyl transformation depending only on the coordinates of the base-space,to gauge-fix the condition d † A = 0. This immediately implies that the torsion S is closed, andthe resulting four-dimensional structure is called a closed HKT manifold. Let us mention,even though it will not be needed, that the coordinate transformation v → v + Λ( y ), inducesthe transformation (cid:36) → (cid:36) + D Λ.Thus far, the analysis has shown that the pair ( g, A ) admits a solution to eq. (6.1) iff g is the metric of a Kundt wave whose base-space is a u -dependent family of HKT spaces.Given such a family of spaces we can find the 1-form (cid:36) by imposing self-duality of the 2-formΩ in eq. (6.55), and then the only indeterminate element of the metric is the wave profile6 CHAPTER 6. ‘SUPERSYMMETRIC’ EINSTEIN-WEYL SPACES H . This study has given us the necessary conditions for the existence of a non-trivial spinorsatisfying eq. (6.1). It remains to be checked that they are also sufficient, which we do bydirect substitution into (6.1).A quick calculation of the ( − ) component leads to θ − (cid:15) = 0, whence the spinor is v -independent. The (+)-component, after using the constraint γ + (cid:15) = 0, leads to ∂ u (cid:15) = − T ij γ ij (cid:15) = 0 , (6.58)where the last step follows from the self-duality of T (cf. eq. (C.36)) T ij = v F ij − [ D (cid:36) ] ij , (6.59)and the chirality of the spinor (cid:15) . We thus conclude that the spinor is also u -independent.Giving the i components of eq. (6.1) a similar treatment we end up with ∇ ( λ ) i (cid:15) = ˜ γ ij A j (cid:15) , (6.60)where we have defined ˜ γ i ≡ iγ i , so { ˜ γ i , ˜ γ j } = 2 δ ij , in order to obtain a purely Riemannianspinorial equation.As one can readily see from eq. (6.1), the above equation is nothing more than its Rie-mannian version for four-dimensional spaces. This kind of spinorial equations was studied byMoroianu in [132], who investigated Riemannian Weyl geometries admitting spinorial fieldsparallel w.r.t. the Weyl connection. For d (cid:54) = 4 it was found that any such Weyl structure wasclosed, whereas in d = 4 the HKT structure outlined above was prescribed. Furthermore,it was shown that if the four-dimensional space is compact, then the HKT structure is con-formally related to either a flat torus, a K3 manifold or the Hopf surface S × S with thestandard locally-flat metric (see e.g. [133]).The integrability condition for eq. (6.60) implies that the Ricci tensor of the metric h hasto satisfy R ( h ) ij = 2 ∇ ( λ )( i A j ) + 2 A i A j + h ij (cid:16) ∇ ( λ ) i A i − A (cid:17) , (6.61)which by comparison with eq. (6.4) it is equivalent to saying that the pair ( h, A ) forms aRicci-flat Weyl geometry i.e. W ( ij ) = 0.As we did in section (6.3), we impose the Einstein-Weyl equations in those directionsin which they are not trivially satisfied, i.e. in the (++)-direction. This fixes the function H , which was otherwise unknown. Furthermore, we would like to impose the simplifyingrestriction that the HKT structure on the base-space does not depend on u . We take thisroute because of the difficulty of finding analytic solutions to the differential equation resultingfrom a u -dependent base-space. A calculation of the (++)-components of the EW equationsthen shows that2 θ + θ − H + ( θ − H ) = (cid:16) ∇ ( λ ) i − S i θ − − A i (cid:17) ( ∂ i − S i θ − − A i ) H , (6.62)where we have allowed for a u -dependence of H .To summarise, any solution to the N = (1 , d = 6 null problem is once again prescribedby a Kundt wave of the form eq. (6.42) constrained by eqs. (6.50), (6.55) and (6.62), whosefour-dimensional base-space is given by a v -independent closed HKT manifold subject toeqs. (6.61), and the gauge connection being an R -instanton. .5. REMAINING NULL CASES Having treated the null cases in d = 4 and d = 6, we now make some general comments for thenull class in other dimensions. Since performing the complete bilinear analysis is a dauntingtask, and in view of the wave-like nature of the null case solutions above, we shall write downa similar Ansatz for the metric. Furthermore, as was pointed out in section (6.3), as long as d (cid:54) = 6 we can use a Weyl transformation to introduce a coordinate u such that ˆ L = du and A = Υ ˆ L . Choosing then the coordinate v to be aligned with the flow of L (= ∂ v ), one canintroduce base-space coordinates y m ( m = 1 , . . . , d −
2) and a Vielbein similar to the one ineq. (C.29), so that the metric is of the form ds d ) = 2 du ( dv + Hdu + S m dy m ) − h mn dy m dy n , (6.63)where h mn ≡ e im e in . This is by definition a Kundt metric, and evaluating the symmetric partof eq. (6.8) in this coordinate system, we get the following restrictionsΥ = − d − ∂ v H , ∂ v S m = 0 , ∂ v h mn = 0 , (6.64)so that the whole v -dependence resides exclusively in H and Υ, which implies that eq. (6.63)is actually in the Walker form [97]. Following the notation of section 6.4, we shall call the v -independent part of ˆ S by (cid:36) , so that in the d (cid:54) = 6 case we have ˆ S = (cid:36) .With this information and the constraint of u -independence imposed, we can proceed toanalyse the spinorial rule. The KSE in the v -direction is automatically satisfied, i.e. ∂ v (cid:15) = 0,and the remaining directions read 0 = ∇ ( λ ) i (cid:15) , (6.65) ∂ u (cid:15) = [ d (cid:36) ] ij γ ij (cid:15) . (6.66)Eq. (6.65) clearly states that the base-space must be a Riemannian manifold of special holon-omy. The integrability condition of the above two equations is0 = (cid:104) ∇ ( λ ) i ( d (cid:36) ) kl (cid:105) γ kl (cid:15) , (6.67)which implies (cid:16) d † d (cid:36) (cid:17) i γ i (cid:15) = 0 , (6.68)so that d † d (cid:36) = 0. Using the coordinate transformation v → v + Λ( y ), one can always take d † (cid:36) = 0, whence (cid:36) ∈ Harm ( B ), i.e. (cid:36) is a harmonic 1-form on the base-space .Given this input, the condition for such a pair ( g, A ) to be an Einstein-Weyl manifold is2 ∂ u ∂ v H − H∂ v H + 2 d − d − ∂ v H ) = − (cid:16) ∇ ( λ ) − (cid:36) (cid:17) i θ i H . (6.69)The factor on the RHS of the above equations becomes, in the (cid:36) = 0 limit, the d’Alembertianon the base-space, and it makes contact with eq. (6.30). This shows that the d = 4 case is asubcase of the general one studied in this section, where one was allowed to use the 2-formΦ to get rid of ˆ S . d = 6, however, is an independent case, where the characteristic behaviourof the theory in that dimension (see e.g. eq. (6.9)) nurtures the closed HKT structure. The same constraint can be obtained through an explicit evaluation of the Einstein-Weyl equations. Bochner’s theorem states that any harmonic 1-form on a compact oriented Ricci-flat manifold is parallel,which by eq. (6.66) implies that in such a case the Killing spinor is u -independent. In the non-compact casehowever there is no such a theorem, as can be seen by taking the base-space to be R d − and 2 (cid:36) ≡ f mn x m dx n ,where the f mm are constants. CHAPTER 6. ‘SUPERSYMMETRIC’ EINSTEIN-WEYL SPACES
In this chapter we have presented a characterisation of supersymmetric Einstein-Weyl spaceswith Lorentzian signature in arbitrary dimensions. We have done this by making use of thetechniques developed for the classification of SUGRA solutions. In particular, we assumedthe existence of a spinor (cid:15) satisfying eq. (6.1). It is in this sense that our solutions have asupersymmetric character. We then proceeded to build and analyse the bilinears that can beconstructed from (cid:15) , which shape the resulting geometry.We have found that for most dimensions those spaces arising from a vector bilinear whichis timelike are trivial, in the sense that they are conformally related to a space admitting aparallel spinor. The odd duck in the pond is the four-dimensional case, for which the onlytimelike solution actually turns out to be Minkowski space, which coincides with which wasalready known [123]. The null class solutions are given by a Kundt metric and a prescribedWeyl gauge field. For the d = 6 case, we also find that the base-space is a closed Hyper-K¨ahler Torsion manifold. Furthermore, it is inspiring that our d = 3 characterisation containsa solution to a three-dimensional Weyl-scalar-flat Lorentzian EW space that was previouslypresented in [128].In said article, Calderbank and Dunajski derived the possible forms of three-dimensionalscalar-flat Lorentzian EW spaces, and found that, up to Weyl and coordinate transformations,there are only two possible metric classes. In our construction, due to eq. (6.5) and the resultsobtained so far, one can see that the scalar curvature is given byW = 2 dd − ∂ v H , (6.70)so that the scalar-flat supersymmetric Lorentzian EW spaces are given by a function H whichis at most linear in v . It then follows that the d = 3, W = 0 approximation of eqs. (6.63)and (6.64) is one of the cases presented by the authors, namely the h in [128, Prop. 2]. Asshown in [134, sec. 10.3.1.3], this is the unique class of (1,2)-dimensional EW spaces admittinga weighted covariantly constant null vector. This supersymmetric class can be obtained bythe Jones-Tod construction [107] on a conformal space of neutral signature admitting anantiself-dual Null-K¨ahler structure [134], a geometric structure which admits a Killing spinor.Furthermore, once one takes into account that one can perform coordinate transformationssuch that h = 1 and (cid:36) = 0, eq. (6.69) for d = 3 then corresponds to the dispersionlessKadomtsev-Petviashvili equation [135, 136].The fact that we only obtain one of the solution classes from said article is indicative thatthe world of Lorentzian EW spaces is richer than the ones obtainable as solutions to eq. (6.1)( i.e. that there are also ‘non-supersymmetric’ EW spaces), and hence our classification doesnot aim at solving the mathematical problem of characterising the whole spectrum. A naturalquestion is whether there are equations like (6.1) whose integrability condition leads to moregeneral EW spaces. In any case, the construction shows that the supersymmetric formalismallows us to generalise and extend solutions that were obtained through other geometricmethods. nexo I Introducci´on
El trabajo recogido en esta tesis doctoral ha estado centrado en la obtenci´on de soluciones avarias teor´ıas de inter´es f´ısico-te´orico y matem´atico, a trav´es de t´ecnicas e ideas caracter´ısticasdel campo de la Supergravedad (SUGRA). Estas t´ecnicas surgen originariamente del programade clasificaci´on de soluciones supersim´etricas a las mismas, que se explican brevemente en elanexo II. Pasamos ahora a ofrecer una breve introducci´on sobre la SUGRA, en el contexto dela f´ısica contempor´anea de Gravitaci´on y Part´ıculas.De forma muy b´asica, la Supergravedad se puede definir como una construcci´on te´oricadonde se junta la Relatividad General con la Supersimetr´ıa (SUSY). La Relatividad Generales la teor´ıa de gravitaci´on que Einstein propuso en 1915, donde se enfatiza el papel que laGeometr´ıa (Matem´aticas) juega en la F´ısica, y en que las transformaciones de coordenadas encada punto (difeomorfismos locales) son esencia misma de la teor´ıa. La SUSY fue introducidade forma independiente por los investigadores Golfand & Likhtman, Volkov & Akulov y Wess& Zumino entre 1971 y 1974, y es una simetr´ıa que relaciona part´ıculas bos´onicas (aquellasque tienen un n´umero de esp´ın entero) con part´ıculas fermi´onicas (con n´umero de esp´ın= n/ n es un n´umero impar, e.g. 1/2, 3/2, ...).Las primeras construcciones de SUGRA se pueden trazar en torno a 1973-1975, cuando loscient´ıficos Volkov, Akulov y Soroka, a lo largo de varios art´ıculos, propusieron unos modelospara gaugear las transformaciones supersim´etricas. Esto significa que estas transformaciones,que hasta ese momento eran globables o r´ıgidas, pasan a ser locales, de forma que en cadapunto del espacio-tiempo se puede hacer una transformaci´on distinta. Esto, unido a unmecanismo de Higgs que desarrollaron para SUGRA (en que los ‘fermiones de Goldstone’,de esp´ın 1/2, son comidos) da lugar a una part´ıcula masiva de esp´ın 3/2, llamada el campode Rarita-Schwinger, y supone una realizaci´on no-lineal de las transformaciones. El primermodelo de SUGRA cuatri-dimensional lineal lo construyeron en 1976 Ferrara, Freedman yVan Nieuwenhuizen, y tambi´en Deser y Zumino. En sus respectivos art´ıculos,estudiaron lateor´ıa de campos (con interacciones) de lo que especularon ser´ıan la part´ıcula que mediar´ıa enla interacci´on gravitacional, el gravit´on, de esp´ın 2. ´Este ya hab´ıa aparecido con anterioridaden la teor´ıa de representaciones del ´algebra global de Super-Poincar´e (que conten´ıa adem´as sucompa˜nero supersim´etrico de esp´ın 3/2, tambi´en conocido como gravitino), pero lo novedosoen este caso es que adem´as demostraban que su acci´on eran invariante bajo transformacioneslocales de supersimetr´ıa.La SUGRA gan´o mucha fama cuando se pens´o que su versi´on en 11 dimensiones, propuestapor Cremmer, Julia y Scherk, pod´ıa servir como candidata a una Teor´ıa del Todo . En ´ultima9900
ANEXO I. INTRODUCCI ´ON instancia se demostr´o que esto no pod´ıa ser as´ı, y que en realidad la SUGRA ten´ıa que serun l´ımite de bajas energ´ıas de la
Teor´ıa de Cuerdas . En concreto, las SUGRAs en 10 y11 dimensiones son teor´ıas de campos efectivas para part´ıculas sin masa, y a nivel ´arbol,y por tanto sus soluciones describen el comportamiento de materia cuerdosa en distanciaslargas ( i.e. baja energ´ıa). Es por esto que el estudio de las construcciones de SUGRA hansido, y siguen siendo, muy importantes en Teor´ıa de Cuerdas, ya que sus soluciones son unpilar fundamental para el desarollo de la misma. Pero incluso de forma independiente delas cuerdas, las teor´ıas de SUGRA parecen ser de gran utilidad. Investigaciones recientesparecen sugerir la viabilidad de determinadas teor´ıas extendidas de SUGRA como teor´ıasde gravedad cu´antica [137]. Esto ser´ıa un gran avance, ya que el problema de la gravedadcu´antica lleva intentando resolverse desde tiempos de Einstein. Es por tanto que la SUGRA,y sus soluciones, siguen ofreciendo una puntera l´ınea de investigaci´on a trav´es de la cual poderdilucidar algunos de los misterios de la Naturaleza.Por ejemplo, las teor´ıas de Supergravedad se han analizado para obtener soluciones cl´asicasy supersim´etricas, que representan agujeros negros en 4 y 5 dimensiones. Estas teor´ıas (de lasque el agujero negro es soluci´on) se pueden relacionar con teor´ıas de SUGRA de dimensionesm´as altas por medio de una t´ecnica conocida como oxidaci´on dimensional. Seg´un hemosexplicado, estas SUGRAs de dimensiones altas (d=10 o d=11, generalmente) representanun l´ımite de bajas energ´ıas de teor´ıas de supercuerdas de la misma dimensionalidad, lo quepermite relacionar la soluci´on del agujero negro original con un vac´ıo de cuerdas. Operar enesta teor´ıa de supercuerdas (que es una teor´ıa cu´antica de campos) no es una tarea sencilla,ya que las correciones cu´anticas que surgen en dicha teor´ıa lo complica enormemente. Tantoque a d´ıa de hoy todav´ıa no se sabe bien c´omo contar el n´umero de estados microsc´opicos desus vac´ıos. Sin embargo, las soluciones supersim´etricas salen al rescate, ya que precisamente´estas son estables bajo dichas correciones cu´anticas, lo que permite poder contar el n´umero deestados microsc´opicos de tal vac´ıo. Esto es un muy notable c´alculo de la entrop´ıa microsc´opicade agujeros negros supersim´etricos, que supuso un verdadero hito cuando se realiz´o parauna clase de agujeros negros extremos en 5 dimensiones, ya que coincid´ıa con el valor dela entrop´ıa macrosc´opica predicha por Bekenstein y Hawking [14]. En vista de esto, y otrasimpactantes e influyentes ideas, como la existencia de dualidades no perturbativas entre teor´ıasde supercuerdas, o la conjetura
AdS/CF T y sus generalizaciones, es l´ogico que a lo largode los a˜nos se hayan realizado numerosos esfuerzos en el estudio de teor´ıas de SUGRA.Para m´as m´erito, si la Supersimetr´ıa es una simetr´ıa real de la Naturaleza , algo de lo queprobablemente haya indicios en el LHC del CERN en los pr´oximos a˜nos, la Supergravedadgozar´ıa de un papel fundamental. En dicho caso las soluciones supersim´etricas sean segura-mente buenas aproximaciones al vac´ıo o los vac´ıos de la Teor´ıa de Cuerdas que describan larealidad a escalas de energ´ıas altas, y esto har´ıa adem´as de la SUGRA una pieza clave en laconstrucci´on de modelos fenomenol´ogicos de F´ısica de Part´ıculas, y en la unificaci´on de lasinteracciones. ´Esta establece una correspondencia entre una teor´ıa de cuerdas sobre una soluci´on maximalmente sim´etricay maximalmente supersim´etrica (el producto geom´etrico entre el espacio AdS y una 5-esfera), y una teor´ıamaximalmente supersim´etrica de tipo Yang-Mills sobre el producto de una l´ınea y una 3-esfera, que unoidentifica como el contorno del espacio AdS . Se la llama tambi´en la correspondencia Gravedad/ Teor´ıasgauge , o
Correspondencia de Maldacena [18]. Aunque ´esta est´e rota a nuestra escala de energ´ıas. nexo II
Resumen
En este anexo comentamos brevemente las t´ecnicas e ideas que dan lugar al trabajo expuestoen esta tesis doctoral. Comenzamos con un resumen sobre el ‘formalismo bilineal’ y el ‘formal-ismo de la geometr´ıa espinorial’, que motivan los problemas estudiados durante mi doctorado,para terminar describiendo los resultados de los cap´ıtulos 3, 4, 5 y 6.En el anexo I hemos discurrido sobre el inter´es de obtener soluciones supersimetr´ıcas. En elart´ıculo [1] se propuso un m´etodo para obtener este tipo de soluciones en la teor´ıa de SUGRAcinco-dimensional m´ınima, bas´andose en la existencia de un espinor de Killing (tal espinorno debe entenderse como un campo fermi´onico, si no que se trata de un espinor cl´asico, decar´acter geom´etrico). ´Este sirve de par´ametro respecto al cual la variaci´on supersim´etrica delgravitino es cero, dando lugar a la conocida como ‘ecuaci´on del espinor de Killing’ (KSE). Ent´erminos puramente geom´etricos, ´esta no es m´as que una condici´on de parelizaci´on del espinor,que surge a ra´ız del ´algebra de SUSY. El siguiente paso es construir campos bilineales en talespinor, y usar las identidades de Fierz para obtener relaciones algebraicas entre esos campos.Esto permite obtener la forma m´as general de la geometr´ıa y los flujos de las soluciones. ´Estasson supersim´etricas por construcci´on (ya que asumen la existencia de al menos un espinorde Killing), y est´an completamente determinadas en funci´on de dichas condiciones. Se puedeencontrar un resumen de este m´etodo en la secci´on 2.1 de estas p´aginas.As´ı mismo, en [2] se propuso un m´etodo cuyo objetivo era el mismo, pero en el que seelimina la complejidad que supone usar matrices Gamma en escenarios multi-dimensionales.Este formalismo interpreta los campos espinoriales de la teor´ıa en funci´on de formas diferen-ciales (m´as una acci´on de las matrices Gamma sobre ellas, tambi´en expresada en funci´on deformas). En lugar de construir bilineales de los que extraer las condiciones de geometrizaci´onde las soluciones, el m´etodo implica resolver las KSEs directamente, y leer de ellas la ge-ometr´ıa y los flujos resultantes. La secci´on 2.2 ofrece un resumen de este m´etodo, adem´as decontener una resoluci´on del mismo problema descrito arriba, esta vez usando esta alternativa.Se puede ver que los resultados obtenidos son los mismos.Ambos m´etodos han sido extensamente usados para el estudio y clasificaci´on de solucionessupersim´etricas de m´ultiples teor´ıas, con y sin mater´ıa, dando lugar a un buen n´umero deresultados, e.g. [138, 31, 120, 139, 60 - 63, 82, 140, 64 - 67, 125, 68, 69]. Sin embargo, el usode estos m´etodos durante mis estudios doctorales ha estado encaminado a objetivos distintos.En lugar de obtener Ans¨atze de soluciones supersim´etricas a partir de la KSE, proponemosuna condici´on de paralelizaci´on alternativa para los espinores. ´Esta es elegida de modo quela acci´on de la teor´ıa tenga una constante cosmol´ogica de tipo De Sitter (signo positivo), en10102
ANEXO II. RESUMEN lugar de las Minkowski (valor cero) y AdS (signo negativo) que, salvo casos contados, sonlas ´unicas posibilidades que hay en las teor´ıas de SUGRA genuinas [32, 33]. Esto implicaque las teor´ıas detr´as de estas construcci´ones no son supersim´etricas, y han sido previamentebautizadas como Teor´ıas de Supergravedad falsas (fakeSUGRA o fSUGRA). Sus soluciones,que no son supersim´etricas, son en cualquier caso de gran inter´es f´ısico, ya que resuelvenlas ecuaciones de Einstein-Maxwell-De Sitter. Esto es interesante puesto que resolver estasecuaciones no es una tarea sencilla; tanto que a pesar de que el problema se conoce desde hacem´as de 60 a˜nos, apenas hay ejemplos que lo hagan. Adem´as, seg´un modelos cosmol´ogicosrecientes, el Universo estar´ıa descrito por un Lagrangiano con una constante cosmol´ogica deeste tipo, con un valor absoluto muy peque˜no.En este sentido, hemos usado ambos formalismos para caracterizar soluciones de teor´ıasen cuatro y cinco dimensiones. Al igual que ocurre con la clasificaci´on de soluciones super-sim´etricas en SUGRA, tambi´en aqu´ı separamos nuestro estudio en funci´on de la norma delvector que se construye como un bilineal de los espinores que satisfacen la KSE. As´ı, distin-guimos entre el caso timelike , cuando la norma del vector es positiva, o caso nulo , cuando lamisma es cero. En el caso de fSUGRA cuatri-dimensional acoplada a vectores no-Abelianos,estudiada en el cap´ıtulo 3, encontramos que las soluciones para el caso timelike vienen dadaspor una m´etrica de tipo conforma-estacionaria ds = 2 | X | ( dτ + ω ) − | X | h mn dy m dy n , (II.1)con la particularidad de que ´esta conlleva una dependencia en el tiempo. Dicha dependenciacoincide con la que propusieron Behrndt y Cvetiˇc en [28] para generar soluciones a partirde las supersim´etricas conocidas en N = 2 d = 4 SUGRA. Adem´as, encontramos que elespacio-base con m´etrica h mn es una subclase de espacios Einstein-Weyl tri-dimensionales,llamados espacios hyper-CR o Gauduchon-Tod [28]. Para el caso nulo encontramos que lassoluciones tienen holonom´ıa dentro del grupo Sim (2), cuyo inter´es est´a fundamentado enque las soluciones puramente gravitacionales con esta caracter´ıstica no reciben correcionescu´anticas [105].As´ı mismo, en el cap´ıtulo 4 se estudia la clasificaci´on del caso nulo a la teor´ıa de fSUGRAm´ınima en cinco dimensiones. La soluci´on viene dada por las ecuaciones (4.72), (4.73), y, aligual que en el caso nulo en cuatro dimensiones, ´esta incluye el caso especial donde el vectornulo N es recurrente, i.e. ∇ µ N ν = C µ N µ , (II.2)que implica que la geometr´ıa resultante (la conexi´on de Levi-Civit`a) tiene holonom´ıa especial Sim (3). Por su parte, el cap´ıtulo 5 contiene el an´alisis de la clasificaci´on de la teor´ıa defSUGRA cinco-dimensional acoplada a vectores Abelianos. ´Esta es una generalizaci´on de laestudiada en el cap´ıtulo anterior, y en este sentido la contiene. Obtenemos que la estructurageom´etrica viene determinada por las ecuaciones (5.137), (5.138), y que la condicion de recur-rencia para ∇ L.C. est´a dada por B = 0. En dicho caso, el valor de la norma de X determinasi el espacio Gauduchon-Tod tri-dimensional es plano o por el contrario la 3-esfera. Adem´as,encontramos que para un valor de X − Q IJ V I V J constante volvemos a tener invariantesescalares algebraicos, en el sentido de [105].Para finalizar, discurrimos sobre c´omo utilizar el mismo concepto en otros contextos,cambiando la condici´on de paralelizaci´on del espinor para obtener distintos resultados. Eneste sentido, el cap´ıtulo 6 contiene una clasificaci´on de espacios Lorentzianos de tipo Einstein-Weyl (EW), que son de inter´es entre la comunidad matem´atica. Nuestras soluciones se basan03en la existencia de un espinor paralelo bajo una determinada conexi´on, de tal modo que sucondici´on de integrabilidad da lugar a la ecuaci´on para espacios EW. Aplicamos entonces lasmismas t´ecnicas (caso timelike y caso nulo), que nos permiten as´ı encontrar que para el casotimelike en cualquier dimensi´on menos en d = 4, estos son conformes a espacios Lorentzianosque admiten espinores paralelos. Estos fueron parcialmente clasificados por Bryant en [123].El caso cuatri-dimensional viene dado por R , , como ya se explicaba en dicha referencia. Lassoluciones en la clase nula vienen dadas por m´etricas Kundt y un vector gauge de tipo Weyl,distintos en funci´on de la dimensionalidad de la teor´ıa.Un punto interesante es que nuestro an´alisis en tres dimensiones contiene una general-izaci´on de una de las dos m´etricas con escalar de Weyl plano estudiadas por Calderbanky Dunajski en [128]. No contiene sin embargo la otra, ya que nuestro m´etodo se basa enla existencia de un espinor paralelo, y esto limita el espectro de soluciones que se puedenobtener. En cualquier caso, se tratan de resultados nunca antes publicados, a los que se hapodido acceder por medio de t´ecnicas caracter´ısticas del estudio de soluciones supersim´etricasa teor´ıas de SUGRA.El ap´endice A contiene las convenciones usadas en los cap´ıtulos centrales, tanto la relativaal c´alculo tensorial como sobre las estructuras espinoriales y los bilineales espinoriales e iden-tidades de Fierz. El ap´endice B ofrece informaci´on sobre las distintas variedades de escalaresrelevantes en las teor´ıas estudiadas. El C contiene detalles t´ecnicos sobre la geometr´ıa delos casos nulos, as´ı como sobre las m´etricas Kundt, que aparecen de forma reiterada comosoluciones en estas clases. El ap´endice D ofrece una peque˜na introducci´on a la geometr´ıa deWeyl y a los espacios Einstein-Weyl y Gauduchon-Tod, mientras que el E contiene detallest´ecnicos sobre el grupo Sim , as´ı como sobre su relaci´on con teor´ıas con vectores recurrentes,cuya conexi´on tiene un grupo de holonom´ıa que es un subgrupo del mismo. Por ´ultimo, elap´endice F es una peque˜na introducci´on a los grupos de Lorentz y esp´ın, y su relaci´on 2 a 1.04
ANEXO II. RESUMEN ppendix A
Conventions
The conventions and definitions used in this thesis are taken from the respective researcharticles which conform the central chapters of this work. They are contained in this appendixfor reference and completeness. Appendix A.1 gives those related to the tensorial calculus.Appendix A.2 presents the spinorial structures of the theories considered. In particular, itcontains the conventions used for the gamma matrices and the spinors. The spinor bilinearsand Fierz identities, which are fundamental to the first of the formalisms, are presented inappendix A.3.
A.1 Tensor conventions
Different notations and conventions have been used for different chapters. This is due tohaving used different methods to attack the classification of fSUGRAs. While in principlethere is no relation between the method employed and the notations/ conventions used, theanalyses considered previous published works in order to reduce the load of work. Hence theconventions taken are those of the previous articles, which unfortunately do not agree. Wepresent both sets for completeness.
A.1.1 Tensor conventions for the bilinear formalism
Chapters 3, 4 and 6 all employ the notation of [29]. The conventions, however, are slightlydifferent. Those of chapter 3 are taken from [127], which in turn are based on [74], to whichwe have adapted the formulae of [29]. The difference between the two is the sign of thespin connection, of the completely antisymmetric tensor (cid:15) abcd and of γ . Thus, chiralities arereversed and self-dual tensors are replaced by antiself-dual tensors, and viceversa. The cur-vatures are identical. Also, all fermions and supersymmetry parameters from [29] have beenrescaled by a factor of 1 /
2, which introduces additional factors of 1 / n ! (see eq. (A.4) below for our choice), which induces an additional factor of 2 inthe supersymmetry transformations of the vectors.The five-dimensional conventions of chapter 4 are those used in [108]. Those, in turn,come from [141], having changed the sign of the metric to have mostly minus signature,multiplying all the gamma matrices γ a by + i and all the γ a by − i and setting κ = 1 / √ APPENDIX A. CONVENTIONS coordinate. We proceed to describe them.We use Greek letters µ, ν, ρ, . . . to denote curved tensor indices in a coordinate basis,and Latin letters a, b, c . . . as flat tensor indices in a Vielbein basis. We symmetrize ( ) andantisymmetrize [ ] with weight one, i.e. dividing by n !, where n is the number of indices insidethe bracket, e.g. A [ a a B b ] ≡ (cid:16) A a a B b + A a b B a + A b a B a − A a a B b − A a b B a − A b a B a (cid:17) . (A.1)We use a mostly minus signature diag(+ , [ − ] d − ). η is the Minkowski metric and we denotea general metric by g . Flat and curved indices are related by tetrads e aµ and their inverses e aµ , thus satisfying e aµ e bν g µν = η ab , e aµ e bν η ab = g µν . (A.2)We define the (Hodge) dual of a completely antisymmetric tensor of rank k , F ( k ) , by (cid:63) F ( k ) µ ··· µ ( d − k ) = k ! √ | g | (cid:15) µ ··· µ ( d − k ) µ ( d − k +1) ··· µ d F ( k ) µ ( d − k +1) ··· µ d . (A.3)Differential forms of rank k are normalized as follows: F ( k ) ≡ k ! F ( k ) µ ··· µ k dx µ ∧ · · · ∧ dx µ k . (A.4) ∇ is the total (general and Lorentz) covariant derivative, whose action on tensors andspinors ψ we choose to be given by ∇ µ φ = ∂ µ φ , ∇ µ ξ ν = ∂ µ ξ ν + Γ µρ,ν ξ ρ , ∇ µ w ν = ∂ µ w ν − w ρ Γ µν,ρ , (A.5) ∇ µ ξ a = ∂ µ ξ a + ω µ,ba ξ b , ∇ µ w a = ∂ µ w a − w b ω µ,ab , ∇ µ ψ = ∂ µ ψ − ω µ,ab γ ab ψ , (A.6)where γ ab is the antisymmetric product of two gamma matrices, i.e. γ ab ≡ γ [ a γ b ] = 12 ( γ a γ b − γ b γ a ) , (A.7) ω µ,ba is the spin connection (antisymmetric in the last two indices) and Γ µρ,ν is the affineconnection (symmetric in the first two indices). From this point on we shall not include thecomma, in order to avoid cluttering of indices. .1. TENSOR CONVENTIONS ∇ µ , ∇ ν ] ξ ρ = R µνσρ (Γ) ξ σ , [ ∇ µ , ∇ ν ] ξ a = R µνba ( ω ) ξ b , [ ∇ µ , ∇ ν ] ψ = − R µν ab ( ω ) γ ab ψ , (A.8)and given in terms of the connections by R µνρσ (Γ) = 2 ∂ [ µ Γ ν ] ρσ + 2Γ [ µ | λσ Γ ν ] ρλ ,R µνab ( ω ) = 2 ∂ [ µ ω ν ] ab − ω [ µ | ac ω | ν ] cb . (A.9)These two connections are related by ω µab = Γ µab + e ν b ∂ µ e aν , (A.10)where the Vielbein postulate ∇ µ e aν = 0 (A.11)has been used. This, in turn, implies that the curvatures are related by R µνρσ (Γ) = e ρa e σb R µνab ( ω ) . (A.12)Finally, metric compatibility and torsionlessness fully determine the connections to be of theform Γ µν ρ = g ρσ { ∂ µ g νσ + ∂ ν g µσ − ∂ σ g µν } ,ω abc = − Ω abc + Ω bca − Ω cab , where Ω abc = e aµ e bν ∂ [ µ e cν ] . (A.13)The indices used in these chapters are explained in the following table.Type Associated structure µ , ν , . . . Curved space a , b , . . . Tangent space m, n, . . .
Cartesian R indices i, j, . . . ; i ∗ , j ∗ , . . . Complex scalar fields and their conjugates.Λ , Σ , . . . sp ( n + 1) indices I, J, . . . N spinor indicesTable A.1: Meaning of the indices employed in the bilinear formalism. Further conventions for d = 4The particularities of the four-dimensional theory induce the existence of (anti) self-dualforms. This implies that there are certain formulae which only hold in this case, and whichhave been used in chapter 3. The four-dimensional fully antisymmetric tensor is defined inflat indices by (cid:15) ≡ +1 ⇒ (cid:15) = − , (A.14)08 APPENDIX A. CONVENTIONS and in curved indices by (cid:15) µ ··· µ = (cid:112) | g | e µ a · · · e µ a (cid:15) a ··· a ; (A.15)thus, with upper indices, it is independent of the metric and has the same value as with flatindices.For any 2-form, we define the imaginary (anti-)selfdual 2-forms F ± ≡ ( F ± i (cid:63)F ) , ± i (cid:63)F ± = F ± . (A.16)For any two 2-forms F , G we have F ± µν G ∓ µν = 0 , F ± [ µρ G ∓ ν ] ρ = 0 . (A.17)A small identity that comes in useful when considering the stress-energy tensor for a numberof field strengths is T (ΛΣ) µν ≡ F (Λ µρ F Σ) ν ρ − g µν F Λ σρ F Σ σρ = 2 F Λ+( µ | ρ F Σ −| ν ) ρ . (A.18)Given any non-null 2-form F , i.e. F µν F µν (cid:54) = 0, and a non-null 1-form ˆ V = V µ dx µ , we canexpress F in the form F = − V − [ E ∧ ˆ V − (cid:63) ( B ∧ ˆ V )] , (A.19)where E ≡ ı V F ( ≡ V µ F µν dx ν ) and B ≡ ı V (cid:63) F . For the complex combinations F ± we have F ± = − V − [ C ± ∧ ˆ V ± i (cid:63) ( C ± ∧ ˆ V )] , (A.20)with C ± ≡ ı V F ± .If we have a (real) null vector l µ , we can always add three more null vectors n µ , m µ , m ∗ µ to construct a complex null tetrad such that the local metric in this basis takes the form −
10 0 − (A.21)with the ordering ( l, n, m, m ∗ ). For the local volume element we obtain (cid:15) lnmm ∗ = i . Withthe dual basis of 1-forms (cid:16) ˆ l, ˆ n, ˆ m, ˆ m ∗ (cid:17) we can construct three independent complex self-dual2-forms that we choose to normalize as follows:ˆΦ (1) = ˆ l ∧ ˆ m ∗ , ˆΦ (2) = [ˆ l ∧ ˆ n + ˆ m ∧ ˆ m ∗ ] , ˆΦ (3) = − ˆ n ∧ ˆ m . (A.22)Any self-dual 2-form F + can be written as a linear combination of these, with complexcoefficients F + = c i ˆΦ ( i ) . (A.23) .2. SPINORIAL STRUCTURES c i can be found by contracting F + with l µ , n µ , m µ , m ∗ µ , i.e. l ν F + νµ = − c l µ − c m µ ,n ν F + νµ = c m ∗ µ + c n µ ,m ν F + νµ = c l µ + c m µ ,m ∗ ν F + νµ = − c m ∗ µ − c n µ . (A.24) A.1.2 Tensor conventions for the spinorial geometry formalism
Chapter 5 employs the spinorial geometry method. The notation is different from that of thespinor-bilinear case. In particular, the following equivalences hold
Spinorial geometry notation Spinor-bilinear notation index M index a index I index Λ C IJK ( − / √ C IJK X I ( − / √ h I X I ( −√ / h I Q IJ a IJ Q IJ (1 / a IJ V I (cid:64) g V I − (4 g/ ξ I F ab (1 / F ab Table A.2: Equivalence between notation employed in the two formalisms.The conventions used are those of [142]. In particular, the form of the covariant derivativetakes on the form ∇ µ ξ ν = ∂ µ ξ ν + Γ ν ρµ ξ ρ , (A.25)where Γ ν ρµ = 12 g νσ ( ∂ µ g ρσ + ∂ ρ g µσ − ∂ σ g ρµ ) , (A.26)and the symmetry is clearly on the last two indices. The Riemann curvature tensor is definedas R ρσµν = ∂ µ Γ ρνσ − ∂ ν Γ ρµσ + Γ ρµλ Γ λνσ − Γ ρνλ Γ λµσ . (A.27)This implies that the curvature scalar R has the opposite sign to that of chapters 3, 4 and 6. A.2 Spinorial structures
We give the conventions used for the spinorial structure of a general Lorentzian manifoldof dimension (1 , d − APPENDIX A. CONVENTIONS positive-semidefiniteness (for Lorentzian spacetimes) of the vector bilinear, which in turnexplain why we consider timelike and null cases in our classification. Notice that chapter 6has a Dirac conjugation matrix equal to D = γ , while in chapters 3 and 4 this is D = iγ .The particularities that apply to these latter cases are given in appendix A.2.1 for d = 4, andappendix A.2.2 for d = 5.On R ,n − we shall put the mostly-negative metric η = diag(+ , [ − ] n − ) and take theGamma matrices to satisfy { γ a , γ b } = 2 η ab . (A.28)We use a unitary representation for these matrices, which implies that γ † = γ and γ † i = − γ i .In chapter 6 we have chosen the Dirac conjugation matrix D = γ , and we define theDirac conjugate of a spinor ψ by ψ ≡ ψ † D , to obtain D γ a D − = γ † a , D γ ab D − = − γ † ab . (A.29)Defining the 1-form L = L a e a by means of L a ≡ ψγ a ψ , it is then automatically real, i.e. L ∗ a = ψ T ( D γ a ) ∗ ψ ∗ = ψγ a ψ D = ψ † ( D γ a ) † ψ = ψ D − γ † a D † ψ = ψγ a ψ = L a , (A.30)where a perhaps expected − i.e. commuting) spinors. In terms of the components we have that L a = (cid:15) † D γ a (cid:15) and it isclear that L = (cid:15) † (cid:15) . Furthermore, we can always rotate the spatial components of L in sucha way that only the first component is non-vanishing. This then implies that g ( L, L ) = L − L . (A.31) L = (cid:15) † γ (cid:15) and if we combine this with γ † = γ , γ = 1 and Tr( γ ) = 0 we can use aSO( (cid:98) n/ (cid:99) ) rotation to write γ = diag([+] (cid:98) n/ (cid:99) , [ − ] (cid:98) n/ (cid:99) ). Decomposing the spinor w.r.t. thestructure of γ as (cid:15) t = ( v, w ), where v and w are vectors in C (cid:98) n/ (cid:99) , we see that L = | v | + | w | , L = | v | − | w | −→ g ( L, L ) = 4 | v | | w | , (A.32)which implies the positive-definiteness of | L | . A.2.1 Gamma matrices and spinors in
SO(1 , We work with a unitary, purely imaginary representation of the Gamma matrices γ a ( γ a ) ∗ = − γ a , (A.33)and the same convention for their anticommutator as above. Thus, γ γ a γ = γ a † = ( γ a ) − ≡ γ a . (A.34)Because of the even dimensionality of the theory, there is a chirality matrix, which is definedby γ ≡ − iγ γ γ γ = i (cid:15) abcd γ a γ b γ c γ d , (A.35) .2. SPINORIAL STRUCTURES γ † = − γ ∗ = γ , ( γ ) = 1 . (A.36)With such a chirality matrix, we have the identity γ a ··· a d = ( − (cid:98) d/ (cid:99) i (4 − d )! (cid:15) a ··· a d b ··· b − d γ b ··· b − d γ . (A.37)Our convention for Dirac conjugation is ψ D ≡ ¯ ψ = ψ † D , (A.38)where in this case we have taken D = iγ . We use 4-component chiral spinors. In N = 2theory, the chirality is related to the position of the SU (2)-indices and Sp ( n h / γ ψ I µ = − ψ I µ , γ λ Ii = + λ Ii , γ (cid:15) I = − (cid:15) I . (A.39)Both (chirality and position of the index) are reversed under complex conjugation: γ ψ I µ = ψ I µ , γ λ I i ∗ = − λ I i ∗ , γ ζ α = + ζ α , γ (cid:15) I = (cid:15) I . (A.40)We take this fact into account when Dirac-conjugating chiral spinors:¯ (cid:15) I ≡ i ( (cid:15) I ) † γ , ¯ (cid:15) I γ = +¯ (cid:15) I , etc . (A.41) A.2.2 Gamma matrices and spinors in
SO(1 , In five dimensions, the first four of the Gamma matrices are taken to be identical to thefour-dimensional purely imaginary Gamma matrices γ , ..., γ satisfying { γ a , γ b } = 2 η ab , (A.42)and the fifth is γ = − γ , so it is purely real. The Clifford algebra anticommutator (A.28)is still valid for a = 0 , · · · ,
4. Furthermore, γ a ··· a = (cid:15) a ··· a and in general γ a ··· a d = ( − (cid:98) d/ (cid:99) (5 − d )! (cid:15) a ··· a d b ··· b d − γ b ··· b − d . (A.43) γ is Hermitean and the other Gammas are anti-Hermitean, thus having a unitary represen-tation.To explain our convention for symplectic-Majorana spinors, let us start by defining theDirac, complex and charge conjugation matrices D ± , B ± , C ± . By definition they satisfy [74] D ± γ a D − ± = ± γ a † , B ± γ a B − ± = ± γ a ∗ , C ± γ a C − ± = ± γ a T . (A.44)The natural choice for a real Dirac conjugation matrix is D = iγ , (A.45)which corresponds to D = D + . The other conjugation matrices are related to it by C ± = B T ± D , (A.46)12 APPENDIX A. CONVENTIONS but it can be shown that in this case only C = C + and B = B + exist and are both antisym-metric. We take them to be C = iγ , B = γ ⇒ B ∗ B = − . (A.47)The Dirac conjugate is defined by ψ D ≡ ¯ ψ = ψ † D = iψ † γ , (A.48)and the Majorana conjugate by ψ M ≡ ψ T C = iψ T γ . (A.49)The Majorana condition (Dirac conjugate = Majorana conjugate) cannot be consistentlyimposed because it requires B ∗ B = +1. Therefore, we introduce the symplectic-Majoranaconjugate in pairs of spinors by using the corresponding symplectic matrix, e.g. ε ij in ψ i SM ≡ ε ij ψ j T C ; (A.50)then the symplectic-Majorana condition is ψ i ∗ = ε ij γ ψ j . (A.51)To impose the symplectic-Majorana condition on hyperinos ζ A , the only thing we have to dois to replace the matrix ε ij by C AB , which is the invariant metric of Sp ( n h ).Our conventions on SU (2) indices are intended to keep manifest the SU (2)-covariance.In SU (2), besides the preserved metric, there is the preserved tensor ε ij . We also introduce ε ij , normalised as ε = ε = +1. Therefore, we may construct new covariant objects byusing ε ij and ε ij , e.g. ψ i ≡ ε ij ψ j , ψ j = ψ i ε ij . Using this notation the symplectic-Majoranacondition can be simply stated as ψ i ∗ = γ ψ i . (A.52)We use the bar on spinors to denote the (single) Majorana conjugate:¯ ψ i ≡ ψ iT C , (A.53)which transforms under SU (2) in the same representation as ψ i does. We can also lower its SU (2) index as ¯ ψ i ≡ ε ij ¯ ψ j . In terms of single Majorana conjugates the symplectic Majoranacondition reads (cid:0) ¯ ψ i (cid:1) ∗ = ¯ ψ i γ . (A.54)Finally, observe that after imposing the symplectic Majorana condition the following simplerelation between the single Dirac and Majorana conjugates holds: ψ i † D = ¯ ψ i , (A.55)which is very useful if one prefers to use the Dirac conjugate instead of the Majorana one. A.3 Spinor bilinears and Fierz identities
In this appendix we provide information on the spinor bilinears constructed out of the (cid:15) ’s.This is primary in the characterisation of the solutions based on the spinor-bilinear methodused in chapters 3, 4 and 6. We give as well the Fierz identities, with a proof of theirderivation. .3. SPINOR BILINEARS AND FIERZ IDENTITIES
A.3.1 Spinor bilinears in d = 4 These bilinears are based on (but not equal to) those of [82]. The scalar ones are defined by X = ε IJ ¯ (cid:15) I (cid:15) J , ¯ (cid:15) I (cid:15) J = ε IJ X ,X = ε IJ ¯ (cid:15) I (cid:15) J , ¯ (cid:15) I (cid:15) J = ε IJ X . (A.56)The vector bilinears are defined by V Ia J ≡ i ¯ (cid:15) I γ a (cid:15) J = V a δ I J + V xa ( σ x ) I J , (A.57)which can be inverted to V a = V Ia I and V xa = ( σ x ) I J V Ia J . (A.58)Finally we have 3 imaginary-selfdual 2-forms defined byΦ IJ ab ≡ ¯ (cid:15) I γ ab (cid:15) J = Φ xab i ( σ x ) IJ −→ Φ x = i ( σ x ) IJ Φ IJ . (A.59)The anti-imaginary-self-dual 2-forms are defined by complex conjugation.From the Fierz identities eq. (A.83) we can then derive that η ab = 14 | X | [ V a V b − V xa V xb ] , (A.60)and consistently with the above that ı V V x = 0 , g ( V, V ) = 4 | X | , g ( V x , V y ) = − | X | δ xy . (A.61)A result that is harder to be found is X Φ xab = − i (cid:104) V [ a V xb ] + i ε abcd V c V xd (cid:105) , (A.62)which translates to X Φ x = 12 i [ V ∧ V x + i (cid:63) ( V ∧ V x )] (A.63)in form notation.In the null case, i.e. for X = 0, the V x are proportional to V and the Φs become lineardependent, severely limiting the utility of the bilinears. In section (3.3), we will, following[21], introduce an auxiliar spinor which leads to Fierz identities similar to the ones above. A.3.2 Spinor bilinears in d = 5 With one commuting symplectic-Majorana spinor (cid:15) i we can construct the following indepen-dent, SU (2)-covariant bilinears:¯ (cid:15) i (cid:15) j : It is easy to see that ¯ (cid:15) i (cid:15) j = − ε jk (¯ (cid:15) k (cid:15) l ) ε li , (¯ (cid:15) i (cid:15) j ) ∗ = − ¯ (cid:15) j (cid:15) i , (A.64)The first equation implies that this matrix is proportional to δ ij and the second equationimplies that the constant is purely imaginary. Thus, we define the SU (2)-invariant scalar˜ f ≡ i ¯ (cid:15) i (cid:15) i = i ¯ (cid:15)σ (cid:15) , ¯ (cid:15) i (cid:15) j = − i ˜ f δ ij . (A.65)All the other scalar bilinears i ¯ (cid:15)σ r (cid:15) ( r = 1 , ,
3) vanish identically.14
APPENDIX A. CONVENTIONS ¯ (cid:15) i γ a (cid:15) j : This matrix satisfies the same properties as ¯ (cid:15) i (cid:15) j , and so we define the vector bilinear V a ≡ i ¯ (cid:15) i γ a (cid:15) i = i ¯ (cid:15)γ a σ (cid:15) , ¯ (cid:15) i γ a (cid:15) j = − i δ ij V a . (A.66)which is also SU (2)-invariant, the other vector bilinears being automatically zero.¯ (cid:15) i γ ab (cid:15) j : In this case ¯ (cid:15) i γ ab (cid:15) j = + ε jk (¯ (cid:15) k γ ab (cid:15) l ) ε li , (¯ (cid:15) i γ ab (cid:15) j ) ∗ = ¯ (cid:15) j γ ab (cid:15) i , (A.67)which means that these 2-form matrices are traceless and Hermitean and we have threenon-vanishing real 2-formsΦ r ab ≡ σ rij ¯ (cid:15) j γ ab (cid:15) i , ¯ (cid:15) i γ ab (cid:15) j = σ rij Φ r ab . (A.68)for r = 1 , ,
3, which transform as a vector in the adjoint representation of SU (2), andthe fourth is ¯ (cid:15)γ ab σ (cid:15) = 0.Using the Fierz identities eq. (A.83) for commuting spinors we get, among other identities, V a V a = ˜ f , (A.69) V a V b = η ab ˜ f + (cid:88) r Φ rac Φ rcb , (A.70) V a Φ rab = 0 , (A.71) V a ( (cid:63) Φ r ) abc = − ˜ f Φ rbc , (A.72)Φ rac Φ scb = − δ rs ( η ab ˜ f − V a V b ) − ε rst ˜ f Φ tab , (A.73)Φ r [ ab Φ scd ] = − ˜ f δ rs ε abcde V e , (A.74) V a γ a (cid:15) i = ˜ f (cid:15) i , (A.75)Φ rab γ ab (cid:15) i = 4 i ˜ f (cid:15) j σ rji . (A.76) A.3.3 Fierz identities
The bilinears that can be constructed from Killing spinors in our four- and five-dimensionalstudies are 2 × σ ˆ r (ˆ r = 0 , . . . ,
3) where σ = . It is natural, then, that we want to use the equation firstderived by Fierz to help us obtain important identities on the bilinears.The general Fierz identity arises from the completeness of the antisymmetric product ofDirac gamma matrices γ a as a basis for 2 (cid:98) d/ (cid:99) × (cid:98) d/ (cid:99) matrices. The 2 (cid:98) d/ (cid:99) -dimensionalcanonical basis for the vector space of these matrices is {O I } = {O , O , ..., O d } = r I { , γ a , γ a a , ... , γ a a ... a d } , (A.77) .3. SPINOR BILINEARS AND FIERZ IDENTITIES r I = 1 or r I = i according to whether (cid:98) I/ (cid:99) is (respectively) even or odd. We canconstruct an orthogonal dual basis {O I } = r I { , γ a , γ a a , ... , γ a a ... a d } , (A.78)where r I = r I ; one can see that O I O J ≡ tr( O I O J ) = 2 (cid:98) d/ (cid:99) δ IJ , such that any 2 (cid:98) d/ (cid:99) × (cid:98) d/ (cid:99) matrix A can be written as a linear combination of the elements in O I . This implies that A = a I O I , and hence tr( O I A ) = 2 (cid:98) d/ (cid:99) a I ; furthermore A αβ = 2 −(cid:98) d/ (cid:99) (cid:88) I A σρ ( O I ) ρσ ( O I ) αβ . (A.79)By considering a general bilinear of spinors (cid:0) ¯ λM ϕ (cid:1) (cid:0) ¯ ψN χ (cid:1) ≡ ¯ λ α M αβ ϕ β ¯ ψ σ N σρ χ ρ (A.80)and eq. (A.79) on M αβ N σρ as a A σβ -matrix, we obtain the Fierz identity (cid:0) ¯ λM ϕ (cid:1) (cid:0) ¯ ψN χ (cid:1) = p −(cid:98) d/ (cid:99) (cid:88) I (¯ λM O I N χ )( ¯ ψ O I ϕ ) , (A.81)where p = +1 for commuting spinors, and p = − anti-commuting spinors.Moreover, to obtain the general form in d = 4, we consider eq. (A.37), that relates productsof n matrices with products of ( d − n ) matrices. This gives, p (¯ λM χ )( ¯ ψN ϕ ) = (¯ λM N ϕ )( ¯ ψχ ) + (¯ λM γ a N ϕ )( ¯ ψγ a χ ) − (¯ λM γ ab N ϕ )( ¯ ψγ ab χ ) − (¯ λM γ a γ N ϕ )( ¯ ψγ a γ χ ) + (¯ λM γ N ϕ )( ¯ ψγ χ ) . (A.82)For d = 5 we use eq. (A.43), as well as an extended basis of Pauli matrices (ˆ r = { , , , } )to express the Sp (1)-structure succinctly. This gives (cid:0) ¯ λM ϕ (cid:1) (cid:0) ¯ ψN χ (cid:1) = p (cid:88) ˆ r =0 (cid:18) (cid:16) ¯ λM σ ˆ r N χ (cid:17) (cid:16) ¯ ψσ ˆ r ϕ (cid:17) + (cid:16) ¯ λM γ a σ ˆ r N χ (cid:17) (cid:16) ¯ ψγ a σ ˆ r ϕ (cid:17) − (cid:0) ¯ λM γ ab σ ˆ r N χ (cid:1) (cid:0) ¯ ψγ ab σ ˆ r ϕ (cid:1) (cid:19) . (A.83)16 APPENDIX A. CONVENTIONS ppendix B
Scalar manifolds
In this appendix we present the technical information about the scalar manifolds which playa part in the theories studied. These geometries are different according to the dimensionalityof the theory in question, and its field content. K¨ahler geometry, for example, is relevantwhen considering a four-dimensional sigma model with N = 1 global supersymmetry [143].To introduce local SUSY, however, ones needs to consider the subclass of K¨ahler-Hodgemanifolds (see e.g. [144, ch. 3]). An introduction to these, including the conventions used, isgiven in appendix B.1.For a lagrangian in d = 4 with N = 2 global SUSY, containing only massless scalars, thecouplings among these fields are given in terms of a hyper-K¨ahler manifold [145]. Addinggravity implies that the scalar fields will parametrise instead a quaternionic-K¨ahler manifold[146 - 148]. This geometry is treated at some length in appendix B.3. When introducing vectormultiplets as well, the self-couplings between the complex scalars, and those relating complexscalars and vectors are dictated by Special K¨ahler geometry [149, 80], which is discussed inappendix B.2. The complete target space of the non-linear sigma model for the scalar fields isthus given by the product of a Special K¨ahler manifold and a quaternionic-K¨ahler manifold.Real Special geometry, on the other hand, arises when considering the coupling of pure d = 5 SUGRA to vector multiplets [150]. This is given in appendix B.4. A more involvedtheory not considered here is that formed by also considering hyperscalar multiplets. Thecomplete target space is then given by the product of a Real Special and a quaternionic-K¨ahlermanifold.Before starting with K¨ahler geometry we give a short scholium on complex and hyper-complex (quaternionic) structures. Complex and quaternionic structures
We say that a manifold M is almost-complex when it is endued with an almost-complexstructure J . This is a linear map defined at each patch of the manifold such that J : T p M → T p M , with J = − . (B.1)We call it almost-complex, as opposed to regularly complex, since we lack an integrabilitycondition connecting the different patches of M . Usually such integrability condition can belinked to the vanishing of the Nijenhuis tensor. In the context of supergravity theories, it iscommon to find the condition of parallelity of J w.r.t. the Levi-Civit`a connection ∇ a J bc = 0 . (B.2)11718 APPENDIX B. SCALAR MANIFOLDS
Eq. (B.2) implies the vanishing of the Nijenhuis tensor, whence M is a complex manifold.This almost-complex structure J need not necessarily be compatible with the metric H ,such that J ◦ H ( ∂ a , ∂ b ) = H ( J ◦ ∂ a , ∂ b ) + H ( ∂ a , J ◦ ∂ b ) = J ab + J ba = 0 . (B.3)If it is, we call the metric Hermitean and we define a 2-form K ( X, Y ) ≡ H ( J ◦
X, Y ), whichis identified with the K¨ahler 2-form. If the form K is closed, i.e. d K = 0, then M is said tobe a K¨ahler manifold. Clearly this is the case if J is parallel `a la (B.2).If instead of one J we have a triplet of them J r ( r = 1 , , i.e. ( J r · J s ) ab = − δ rs δ ab + ε rst ( J t ) ab , (B.4)we have what is called an almost-quaternionic structure (it is also often called an almost-hyper-complex structure). Again, if the three maps are compatible with the metric, in which case themetric is said to be hyper-Hermitean, we can define the three 2-forms K r ( X, Y ) ≡ H ( J r X, Y ).If the condition d K r = 0 is satisfied, then the two-forms are closed and M is said to be ahyper-K¨ahler manifold. B.1 K¨ahler geometry
A K¨ahler manifold M is a complex manifold on which there exist complex coordinates Z i and Z ∗ i ∗ = ( Z i ) ∗ and a real function K ( Z, Z ∗ ), called the K¨ahler potential , such that the lineelement is ds = 2 G ii ∗ dZ i dZ ∗ i ∗ , with G ii ∗ = ∂ i ∂ i ∗ K . (B.5)The K¨ahler (connection) 1-form Q is defined by Q ≡ i ( dZ i ∂ i K − dZ ∗ i ∗ ∂ i ∗ K ) , (B.6)and the K¨ahler 2-form K is its exterior derivative K ≡ d Q = i G ii ∗ dZ i ∧ dZ ∗ i ∗ . (B.7)The choice of complex coordinates is such that the complex structure is thus trivial, i.e. J ii = −J i ∗ i ∗ = i . The Levi-Civit`a connection on a K¨ahler manifold can be shown to beΓ jki = G ii ∗ ∂ j G i ∗ k , Γ j ∗ k ∗ i ∗ = G i ∗ i ∂ j ∗ G k ∗ i . (B.8)The only non-vanishing components of the Riemann curvature tensor are given by R ij ∗ kl ∗ ,but we shall not be needing its explicit expression. The Ricci tensor is given by R ij ∗ = ∂ i ∂ j ∗ (cid:0) log det G (cid:1) . (B.9)As can be easily seen from inspection of eq. (B.5), the K¨ahler potential is not unique; itis defined up to K¨ahler transformations of the form K (cid:48) ( Z, Z ∗ ) = K ( Z, Z ∗ ) + f ( Z ) + f ∗ ( Z ∗ ) , (B.10) .1. K ¨AHLER GEOMETRY f ( Z ) is a holomorphic function of the complex coordinates Z i . Under these transfor-mations, the K¨ahler metric and K¨ahler 2-form are invariant, while the components of theK¨ahler connection 1-form transform according to Q (cid:48) i = Q i − i ∂ i f . (B.11)By definition, objects with K¨ahler weight ( q, ¯ q ) transform under the above K¨ahler transfor-mations with a factor e − ( qf +¯ qf ∗ ) / and the K¨ahler-covariant derivative D acting on them isgiven by D i ≡ ∇ i + iq Q i , D i ∗ ≡ ∇ i ∗ − i ¯ q Q i ∗ , (B.12)where ∇ is the covariant derivative associated to the Levi-Civit`a connection on M . The Ricciidentity for this covariant derivative, for a weight ( q, ¯ q ) scalar object, reads[ D i , D j ∗ ] φ ( q, ¯ q ) = − ( q − ¯ q ) G ij ∗ φ ( q, ¯ q ) . (B.13) B.1.1 K¨ahler-Hodge manifolds
When ( q, ¯ q ) = (1 , − L → M over the K¨ahler manifold M , whose first (and only) Chern class equals the K¨ahler 2-form K . A complex line bundlewith this property is known as a K¨ahler-Hodge (KH) manifold. These are the manifoldsparametrized by the complex scalars of the chiral multiplets of N = 1 , d = 4 Supergravity.Furthermore, objects such as the superpotential and the spinors of the theory have a well-defined K¨ahler weight. On the other hand, manifolds parametrized by the complex scalarsof the vector multiplets of N = 2 , d = 4 Supergravity are also KH manifolds, but mustsatisfy further constraints that define what is known as Special K¨ahler Geometry (or SpecialGeometry for short), described in appendix B.2.If one is interested on the spacetime pullback of the K¨ahler-covariant derivative on tensorfields with K¨ahler weight ( q, − q ) (weight q , for short), this takes the simple form D µ = ∇ µ + iq Q µ , (B.14)where ∇ µ is the standard spacetime covariant derivative plus possibly the pullback of theLevi-Civit`a connection on M , and Q µ is the pullback of the K¨ahler 1-form, i.e. Q µ = i ( ∂ µ Z i ∂ i K − ∂ µ Z ∗ i ∗ ∂ i ∗ K ) . (B.15) B.1.2 Gauging holomorphic isometries
We now proceed to review some of the basics of the gauging of holomorphic isometries ofK¨ahler-Hodge manifolds that occur in N = 1 and N = 2 , d = 4 supergravities. We will firststudy the general problem in complex manifolds. This is enough for purely bosonic theories,in which only the complex structure is relevant. In the presence of fermions, however, theK¨ahler-Hodge structure becomes necessary, and only those transformations that preserve itwill be symmetries (of the full theory) that can be gauged. We study this problem next.The special-K¨ahler structure is necessary in N = 2 , d = 4 Supergravity and, again, onlythose transformations that preserve it are symmetries that can be gauged. This problem willbe studied in appendix B.2.2, after we define special-K¨ahler manifolds.20 APPENDIX B. SCALAR MANIFOLDS
Gauging in complex manifolds
We start by assuming that the Hermitean metric G ij ∗ (we will use the K¨ahler-Hodge structurelater) admits a set of Killing vectors { K Λ = k Λ i ∂ i + k ∗ Λ i ∗ ∂ i ∗ } satisfying the Lie algebra[ K Λ , K Σ ] = − f ΛΣΩ K Ω , (B.16)of the group G V that we want to gauge. Hermiticity implies that the components k Λ i and k ∗ Λ i ∗ of the Killing vectors are, respectively, holomorphic and antiholomorphic, and satisfy(separately) the above Lie algebra. Once (anti-)holomorphicity is taken into account, theonly non-trivial components of the Killing equation are £ Λ G ij ∗ = ∇ i ∗ k ∗ Λ j + ∇ j k Λ i ∗ = 0 , (B.17)where £ Λ stands for the Lie derivative w.r.t. K Λ .The standard σ -model kinetic term G ij ∗ ∂ µ Z i ∂ µ Z ∗ j ∗ is then automatically invariant underinfinitesimal reparametrizations of the form δ α Z i = α Λ k Λ i ( Z ) , (B.18)where the infinitesimal variation parameters α Λ are constants. If instead they become ar-bitrary functions of the spacetime coordinates, i.e. α Λ = α Λ ( x ), we need to introduce acovariant derivative using as connection the vector fields present in the theory. We write thecovariant derivative as D µ Z i = ∂ µ Z i + gA Λ µ k Λ i , (B.19)which transforms as δ α D µ Z i = α Λ ( x ) ∂ j k Λ i D µ Z j = − α Λ ( x )( £ Λ − K Λ ) D µ Z j , (B.20)provided that the gauge potentials transform as δ α A Λ µ = − g − D µ α Λ ≡ − g − ( ∂ µ α Λ + gf ΣΩΛ A Σ µ α Ω ) . (B.21)The gauge field strength is given by F Λ µν = 2 ∂ [ µ A Λ ν ] + gf ΣΩΛ A Σ[ µ A Ω ν ] , (B.22)and changes under gauge transformations as δ α F Λ µν = − α Σ ( x ) f ΣΩΛ F Ω µν . (B.23)Now, to make the σ -model kinetic term gauge invariant, it is enough to replace the partialderivatives by covariant derivatives G ij ∗ ∂ µ Z i ∂ µ Z ∗ j ∗ −→ G ij ∗ D µ Z i D µ Z ∗ j ∗ . (B.24)For any tensor field Φ transforming covariantly under gauge transformations, i.e. trans-forming as δ α Φ = − α Λ ( x )( L Λ − K Λ )Φ , (B.25) The index Λ always takes values from 1 to n V (¯ n = n V + 1) in N = 1 ( N = 2) Supergravity , but some(or all) the Killing vectors may actually be zero. Where spacetime ( µ, ν, . . . ), gauge (Λ , Σ , . . . ) and target space tensor ( i, i ∗ , . . . ) indices are not explicitlyshown. .1. K ¨AHLER GEOMETRY Lie covariant derivative L Λ as L Λ ≡ £ Λ − S Λ , (B.26)and S Λ represents a symplectic rotation, the gauge covariant derivative is given by D µ Φ = {∇ µ + D µ Z i Γ i + D µ Z ∗ i ∗ Γ i ∗ − gA Λ µ ( L Λ − K Λ ) } Φ . (B.27)In particular, on D µ Z i D µ D ν Z i = ∇ µ D ν Z i + Γ jki D µ Z j D ν Z k + gA Λ µ ∂ j k Λ i D ν Z j , (B.28)which implies [ D µ , D ν ] Z i = gF Λ µν k Λ i . (B.29)An important case is that of fields Φ which only depend on the spacetime coordinatesthrough the complex scalars Z i and their complex conjugates, so that ∇ µ Φ = ∂ µ Φ = ∂ µ Z i ∂ i Φ + ∂ µ Z ∗ i ∗ ∂ i ∗ Φ . (B.30)Φ is then an invariant field if L Λ Φ ≡ ( £ Λ − S Λ )Φ = 0 . (B.31)Only if all the fields present in the theory are invariant fields, can the theory be gauged.Only in such a case ∇ µ Φ = ∂ µ Φ = ∂ µ Z i ∂ i Φ + ∂ µ Z ∗ i ∗ ∂ i ∗ Φ can be true irrespectively of gaugetransformations. These fields transform under gauge transformations according to δ α Φ = − α Λ ( L Λ − K Λ )Φ = α Λ K Λ Φ , (B.32)and their covariant derivative is given by D µ Φ = { ∂ µ + D µ Z i Γ i + D µ Z ∗ i ∗ Γ i ∗ + gA Λ µ K Λ } Φ , (B.33)which is always the covariant pullback of the target covariant derivative: D µ Φ = D µ Z i ∇ i Φ + D µ Z ∗ i ∗ ∇ i ∗ Φ . (B.34)As an example, consider the holomorphic kinetic matrix f ΛΣ ( Z ) in N = 1, d = 4 Su-pergravity or the period matrix N ΛΣ ( Z, Z ∗ ) in N = 2, d = 4 Supergravity, both of whichare symmetric matrices that codify the couplings between the complex scalars and the vectorfields. These matrices transform under global rotations of the vector fields δ α A Λ µ = − α Σ f ΣΩΛ A Ω µ (B.35) We will extend this definition to fields with non-zero K¨ahler weight after we study the symmetries of theK¨ahler structure. For the moment we only consider tensors of the Hermitean space with metric G ij ∗ , possiblywith gauge and spacetime indices. Alternatively, we could say that it is a field invariant under reparametrizations up to rotations. APPENDIX B. SCALAR MANIFOLDS according to δ α f ΛΣ ≡ − α Ω S Ω f ΛΣ = 2 α Ω f Ω(ΛΠ f Σ)Π , (B.36)(analogously for N ΛΣ ) and under the reparametrizations of the complex scalars, eq. (B.18),as δ α f ΛΣ = − α Ω £ Ω f ΛΣ − α Ω k Ω i ∂ i f ΛΣ . (B.37)These transformations will only be a symmetry of the theory if their values coincide, i.e. if( £ Ω − S Ω ) f ΛΣ = L Ω f ΛΣ = 0 , (B.38)this is, only if f ΛΣ ( Z ) is an invariant field according to the above definition. Its covariantderivative is given by D µ f ΛΣ = D µ Z i ∂ i f ΛΣ , (B.39)on account of its holomorphicity. Gauging in K¨ahler-Hodge manifolds
Let us now assume that the scalar manifold is not just Hermitean, but rather K¨ahler-Hodge,and proceed to study how the K¨ahler structure is preserved. The transformations generatedby the Killing vectors will preserve the K¨ahler structure if they leave the K¨ahler potentialinvariant up to K¨ahler transformations, i.e. for each Killing vector K Λ £ Λ K ≡ k Λ i ∂ i K + k ∗ Λ i ∗ ∂ i ∗ K = λ Λ ( Z ) + λ ∗ Λ ( Z ∗ ) . (B.40)From this condition it follows that £ Λ λ Σ − £ Σ λ Λ = − f ΛΣΩ λ Ω . (B.41)On the other hand, the preservation of the K¨ahler structure implies the conservation ofthe K¨ahler 2-form K £ Λ K = 0 . (B.42)The closedness of K implies that £ Λ K = d ( i k Λ K ) and therefore the preservation of the K¨ahlerstructure implies the existence of a set of real 0-forms P Λ known as momentum maps , suchthat i k Λ K = d P Λ . (B.43)A local solution for this equation is provided by i P Λ = k Λ i ∂ i K − λ Λ , (B.44)which, on account of eq. (B.40), it is equivalent to i P Λ = − ( k ∗ Λ i ∗ ∂ i ∗ K − λ ∗ Λ ) , (B.45)or P Λ = i k Λ Q − i ( λ Λ − λ ∗ Λ ) . (B.46)The momentum map can be used as a prepotential from which the Killing vectors can bederived k Λ i ∗ = i∂ i ∗ P Λ . (B.47) .1. K ¨AHLER GEOMETRY Killing prepotentials .In principle, the momentum maps are defined up to an additive real constant. In N = 1, d = 4 theories (but not in N = 2, d = 4) it is possible to have non-vanishing, constantmomentum maps with i P Λ = − λ Λ for vanishing Killing vectors. In this case, no isometry isgauged; instead it is the U (1) symmetry associated to K¨ahler transformations (in K¨ahler-Hodge manifolds) that is gauged. These constant momentum maps are called D- or Fayet-Iliopoulos terms, and appear in the supersymmetry transformation rules of gaugini, in thepotential and in the covariant derivatives of sections that we now discuss.Using Eqs. (B.16),(B.40) and (B.41) one finds £ Λ P Σ = 2 ik [Λ i k ∗ Σ] j ∗ G ij ∗ = − f ΛΣΩ P Ω . (B.48)This equation fixes the additive constant of the momentum map in directions in which anon-Abelian group is going to be gauged. The gauge transformation rule for a section Φ ofK¨ahler weight ( p, q ) is δ α Φ = − α Λ ( x )( L Λ − K Λ )Φ , (B.49)where L Λ stands for the symplectic and K¨ahler-covariant Lie derivative w.r.t. K Λ , and it isgiven by L Λ Φ ≡ { £ Λ − [ S Λ − ( pλ Λ + qλ ∗ Λ )] } Φ , (B.50)where the S Λ are sp (¯ n ) matrices that provide a representation of the Lie algebra of the gaugegroup G V acting on the section Φ: [ S Λ , S Σ ] = + f ΛΣΩ S Ω . (B.51)The gauge covariant derivative acting on these sections is given by D µ Φ = {∇ µ + D µ Z i Γ i + D µ Z ∗ i ∗ Γ i ∗ + ( pk Λ i ∂ i K + qk ∗ Λ i ∗ ∂ i ∗ K )+ gA Λ µ [ S Λ + i ( p − q ) P Λ − ( £ Λ − K Λ )] } Φ . (B.52)Invariant sections are then those for which L Λ Φ = 0 ⇒ £ Λ Φ = [ S Λ − ( pλ Λ + qλ ∗ Λ )]Φ , (B.53)and their gauge covariant derivatives are, again, the covariant pullbacks of the K¨ahler-covariant derivatives D µ Φ = D µ Z i D i Φ + D µ Z ∗ i ∗ D i ∗ Φ . (B.54)The prime example of an invariant field is the covariantly holomorphic section L ( Z, Z ∗ ) of N = 1, d = 4 theories. This is a K¨ahler weight (1 , −
1) section, related to the holomorphicsuperpotential W ( Z ) by L ( Z, Z ∗ ) ≡ W ( Z ) e K / , (B.55)and its covariant holomorphicity follows from the holomorphicity of W D i ∗ L = ( ∂ i ∗ + i Q i ∗ ) L = e K / ∂ i ∗ ( e −K / L ) = e K / ∂ i ∗ W = 0 . (B.56) Cf. eq. (B.11). Again, spacetime and target space tensor indices are not explicitly shown. Symplectic indices are notshown, either. APPENDIX B. SCALAR MANIFOLDS
In order for the global transformation eq. (B.18) to be a symmetry of the full theory thatwe can gauge, L must be an invariant section, i.e. L Λ L = { £ Λ + ( λ Λ − λ ∗ Λ ) }L = 0 ⇒ K Λ L = − ( λ Λ − λ ∗ Λ ) L . (B.57)Then under gauge transformations it will transform according to δ α L = − α Λ ( x )( λ Λ − λ ∗ Λ ) L , (B.58)and its covariant derivative will be given by D µ L = ( ∂ µ + i ˆ Q µ ) L = D µ Z i D i L , (B.59)where we have defined ˆ Q µ ≡ Q µ + gA Λ µ P Λ . (B.60)Observe that this 1-form is, in general, different from the “covariant pullback” of the K¨ahler1-form, which is i D µ Z i ∂ i K + c . c . . (B.61)The difference between this and the correct one is i D µ Z i ∂ i K + c . c . − ˆ Q µ = gA Λ µ (cid:61) m λ Λ , (B.62)and only vanishes when the isometries that have been gauged leave the K¨ahler potentialexactly invariant, i.e. for λ Λ = 0.It should be evident that D i L is also an invariant field, and therefore the part of the N = 1, d = 4 Supergravity potential that depends on the superpotential − |L| + 8 G ij ∗ D i LD j ∗ L ∗ (B.63)is automatically exactly invariant. On the other hand eq. (B.48) proves that the momentummap itself is an invariant field. Then, δ α P Λ = − α Σ ( x ) f ΣΛΩ P Ω , D µ P Λ = ∂ µ P Λ + gf ΛΣΩ A Σ µ P Ω , D µ P Λ = D µ Z i ∂ i P Λ + D µ Z ∗ i ∗ ∂ i ∗ P Λ , (B.64)and the part of the N = 1 , d = 4 Supergravity potential that depends on it, i.e. + g ( (cid:61) m f ) − | ΛΣ P Λ P Σ , (B.65)is also automatically invariant.Finally, let us consider the spinors of the theory. They have a non-vanishing K¨ahler weightwhich is ( − / , /
2) times their chirality. For instance, the gravitino of N = 1, d = 4 theoriestransform as δ α ψ µ = − α Λ ( x )( λ Λ − λ ∗ Λ ) ψ µ , D µ ψ ν = {∇ µ + i ˆ Q} ψ ν . (B.66) .2. SPECIAL GEOMETRY B.2 Special geometry
In this short appendix we shall discuss the geometric structure underlying the couplings ofvector supermultiplets in N = 2 d = 4 Supergravity, which has received the name of SpecialK¨ahler geometry (usually just called Special geometry). The first articles introducing thisstructure were [149, 80] and the formalisation was given in [151]. The essential references are[152 - 157]. After discussing the coordinate independent formulation of special geometry, wemake contact with the original formulation in terms of the prepotential in B.2.1. AppendixB.2.2 discusses the gauging of isometries, and how this is used in order to construct gaugedsupergravities.The formal starting point for the definition of a Special K¨ahler manifold lies in the def-inition of a K¨ahler-Hodge manifold. As explained in the section above, a KH-manifold is acomplex line bundle over a K¨ahler manifold M , such that the first Chern class of the linebundle equals the K¨ahler form. This then implies that the exponential of the K¨ahler potentialcan be used as a metric on the line bundle. Furthermore, the connection on the line bundleis Q = (2 i ) − ( dz i ∂ i K − d ¯ z ¯ ı ∂ ¯ i K ). Let us denote the line bundle by L → M , where thesuperscript is there to indicate that the covariant derivative is D = ∇ + i Q Consider then a flat 2¯ n vector bundle E → M with structure group Sp (¯ n ; R ), and take asection V of the product bundle E ⊗ L → M and its complex conjugate V , which is a sectionof the bundle E ⊗ L − → M . A special K¨ahler manifold is then a bundle E ⊗ L → M , forwhich there exists a section V such that V = (cid:18) L Λ M Λ (cid:19) → (cid:104)V | V(cid:105) ≡ L Λ M Λ − L Λ M Λ = − i , D ¯ ı V = 0 , (cid:104) D i V | V(cid:105) = 0 . (B.67)By defining the objects U i ≡ D i V = (cid:18) f Λ i h Λ i (cid:19) , U ¯ ı = U i , (B.68)it follows from the basic definitions that D ¯ ı U i = G i ¯ ı V , (cid:104)U i | U ¯ ı (cid:105) = i G i ¯ ı , (cid:104)U i | V(cid:105) = 0 , (cid:104)U i | V(cid:105) = 0 . (B.69)Let us now focus on (cid:104) D i U j | V(cid:105) = −(cid:104) U j | U i (cid:105) , where we have made use of the third constraint.As one can see the r.h.s. is antisymmetric in i and j , whereas the l.h.s. is symmetric. Thisthen means that (cid:104) D i U j | V(cid:105) = (cid:104)U j | U i (cid:105) = 0. The importance of this last equation is that ifwe group together E Λ = ( V , U i ), one can see that (cid:104)E Σ | E Λ (cid:105) is a non-degenerate matrix, whichallows one to construct an identity operator for the symplectic indices, such that for a givensection A ∈
Γ ( E, M ) we have A = i (cid:104)A | V(cid:105) V − i (cid:104)A | V(cid:105) V + i (cid:104)A | U i (cid:105)G i ¯ ı U ¯ ı − i (cid:104)A | U ¯ ı (cid:105)G i ¯ ı U i . (B.70) For a complete review refer to [29]. APPENDIX B. SCALAR MANIFOLDS
Furthermore, the inner product with V and U ¯ ı vanishes due to the basic properties. Let usdefine the weight (2 , −
2) object C ijk ≡ (cid:104) D i U j | U k (cid:105) → D i U j = i C ijk G k ¯ l U ¯ l , (B.71)where the last equation is a consequence of eq. (B.70). Since the U are orthogonal, one cansee that C is completely symmetric in its three indices, and one can show that D ¯ ı C jkl = 0 , D [ i C j ] kl = 0 . (B.72)Let us then introduce the concept of a monodromy matrix N , which can be defined throughthe relations M Λ = N ΛΣ L Σ , h Λ i = N ΛΣ f Σ i . (B.73)The relations of (cid:104)U i | V(cid:105) = 0 then imply that N is a symmetric matrix, which hence auto-matically trivialises (cid:104)U i | U j (cid:105) = 0.Observe that as Im ( N ΛΣ ) ≡ Im ( N ) ΛΣ appears in the kinetic term of the (¯ n = n V + 1)vector fields it has to be negative definite, whence also invertible, in order for the kinetic termto be well-defined. One can see that this is implied by the properties of special geometry[80]. As it is invertible, we can use it as a ‘metric’ for raising and lowering Λ-indices, e.g. L Λ ≡ Im ( N ) − | ΛΣ L Σ . Likewise, we shall use G i ¯ to raise and lower K¨ahler-indices. Moreover,from the other basic properties in (B.69) we find L Λ L Λ = − , L Λ f Λ i = 0 , f Λ i ¯ f Λ¯ = − G i ¯ . (B.74)An important identity that one can derive, is given by U ΛΣ ≡ f Λ i G i ¯ ı ¯ f Σ¯ ı = − Im( N ) − | ΛΣ − L Λ L Σ , (B.75)so that U ΛΣ = U ΣΛ .Let us construct the ( n V + 1) × ( n V + 1)-matrices M = ( M Λ , ¯ h Λ ¯ ı ) and L = ( L Λ , ¯ f Λ¯ ı ).With them we can write the defining relations for the monodromy matrix as M ΛΣ = N ΛΩ L ΩΣ ,a system which can be easily solved by putting N = M L − , where L − is the inverse of L .Formally, one finds L − = − (cid:18) L Λ f ¯ ı Λ (cid:19) , (B.76)which is a recursive argument, but useful to derive ∂ ¯ ı N ΛΣ = − i (cid:0) ¯ f Λ¯ ı L Σ + L Λ ¯ f Σ¯ ı (cid:1) (B.77)and ∂ ¯ ı N ΛΣ = 4 C ¯ ı ¯ ¯ k f ¯ Λ f ¯ k Σ . (B.78) .2. SPECIAL GEOMETRY B.2.1 Prepotential: Existence and more formulae
In explicit constructions of the models it is worthwhile to introduce the explicitly holomorphicsection Ω = e −K / V , which allows us to rewrite the system (B.67) asΩ = (cid:18) X Λ F Σ (cid:19) → (cid:104) Ω | Ω (cid:105) ≡ X Λ F Λ − X Λ F Λ = − i e −K ,∂ ¯ ı Ω = 0 , (cid:104) ∂ i Ω | Ω (cid:105) = 0 . (B.79)If we now assume that F Λ depends on Z i through the X ’s, then the last equation aboveimplies that ∂ i X Λ (cid:2) F Λ − ∂ Λ (cid:0) X Σ F Σ (cid:1)(cid:3) = 0 . (B.80)If ∂ i X Λ is invertible as a n V × ( n V + 1) matrix, then we must conclude that F Λ = ∂ Λ F ( X ) , (B.81)where F is a homogeneous function of degree 2, baptised in the literature as the prepotential .Should ∂ i X Λ not be invertible, then, as shown in [153], one can do a symplectic transformationsuch that a prepotential exists.Making use of the prepotential and the definitions (B.73), we then calculate N ΛΣ = F ΛΣ + 2 i Im( F ) ΛΛ (cid:48) X Λ (cid:48) Im( F ) ΣΣ (cid:48) X Σ (cid:48) X Ω Im( F ) ΩΩ (cid:48) X Ω (cid:48) , (B.82)which is manifestly symmetric. From the above expression we can obtain the sometimes usefulresult Im ( N ) − | ΛΣ = − F − | ΛΣ − L Λ L Σ − L Λ L Σ , (B.83)where F − is the inverse of F ΛΣ ≡ Im ( F ΛΣ ). Also, having the explicit form of N we canderive an explicit representation for CC ijk = e K ∂ i X Λ ∂ j X Σ ∂ k X Ω F ΛΣΩ , (B.84)so that the prepotential determines all the structures present in Special geometry. B.2.2 Gauging holomorphic Killing vectors in Special geometry
We are now interested in holomorphic Killing vectors associated to the K¨ahler manifold withmetric G . This is relevant e.g. in chapter 3, where we have considered gauged vector fields infour-dimensional fSUGRA. Consider the real Killing vector K = K i ( Z ) ∂ i + ¯ K ¯ ı ( Z ) ∂ ¯ ı −→ £ K G = 0 . (B.85)For ease of treatment, it is customary to put all the vectors fields and the graviphoton (whichis inside the gravity multiplet) on the same footing. This means that we shall be labelling theKilling vectors by an index like Λ, which runs from 1 to ¯ n (= n V + 1). One then imposes anadditional constrain bringing the number of vectors back to n V . In five-dimensional theory,for example, this is done by eq. (B.141). Here we shall be using one of the fields to gauge28 APPENDIX B. SCALAR MANIFOLDS the R -symmetry, and thus we can at most use at most n V vectors to gauge isometries. Ingeneral, these Killing vectors define a non-Abelian algebra, which we take to be[ K Λ , K Σ ] = − f ΛΣΓ K Γ . (B.86)These isometries need not leave invariant the K¨ahler potential, but only up to a K¨ahlertransformation, i.e. £ Λ K ≡ K Λ K = λ Λ ( Z ) + λ Λ ( Z ) , (B.87)where we employ the notation £ Λ = £ K Λ . It is clear that the K¨ahler transformation param-eters λ have to form a representation under the group that we are gauging, and in fact onesees that £ Λ λ Σ − £ Σ λ Λ = − f ΛΣΩ λ Ω . (B.88)If we also assume that the Killing vectors are compatible with the complex structure J definedon the K¨ahler manifold, and therefore also with the K¨ahler form K ( X, Y ) ∼ G ( J X, Y ), wecan derive, analogously to eq. (B.43) above £ Λ K = d ( ı Λ K ) −→ π ı Λ K = d P Λ , (B.89)where the object P Λ is called the momentum map associated to K Λ . A closed form for themomentum map can be easily seen to be i P Λ = (cid:0) K i Λ ∂ i K − K ¯ ı Λ ∂ ¯ ı K − λ Λ + λ Λ (cid:1) = K i Λ ∂ i K − λ Λ , (B.90)where we made use of eq. (B.87) and fixed a possible constant to be zero. Using this formand eq. (B.88), it is straightforward to show that £ Λ P Σ = − f ΛΣΩ P Ω , (B.91)The action of the Killing vector on the symplectic section is most easily described on the(1 , £ Λ Ω = S Λ Ω − λ Λ Ω , (B.92)where S ∈ sp (¯ n ; R ) and forms a representation of the algebra we are gauging, i.e. [ S Λ , S Σ ] = f ΛΣΓ S Γ . The natural spacetime (not the K¨ahler) connection that acts on this symplecticsection is D Ω = (cid:0) ∇ + ∂Z i ∂ i K + i g A Λ P Λ + g A Λ S Λ (cid:1) Ω , (B.93)which is constructed in such a way that δ α D Ω = α Λ ( S Λ − λ Λ ) D Ω. From the above equation,it is a small calculation to derive the covariant derivative on objects such as V or V . In fact,one can see that, if dealing with a symplectic ( p, q )-weight object, one has δ α Φ ( p,q ) = α Λ (cid:0) S Λ − p λ Λ − q ¯ λ Λ (cid:1) Φ ( p,q ) , (B.94) D Φ ( p,q ) = (cid:104) ∇ + p ∂Z i ∂ i K + q ∂Z ¯ ı ∂ ¯ ı K + i ( p − q ) g A Λ P Λ + g A Λ S Λ (cid:105) Φ ( p,q ) , (B.95) δ α D Φ ( p,q ) = α Λ (cid:0) S Λ − p λ Λ − q ¯ λ Λ (cid:1) D Φ ( p,q ) . (B.96) .3. HYPER-K ¨AHLER AND QUATERNIONIC-K ¨AHLER GEOMETRY K i U i = ( S K + i P K ) V −→ D V = D Z i U i , (B.97)which in turn can be used to obtain D U i = D Z j D j U i + D Z ¯ D ¯ U i and D N = D Z i ∂ i N + D Z ¯ ı ∂ ¯ ı N . (B.98)Equation (B.97) allows us to write down the following identities0 = (cid:104)V | S Λ V(cid:105) , P Λ = (cid:104)V | S Λ V(cid:105) , K Λ¯ ı = i (cid:104)U ¯ ı | S Λ V(cid:105) , (cid:104)U i | S Λ V(cid:105) . (B.99)As done in [81], we consider only a subset the possible gaugings; we restrict to groupswhose embedding into sp (¯ n ; R ) is given by S Λ = (cid:18) [ S Λ ] Σ Ω − [ S Λ ] Σ Ω (cid:19) = (cid:18) f ΛΩΣ − f ΛΣΩ (cid:19) . (B.100)With this restriction on the gaugeable symmetries, we can then derive the following importantidentity 0 = L Λ K i Λ . (B.101)Further identities that follow are L Λ P Λ = 0 , L Λ λ Λ = 0 , ¯ f Λ i P Λ = i L Λ K i Λ . (B.102) B.3 Hyper-K¨ahler and quaternionic-K¨ahler geometry
Quaternionic spaces arise naturally in Supergravity in the context of N = 2, d = 4 theorieswith hyperscalar multiplets [146]. Although these do not make an explicit appearance inthis work, they lie at the heart of the Fayet-Iliopoulos terms (which are used to obtain apositive cosmological constant in fSUGRA), and hence will be briefly discussed. AppendixB.3.1 discusses the gauging of isometries in quaternionic-K¨ahler spaces, and section B.3.2gives a short description of the rˆole of the FI terms in fSUGRA.A quaternionic-K¨ahler manifold is a real 4 m -dimensional Riemannian manifold HM en-dowed with a triplet of complex structures J x : T ( HM ) → T ( HM ) ( x = 1 , ,
3) that satisfythe quaternionic algebra J x J y = − δ xy + ε xyz J z , (B.103)and with respect to which the metric, denoted by H , is Hermitean H ( J x X, J x Y ) = H ( X, Y ) , ∀ X, Y ∈ T ( HM ) . (B.104)This implies the existence of a triplet of 2-forms K x ( X, Y ) ≡ H ( X, J x Y ) globally knownas the su (2)-valued hyper-K¨ahler 2-forms . Observe that the foregoing definition on a realcoordinate base means K xuv = H uw ( J x ) wv ≡ ( J x ) uv . Most of the time we shall use an so (3)-valued notation in order not to have the x -indices floating around; this implies writing e.g. K = K x T x where the generators T x satisfy [ T x , T y ] = ε xyz T z .30 APPENDIX B. SCALAR MANIFOLDS
The structure of a quaternionic-K¨ahler manifold requires an SU (2) bundle to be con-structed over HM with connection 1-form A x , with respect to which the hyper-K¨ahler 2-formis covariantly closed, i.e. D X K ≡ ∇ X K + [ A X , K ] = 0 . (B.105)Then, depending on whether the curvature of this bundle F ≡ d A + A ∧ A −→ F x ≡ d A x + ε xyz A y ∧ A z , (B.106)is zero or proportional to the hyper-K¨ahler 2-form , i.e. F = κ K , κ ∈ R / { } , (B.107)the manifold is a hyper-K¨ahler manifold or a quaternionic-K¨ahler manifold, respectively.The SU (2)-connection acts on objects with vectorial SU (2)indices, such as chiral spinors,as follows D ξ I ≡ dξ I + A I J ξ J , F I J = d A I J + A I K ∧ A K J , D χ I ≡ dχ I + B I J χ J , G I J = d B I J + B I K ∧ B K J . (B.109)Consistency with the raising and lowering of vector SU (2) then indices implies that B I J = − A I J ≡ − ε IK A K L ε LJ , (B.110)whereas compatibility with the raising of indices due to complex conjugation implies B I J = ( A I J ) ∗ . (B.111)These two things together thus means that A I J is an anti-Hermitean matrix, whence weexpand A I J = i A x ( σ x ) I J and B I J = − i A x ( σ x ) I J , (B.112)where the indices of the σ -matrices are raised/ lowered with (cid:15) .At this point, there remains a question about the normalisation of the Pauli matrices,which is readily fixed by imposing that F I J = i F x ( σ x ) I J , (B.113)which implies that ( σ x σ y ) I J = δ xy δ I J − iε xyz ( σ z ) I J . (B.114) Note that in constructions of SUGRA it is the additional constraint given by having SUSY, not the actualgeometry, which fixes the value of κ to κ = − The convention for raising and lowering of SU (2) indices is given by χ I = χ J (cid:15) JI , ψ I = (cid:15) IJ ψ J . (B.108) .3. HYPER-K ¨AHLER AND QUATERNIONIC-K ¨AHLER GEOMETRY σ -matrices that fullfill the above definingrelations ( σ x ) I J : (cid:18) (cid:19) , (cid:18) i − i (cid:19) , (cid:18) − (cid:19) , ( σ x ) I J : (cid:18) (cid:19) , (cid:18) − ii (cid:19) , (cid:18) − (cid:19) , ( σ x ) IJ : (cid:18) − (cid:19) , (cid:18) i i (cid:19) , (cid:18) − − (cid:19) , ( σ x ) IJ : (cid:18) − (cid:19) , (cid:18) i i (cid:19) , (cid:18) (cid:19) , (B.115)where the x = 1 , , δ xy = − σ xIJ σ yIJ , δ I ( K δ J L ) = − σ xIJ σ xKL . (B.116)It is convenient to use a Vielbein on HM having as flat indices a pair ( α, I ) consisting ofan SU (2) index I = 1 , Sp ( m ) index α = 1 , . . . , m U αI = U αI u dq u , (B.117)where q u ( u = 1 , . . . , m ) are real coordinates on HM . We shall refer to this Vielbein U αI as the Quadbein. The Quadbein is related to the metric H uv by H uv = U αI u U βJ v ε IJ C αβ , (B.118)where C is a real antisymmetric 2 m × m matrix and it is in fact the metric for the Sp ( m )group. Furthermore one requires that, in concordance with our rules of raising and loweringindices, U αI ≡ ε IJ C αβ U βJ = (cid:0) U αI (cid:1) (cid:63) . (B.119)The Quadbein thus satisfies all the usual relations that a Vielbein satisfies.In thise sense, the Quadbein satisfies a Vielbein postulate , i.e. they are covariantly constantwith respect to the standard Levi-Civit`a connection Γ uvw , the SU (2)-connection and the Sp ( m )-connection ∆ uαβ :0 = D u U αI v = ∇ u U αI v + B uI J U αJ v + ∆ uαβ U βI v C βγ = 0 . (B.120)This postulate relates the three connections and the respective curvatures, leading to thestatement that the holonomy of a quaternionic-K¨ahler manifold is contained in Sp (1) · Sp ( m ), i.e. R tsuv U αI u U βJ v = − G IJts C αβ − R αβts ε IJ = F IJts C αβ − R αβts ε IJ , (B.121)where we have defined the Sp ( m )-curvature R as R αβ ≡ d ∆ αβ + ∆ αγ ∧ ∆ γβ . (B.122)32 APPENDIX B. SCALAR MANIFOLDS
It is clear that on R m we can define a quaternionic structure which is covariantly constant.In this case, the fact that the Quadbein is covariantly constant means that on a quaternionic-K¨ahler space we can induce a covariantly constant quaternionic structure by inducing the onefrom the tangent space. In fact, this means that we have K xuv = − i σ xIJ C αβ U αIu U βJv . (B.123)Using the above expression for the quaternionic-K¨ahler forms, we can then obtain the identity U αIu C αβ U βJv = H uv (cid:15) IJ − i K xuv σ x IJ . (B.124)As a result of this and eq. (B.107), we can derive the identity F uv I J = κ U αI [ u U αJv ] , (B.125)where once again we would like to point out that Supergravity fixes κ = − B.3.1 Gauging isometries in quaternionic-K¨ahler spaces
As in previous sections, we now discuss the possible isometries of quaternionic-K¨ahler spaces.W.r.t. the real coordinates q u on HM , the Killing vectors are given by K Λ = K u Λ ∂ u , such that £ Λ H = 0 . (B.126)We shall consider fields on HM that transform in the adjoint representation of SO (3). Anexample of such objects is K . Calling such a generic field Ψ = Ψ x T x , it transforms under so (3) as δ λ Ψ = − [ λ, Ψ] , (B.127)where λ is an so (3)-valued transformation parameter. Of course, a covariant derivative iseasily introduced by putting D X Ψ = ∇ X Ψ + [ A X , Ψ] so long as δ λ A = D λ . (B.128)We define an SO (3)-covariant Lie derivative, by postulating L Λ Ψ = £ Λ Ψ + [ W Λ , Ψ] , which must satisfy (cid:26) δ λ L Λ Ψ = − [ λ , L Λ Ψ] , [ L Λ , L Σ ] = − f ΛΣΩ L Ω . (B.129)The last rule is nothing but the usual commutation relations for Lie derivatives, but wherewe have used eq. (B.86) to define the commutation relations for the Killing vectors. The firstconstraint implies δ λ W Λ = L Λ λ , whereas the second implies £ Λ W Σ − £ Σ W Λ + [ W Λ , W Σ ] = − f ΛΣΩ W Ω . (B.130)We go on to introduce the notion of momentum map P Λ by defining W Λ = ı Λ A − P Λ . (B.131)Substituting the above definition into eq. (B.130), one can see that the momentum map hasto satisfy D Λ P Σ − D Σ P Λ − [ P Λ , P Σ ] + κ ı Λ ı Σ K = − f ΛΣΩ P Ω , (B.132) .3. HYPER-K ¨AHLER AND QUATERNIONIC-K ¨AHLER GEOMETRY D Λ = K u Λ D u .So far we have discussed the Killing vectors and their transformations, and we shall nowconsider their compatibility with the complex structures. This means imposing L Λ K = 0 −→ D ( ı Λ K ) = [ P Λ , K ] . (B.133)The integrability condition for the above equation can be massaged to give DP Λ = − κ ı Λ K , (B.134)which, in view of the similarity with the result in eq. (B.89), justifies the use of the name oftri-holomorphic map for P Λ . The above definition implies[ P Λ , P Σ ] + κ ı Λ ı Σ K = f ΛΣΩ P Ω , (B.135)where eq. (B.132) has been used. Another implication is that the tri-holomorphic map is aninvariant field, i.e. that its covariant Lie derivative is zero0 = L Λ P Σ = £ Λ P Σ + [ W Λ , P Σ ] + f ΛΣΩ P Ω , (B.136)where the derivative includes SO (3) and G terms. B.3.2 A small discussion of the FI terms
This subsection discusses the relevance of hyperscalar multiplets in fakeSupergravity, evenfor theories that do not explicitly contain them. This is because the Wick rotation of the FIterm lies behind the positivity of the cosmological constant (minus sign in the action), givingrise to fSUGRA.Consider the case in which there are no hyperscalars. Eq. (B.135) can be written incomponents as ε xyz P x Λ P y Σ = f ΛΣΩ P z Ω . (B.137)This equation allows for two different solutions, namely U (1) and SU (2). • If we take the gauge group to be Abelian, i.e. f ΛΣΩ = 0, the P x Λ will be given by su (2)-valued tri-holomorphic momentum maps that commute. Thus without loss of generalitywe can take them to be P Λ = P x Λ T x = ξ Λ T ; this expression for the tri-holomorphicmomentum maps is called the U (1) FI term. • If the gauge group is chosen to be SU (2), then f ΛΣΩ = ε ΛΣΩ . In this case a solution toeq. (B.137) is given by P x Λ = δ x Λ and it is called the SU (2) FI term. We shall ignore the SU (2) FI term in the main text, as it induces non-Abelian terms that are difficult towork with.The U (1) FI term is paramount in constructing fSUGRA. Consider e.g. minimal N = 2 d = 4 SUGRA; this theory has ¯ n = 1, n v = 0 and it is defined by a prepotential that reads F = − i X . Including the U (1) FI term into the mix, we find a supersymmetric action thatis S = (cid:90) √ g (cid:2) R − F + ξ (cid:3) , (B.138) The coupling constant g that usually appears in supersymmetric actions, see e.g. eqs. (3.16), (3.15) insection 3.1, has been absorved into the FI term and also into the structure constants, even if this is not visiblehere. This is desirable as this absorption allows for different coupling constants (hence multiple gauge groups),a fact which is not obvious when having only g . APPENDIX B. SCALAR MANIFOLDS which is the action governing Einstein-Maxwell-anti De Sitter. In Supergravity lingo, thistheory is known as minimal gauged N = 2 d = 4 SUGRA.In order for the above action to describe an Einstein-Maxwell-De Sitter theory, the cos-mological constant has to change sign, which can be done by Wick rotating the FI term ξ → iC ; this leads to S = (cid:90) √ g (cid:2) R − F − C (cid:3) . (B.139)If we reinterpret this Wick rotation from a group theory perspective, we thus have P = ξ T → iC T = C ˜ T , (B.140)where in the last step Weyl’s unitarity trick has been used to introduce a non-compact gener-ator ˜ T . One can then see that after a Wick rotation we are no longer gauging a U (1) group,but rather R . B.4 Real Special K¨ahler geometry
This appendix contains some useful information on
Real Special K¨ahler geometry (usually justreferred to as Real Special geometry, or even Very Special geometry, following the originalarticle [150]). This becomes relevant when considering five-dimensional SUGRA with gaugevector fields, and thus applicable to chapter 5.We consider a theory containing n vector multiplets. As commented briefly above, thegeometry of the n physical scalars φ x ( x = 1 , . . . , n ) in these multiplets is fully determinedby a constant real symmetric tensor C IJK ( I, J, K = 0 , , . . . , ¯ n ≡ n + 1). The scalars thusappear through ¯ n functions h I ( φ ) constrained to satisfy C IJK h I h J h K = 1 . (B.141)One defines h I ≡ C IJK h J h K → h I h I = 1 , (B.142)and a metric a IJ that can be use to raise and lower the SO (¯ n ) index h I ≡ a IJ h J , h I ≡ a IJ h J . (B.143)The definition of h I allows one to find a IJ = − C IJK h K + 3 h I h J . (B.144)Next, one defines h Ix ≡ −√ h I ,x ≡ −√ ∂h I ∂φ x , (B.145)along with h Ix ≡ a IJ h Jx = + √ h I,x , (B.146)which satisfy h I h Ix = 0 , h I h Ix = 0 , (B.147) .4. REAL SPECIAL GEOMETRY h I enjoy the following properties of closure and orthogonality (cid:18) h I h Ix (cid:19) (cid:0) h I h yI (cid:1) = (cid:18) δ yx (cid:19) , (cid:0) h I h xI (cid:1) (cid:18) h J h Jx (cid:19) = δ JI . (B.148)Therefore any object with SO (¯ n ) index can be decomposed as A I = (cid:0) h J A J (cid:1) h I + (cid:0) h xJ A J (cid:1) h Ix . (B.149)The metric on the scalar manifold, g xy ( φ ), is the pullback of a IJ g xy = a IJ h Ix h Jy = − C IJK h Ix h Jy h K , (B.150)and can be used to raise/ lower { x, y } indices. Other useful expressions are a IJ = h I h J + h xI h Jx ,C IJK h K = h I h J − h xI h Jx , h I h J = a IJ + C IJK h K ,h xI h Jx = a IJ − C IJK h K . (B.151)We also introduce the Levi-Civit`a covariant derivative associated to such a metric g xy h Ix ; y ≡ h Ix,y − Γ xyz h Iz . (B.152)One can show that h Ix ; y = √ ( h I g xy + T xyz h zI ) , (B.153) h Ix ; y = − √ ( h I g xy + T xyz h Iz ) , (B.154)Γ xyz = h Iz h Ix,y − √ T xyw = 8 h zI h Ix,y + √ T xyw , (B.155)for T xyz = √ h Ix ; y h Iz = −√ h Ix h Iy ; z . (B.156)36 APPENDIX B. SCALAR MANIFOLDS ppendix C
Geometrical data for nul l case solutions
Here we give some explicit geometric information which is of relevance in the classifica-tion of null case solutions. Appendix C.1 gives the spin connection and curvatures for thefour-dimensional case, and hence applied in section 3.3. Appendix C.2 contains the five-dimensional information, which is used in chapter 4. A short scholium on the Kundt wavemetric is given in appendix C.3, since this background appears repeteadly as solution to thenull class.
C.1 Spin connection and curvatures in d = 4 fSUGRA Let us set-up a null-Vierbein by ds null = e + ⊗ e − + e − ⊗ e + − e • ⊗ e ¯ • − e ¯ • ⊗ e • , (C.1)and choose e + = L = du , θ + = N (cid:91) = ∂ u − H∂ v ,e − = N = dv + Hdu + (cid:36)dz + (cid:36)d ¯ z , θ − = L (cid:91) = ∂ v ,e • = M = e U dz , θ • = − M (cid:91) = e − U [ ∂ z − (cid:36)∂ v ] ,e ¯ • = M = e U d ¯ z , θ ¯ • = − M (cid:91) = e − U [ ∂ ¯ z − (cid:36)∂ v ] , (C.2)where, conforming to the results of eq. (3.75), only H = H ( u, v, z, ¯ z ) and U and the (cid:36) sdepend on u , z and ¯ z .The non-vanishing components of the spin connection can be seen to be ω + − = − θ − H e + , (C.3) ω + • = (cid:0) e − U θ + (cid:36) − θ • H (cid:1) e + − (cid:2) θ + U + e − U ( ∂ z (cid:36) − ∂ ¯ z (cid:36) ) (cid:3) e ¯ • , (C.4) ω +¯ • = (cid:0) e − U θ + (cid:36) − θ ¯ • H (cid:1) e + − (cid:2) θ + U − e − U ( ∂ z (cid:36) − ∂ ¯ z (cid:36) ) (cid:3) e • , (C.5) ω • ¯ • = e − U ( ∂ z (cid:36) − ∂ ¯ z (cid:36) ) e + − e • θ • U + e ¯ • θ ¯ • U . (C.6) We define the directional derivatives θ a to be the duals of the frame 1-forms E a , i.e. normalised such that E a ( θ b ) = δ ab . We reserve the notation ∂ x for the directional derivative on the base-space, namely ∂ x ≡ e xm ∂ m . APPENDIX C. GEOMETRICAL DATA FOR NULL CASE SOLUTIONS
A further calculation leads to the Ricci tensor, whose non-vanishing coefficients are R + − = − θ − H , (C.7) R • ¯ • = 2 e − U ∂ z ∂ ¯ z U , (C.8) R + • = e − U θ + ∂ z U − θ • θ − H + θ • (cid:0) e − U [ ∂ z (cid:36) − ∂ ¯ z (cid:36) ] (cid:1) , (C.9) R +¯ • = R + • , (C.10) R ++ = 2 e − U θ e U + 2 θ − H θ + U + e − U ( ∂ z (cid:36) − ∂ ¯ z (cid:36) ) − e − U θ • (cid:2) e U θ ¯ • H (cid:3) − e − U θ ¯ • (cid:2) e U θ • H (cid:3) + e − U ∂ u ( ∂ z (cid:36) + ∂ ¯ z (cid:36) ) . (C.11)Observe that the last term in eq. (C.11) can always be put to zero by the coordinate trans-formation v −→ v + ρ ( u, z, ¯ z ). C.2 Spin connection and curvatures in minimal d = 5 fSUGRA Defining the spin connection Ω ab by means of dE a = Ω ab ∧ E b and imposing it to be metriccompatible Ω ( ab ) = 0 leads toΩ + − = − θ − H E + − θ − ω x E x , (C.12)Ω + x = − ( θ x H − E mx θ + ω m ) E + + θ − ω x E − − (cid:104) E [ ym θ x ] ω m + E m ( y θ + E x ) m (cid:105) E y , (C.13)Ω − x = θ − ω x E + , (C.14)Ω xy = − λ zxy E z − (cid:104) E m [ x θ y ] ω m − E m [ x θ + E y ] m (cid:105) E + , (C.15)where we have defined ð E x = λ xy ∧ E y and λ zy = δ zx λ xy , whereas Ω xy = η xz Ω zy , so that thesign difference is paramount. Observe that a similar condition holds for defining E mx = E mx .Of course, λ is fixed by eq. (4.40) to be λ xy = 2 ξ ℵ x E y − ξ ℵ y E x + √ ξ ε xyz E z . (C.16)As stated above, if ( g, A ) solves the fKSE one only needs to demand M + = 0 and E ++ = R ++ + (cid:37) x (cid:37) x = 0 in order to ensure that ( g, A ) solves all the equations of motion. Wenow treat a simplified case which shows how the GT-geometry appears in the EOMs. C.2.1 The u -independent case with (cid:36) = 0 The non-vanishing components of the Ricci tensor are given by R + − = − θ − H − θ − ω x θ − ω x − ∇ x θ − ω x , (C.17) R ++ = −∇ z [ ∇ z Υ + 2 ξ ℵ z Υ ] − ξ v ℵ x ℵ x , (C.18) R + x = 4 ξ v ℵ x , (C.19) R xy = R ( λ ) xy + 2 ξ ℵ x ℵ y − ξ ∇ ( x ℵ y ) , (C.20)where R ( λ ) is the Ricci tensor for the three-dimensional spin connection λ . .3. KUNDT METRICS E ++ = −∇ z [ ∇ z Υ + 2 ξ ℵ z Υ ] , (C.21) E xy = R ( λ ) xy − ξ ∇ ( x ℵ y ) − ξ ℵ x ℵ y + 4 ξ ℵ z ℵ z δ xy + 6 ξ δ xy . (C.22)Comparing the last equation with the symmetric part of the Ricci tensor for the Weyl con-nection in eq. (D.4) for d = 3, and taking into account that we are dealing with a Gauduchonmetric, one can see that upon identifying θ = 2 ξ ℵ we can rewrite eq. (C.22) as E xy = W ( xy ) + 6 ξ δ xy . (C.23)Comparing this and eq. (4.25) to the results in appendix D.1, we see that the three-dimensionalmanifold is a Gauduchon-Tod space with κ = 2 √ ξ . C.3 Kundt metrics
A Kundt wave [110] is a metric that allows for a non-expanding, shear-free and twist-freegeodesic null vector N . That is, respectively, ∇ µ N µ = 0 , (C.24) ∇ ( µ N ν ) ∇ µ N ν = 0 , (C.25)ˆ N ∧ d ˆ N = 0 , (C.26) ∇ N N = 0 , (C.27)where ˆ N is the 1-form dual to the vector field. They were first studied in the arbitrary d -dimensional case in [111 - 114]. The line element can always be taken to read ds = ˆ E + ⊗ ˆ E − + ˆ E − ⊗ ˆ E + − ˆ E x ⊗ ˆ E x , (C.28)where we have generically introduced the light-cone frame by E + = du , θ + = ∂ u − H∂ v ,E − = dv + Hdu + S m dy m , θ − = ∂ v ,E x = e mx dy m , θ x = e xm ( ∂ m − S m ∂ v ) , (C.29)where the Vielbein on the base-space e xi is independent of v . The only v -dependence residesin H and ˆ S ≡ S m dy m . This is the kind of metric that appears in the characterisation of thenull cases studied above, eqs. (3.76), (4.72), (5.52), (6.42) and (6.63).Moreover, whenever ˆ S does not depend on v , it can be written in the Walker form ds = 2 du ( dv + H ( u, v, y p ) du + S m ( u, y p ) dy m ) + g mn ( u, x p ) dy m dy n , (C.30)where g mn ≡ e mx e nx . Eq. (C.30) is the general d -dimensional metric of a space with holonomycontained in Sim( d −
2) [97].40
APPENDIX C. GEOMETRICAL DATA FOR NULL CASE SOLUTIONS
Defining the spin connection ω ab ≡ E c ω c,ab by means of dE a = ω ab ∧ E b and imposing itto be metric compatible, i.e. ω ( ab ) = 0, leads to ω + − = − θ − H E + − θ − S x E x , (C.31) ω + x = − ( θ x H − e mx θ + S m ) E + + θ − S x E − − (cid:104) T yx + e m ( y θ + e x ) m (cid:105) E y , (C.32) ω − x = θ − S x E + , (C.33) ω xy = − λ zxy E z − (cid:104) T xy − e m [ x θ + e y ] m (cid:105) E + , (C.34)where we have defined d E x = λ xy ∧ E y and also λ zy = δ zx λ xy , whereas ω xy = η xz ω zy , soagain the sign difference is paramount . Furthermore, we define T xy ≡ e [ xm θ y ] S m , (C.35)which for d = 6 reads T ij = v F ij − [ D (cid:36) ] ij . (C.36)If we impose that the only u -dependency resides in H , the non-vanishing components ofthe Ricci tensor become R ++ = −∇ ( λ ) x ∂ x H + θ − H ∇ ( λ ) x S x − H ∇ ( λ ) x θ − S x + 2 S x ∂ x θ − H − θ − S x ∂ x H − S x S x θ − H , (C.37) R + − = − θ − H − θ − S x θ − S x + ∇ ( λ ) x θ − S x , (C.38) R + x = − θ x θ − H − ∇ ( λ ) y T xy + S y θ − T xy + T xy θ − S y , (C.39) R xy = R ( λ ) xy − ∇ ( λ )( x | θ − S | y ) + θ − S x θ − S y , (C.40)The Ricci scalar is then given by R = − θ − H − θ − S x θ − S x + 2 ∇ ( λ ) x θ − S x − R ( λ ) , (C.41)where R ( λ ) is the Ricci scalar curvature of λ . Observe that a similar condition holds for defining e mx = e mx . ppendix D Weyl geometry
In this appendix we present a brief introduction to Weyl geometry, which arises in the contextof theories with a scaling symmetry. Furthermore, we also give a sketch of Einstein-Weylmanifolds, and the special class of Gauduchon-Tod spaces, which appear repeatedly in thecharacterisation of solutions to fakeSupergravity theories.Weyl geometry appeared naturally in an attempt to couple gravity and electromagnetism[158]. A Weyl manifold is a manifold M of dimension d together with a conformal class [ g ]of metrics on M and a torsionless connection D , which preserves the conformal class, i.e. D g = 2 θ ⊗ g , (D.1)for a chosen representative g ∈ [ g ]. Using the above definition, we can express the connection D X Y as D µ Y ν = ∇ gµ Y ν + γ µν ρ Y ρ , with γ µν ρ = g µρ θ ν + g ν ρ θ µ − g µν θ ρ , (D.2)where ∇ g is the Levi-Civit`a connection for the chosen g ∈ [ g ]. We define the curvature ofthis connection through [ D µ , D ν ] Y ρ = − W µνρσ Y σ , using which we define the associated Riccicurvature as W µν ≡ W µρν ρ . The Ricci tensor is not symmetric and we have W [ µν ] = − d F µν , where F ≡ dθ , (D.3) W ( µν ) = R ( g ) µν − ( d − ∇ ( µ θ ν ) − ( d − θ µ θ ν − g µν ( ∇ ρ θ ρ − ( d − θ ρ θ ρ ) . (D.4)The Ricci-scalar is defined as W ≡ W ρρ , which explicitly reads W = R ( g ) − d − ∇ ρ θ ρ + ( d − d − θ ρ θ ρ . (D.5)The 1-form θ acts as gauge field gauging an R -symmetry, which is why we have been talkingabout a conformal class of metrics on M . In fact under a transformation g → e w g we havethat θ → θ + dw and W → e − w W , whereas W µνρσ and W µν are conformally-invariant. We shallcall a Weyl structure trivial/ closed if its curvature tensor is trivially zero, i.e. θ = d Λ , for Λa function.A metric g in the conformal class [ g ] is said to be standard (also referred to as Gauduchon )if it is such that d (cid:63) θ = 0 , or equivalently ∇ ρ θ ρ = 0 , (D.6)where the (cid:63) is taken w.r.t. the chosen metric g .14142 APPENDIX D. WEYL GEOMETRY
D.1 Einstein-Weyl and Gauduchon-Tod spaces
Einstein-Weyl (EW) manifolds are a special generalisation of Einstein manifolds, i.e. mani-folds (
M, g ) that satisfy R µν = k g µν , (D.7)in the context of Weyl geometry. We say a manifold of dimension d is Einstein-Weyl if thecurvature satisfies W ( µν ) = 1 d g µν W . (D.8)Gauduchon proved the existence of a standard metric on a compact EW manifold [159], andTod proved that (on compact EW manifolds) this implies that θ (cid:91) is a Killing vector of thestandard metric g [160].Einstein-Weyl geometries appear repeateadly in the study of fakeSupergravity theories, inthe form of Gauduchon-Tod spaces. These are a subclass of EW manifolds where an additionalgeometric constraint is demanded. In [84], Gauduchon and Tod studied the structure offour-dimensional hyper-Hermitian Riemannian spaces admitting a tri-holomorphic Killingvector, i.e. a Killing vector that is compatible with the three almost complex structures ofthe hyper-Hermitian space. They found that the three-dimensional base-space is determinedby a Dreibein, or orthonormal frame, E x , a 1-form θ and a real function κ satisfying dE x = θ ∧ E x − κ (cid:63) (3) E x , (D.9)where (cid:63) (3) is to be taken w.r.t. the Riemannian metric constructed out of the Dreibein.The underlying geometry imposed by the above equation is that of a specific type of three-dimensional EW space, called hyper-CR or Gauduchon-Tod (GT) space . The restriction(D.9) can equivalently be given by W = − κ , (D.10) (cid:63)dθ = dκ + κ θ . (D.11)The standard example of a GT space is the Berger sphere [84] ds = dφ + sin ( φ ) dϕ + cos ( µ ) ( dχ + cos( φ ) dϕ ) ,θ = sin( µ ) cos( µ ) ( dχ + cos( φ ) dϕ ) , (D.12)which is the unique compact Riemannian GT manifold, and can be seen as a squashed S oran SU (2) group manifold with a U (1)-invariant metric. One can easily see that the metric isGauduchon-Tod with κ = cos( µ ). Thus, in order to use it in the five-dimensional solutions ofchapters 4 and 5, it needs to be rescaled by a constant. Observe that the Jones-Tod construction implies that the three-dimensional GT space, orthogonal to ageneric Killing vector on a four-dimensional hyper-Hermitian space, is always Einstein-Weyl [107]. The sign difference between eq. (D.10) and eq. (S) in [84, prop. 5] is due to a differing definition of theRiemann tensor. .1. EINSTEIN-WEYL AND GAUDUCHON-TOD SPACES ds = dx + 4 | x + h | dzd ¯ z (1 + | z | ) , (D.13) θ = 2Re (cid:16) x + h (cid:17) dx , (D.14) κ = 2Im (cid:16) x + h (cid:17) , (D.15)where h is an arbitrary holomorphic function h = h ( z ). This was used in section 5.3.3 onchapter 5 to propose a solution to the theory. Note that the choice h = − ¯ h results in the3-sphere, and h = ¯ h leads to the flat metric on R with κ = 0.44 APPENDIX D. WEYL GEOMETRY ppendix E
Similitude group and holonomy
In this appendix we present some information on the
Similitude group, which appears in themain chapters of this thesis. We also provide with details on its relationship to the holonomygroup, and recurrent vector fields. For more information, see [161] and references therein.
E.1 Sim and ISim as subgroups of the Caroll and Poincar´egroups
The Galilean (or Galilei) group describes, in Classical Mechanics, how to transform coordi-nates which are measured in two references frames moving relative to each other, at constantspeed. Of course we now know that these transformations are only valid in a regime of lowvelocities, being superseded by Lorentz transformations in most cases of interest for ParticlePhysics. This means that the Galilean group provides a non-relativistic limit to the Poincar´egroup. The Carroll group is constructured similarly, where one chooses a different coordinateto label time (see below for details).Both the Galilean and the Carroll group are symmetry groups of a theory living on aMinkowski spacetime of (1 ,
1) dimensions more [162, 96]. Hence they are best understoodas subgroups of the Poincar´e group. The ( d + 3 d + 2) / iso (1 , d ) are translations, rotations and boosts, subject to[ M ij , M kl ] = i ( η ik M jl − η il M jk − η jk M il + η jl M ik ) , [ M ij , P k ] = i ( η ik P j − η jk P i ) , [ P i , P j ] = 0 . (E.1)However, an alternative description of these is more appropiate to describe the Galilean andCarroll subgroups. By going to the light-cone metric of Minkowski space ds = − dudv + dx i dx i for i = 2 , ..., d (E.2)we find that the elements of the Poincar´e algebra can be cast as space translations androtations P i = ∂ i , M ij = − i ( x i ∂ j − x j ∂ i ), two null translations H = ∂ u , M = ∂ v and 2 d − N = − i ( u∂ u − v∂ v ) , K i = − i ( u∂ i + x i ∂ v ) , V i = − i ( v∂ i + x i ∂ u ).The Bargmann algebra (see [163]) is obtained by considering the subalgebra of iso (1 , d )that commutes with M ; this is the reason why it is often referred to as the ‘central extension14546 APPENDIX E. SIMILITUDE GROUP AND HOLONOMY of the Galilean algebra’, and equivalent to excluding generators N and V i . It has the followingrules[ M ij , M kl ] = i ( η ik M jl − η il M jk − η jk M il + η jl M ik ) , [ M ij , P k ] = i ( η ik P j − η jk P i ) , [ M ij , K k ] = i ( η ik K j − η jk K i ) , [ M ij , H ] = 0 , [ P i , P j ] = 0 , [ P i , H ] = 0 , [ K i , K j ] = 0 , [ K i , H ] = iP i , [ K i , P j ] = iη ij M . (E.3)The Galilean group, in turn, is the Bargmann group when we mod out the action of M . Thismeans that the last commutator above can be put to zero. We shall not be using the Galileangroup in this thesis, but it has gained some popularity in the last few years because of itsrelevance in studies of condensed matter systems through string theory methods.The Carroll algebra was first introduced in [164] by performing a Wigner-˙In¨on¨u contractionon the Poincar´e algebra. This contraction is different to the one used to obtain the Bargmann,and it is obtained by looking for the subalgebra of iso (1 , d ) formed by { P i , M ij , K i , M, N } modulo N , whose full commutator rules are[ M ij , M kl ] = i ( η ik M jl − η il M jk − η jk M il + η jl M ik ) , [ M ij , P k ] = i ( η ik P j − η jk P i ) , [ M ij , K k ] = i ( η ik K j − η jk K i ) , [ M ij , N ] = 0 , [ M ij , M ] = 0 , [ P i , P j ] = 0 , [ P i , N ] = 0 , [ P i , M ] = 0 , [ K i , K j ] = 0 , [ K i , P j ] = iη ij M , [ K i , N ] = iK i , [ K i , M ] = 0 , [ N, N ] = 0 , [ N, M ] = − iM , [ M, M ] = 0 . The
Similitude group
Sim ( d −
1) is obtained as a subgroup of the Carroll group by keeping thesubset { M ij , K i , N } of the Carollian algebra. Its algebra hence has ( d − d + 2) / so (1 , d ).[ M ij , M kl ] = i ( η ik M jl − η il M jk − η jk M il + η jl M ik ) , [ M ij , N ] = 0 , [ M ij , K k ] = i ( η ik K j − η jk K i ) , [ K i , K j ] = 0 , [ N, N ] = 0 , [ K i , N ] = iK i . In fact, the group is isomorphic to the Euclidean group of R d − , augmented by homotheties(similarities) parametrised by a scaling factor, and it is the maximal proper subgroup of theLorentz group SO (1 , d ).If we add the translations operators { P i , H, M } the algebra obtained has ( d + d + 4) / iso (1 , d ). The group then formed is correspondingly labelled ISim ( d −
1) =
Sim ( d − (cid:110) R ,d [165]. For d = 3, this is the symmetry group used in VerySpecial Relativity, a theory allowing a small Lorentz violation, consistent with the observedCP violation [115]. E.2 Holonomy
Holonomy is a measure of how much parallel transportation along a closed loop on a smoothmanifold fails to preserve a certain geometrical quantity. In a nutshell, when parallel-transporting a non-scalar object around a closed loop, it will only remain constant if the .3. RECURRENCY M and a connection ∇ gives the holonomy group. In more formal language, thisgroup is described asHol x ( ∇ ) = (cid:110) P γ ∈ GL ( M x ) , s.t. γ is a closed loop based at x ∈ M (cid:111) , where P γ : M x → M x .Since we deal with problems arising in the context of supergravity theories, we shallparticularly deal with manifolds of special holonomy. These are defined as manifolds admittingthe existence of parallel spinors (w.r.t. Levi-Civit`a), and thus they naturally appear in thecontext of (fake-) supersymmetric studies. The classification of Riemannian irreducible non-symmetric simply-connected holonomy manifolds was given by Berger [166]. Moreover, Wanggave the dimensionalities of the spaces of (non-trivial) parallel spinors [70], thus establishingwhich manifolds have Ricci-flat metrics [4, 167]. Hol. group dim( M ) Associated manifold ( M, g ) dim( { (cid:15) } ) s.t. ∇ (cid:15) = 0 SO ( n ) n ≥ U ( n ) 2 n generic K¨ahler 0 SU ( n ) 2 n special K¨ahler (CY) 2 Sp ( n ) · Sp (1) 4 n quaternionic-K¨ahler 0 Sp ( n ) 4 n hyper-K¨ahler n + 1 G Spin (7) 8 exceptional holonomy 1Table E.1: Berger’s classification of possible holonomy groups for irreducible non-symmetricsimply-connected Riemannian manifolds. Notice how only the third, fifth, sixth and seventhcases are Ricci-flat, thus composing the list of special holonomy manifolds.In this thesis, however, we are interested in Lorentzian spaces, and holonomy is one ofthose features where Riemannian and pseudo-Riemannian geometry take on a very differentform. The interested reader can consult [166, 168, 169] for further information on this topic.We should also note that, since we consider connections that respect lengths, the maximalpossible holonomy group is SO (1 , d ). Given that Sim ( d −
1) is its maximal proper subgroup,this means that the minimal holonomy reduction that can occur in a Lorentzian spacetime M ,d is Hol M = Sim ( d − Sim ( d −
1) and recurrent vector fields.
E.3 Recurrency
We say that a vector field n µ is recurrent if ∇ µ n ν = B µ n ν , (E.4)where B µ is called the recurrence one-form [96]. Geometrically this means that n µ does notchange direction under parallel-transportation. When on top of that n µ is null, i.e. n µ n µ = 0,this implies that the connection has holonomy inside Sim ( d − APPENDIX E. SIMILITUDE GROUP AND HOLONOMY
Considering the musical isomorphism that takes n µ → (cid:91) n ν = g νµ n µ , one can show that dn (cid:91) = B ∧ n (cid:91) , which implies n (cid:91) = du → dn (cid:91) = 0 → ∇ µ n ν = ξn µ n ν , for some function ξ . This last equation says that n µ generates a geodesic null congruence thatis hypersurface orthogonal, non-expanding and shear-free. We parametrise this congruenceby v , and hence n = n µ ∂ µ = ∂∂v .The components of n and n (cid:91) allow us to write explictly the metric in Walker form [97] ds = H ( u, v, x ) dudu + 2 dudv + 2 A i ( u, x ) dudx i + g ij ( u, x ) dx i dx j , (E.5)where we have taken the function u , parameter v and the ( d −
1) transverse x i as coordinates.One can then use Ricci identity to obtain R µνρσ n σ = ( dB ) µν n ρ . (E.6)If we interpret the curvature tensor R µνρσ as a Lorentz algebra-valued two-form, eq. (E.6)means that the rotation performed on n for a given loop γ = ( X, Y ) is related to the fieldstrength of the recurrence one-form, i.e. ( O ( X,Y ) ◦ n ) ρ = F ( B ) ( X, Y ) n ρ . (E.7)It is an easy matter to see that R µν + i = 0. This implies that ( R γ ) + i = 0 and hence that thelocal holonomy group is generated by ( d −
1) generators less than those of SO(1 , d ). In otherwords, the maximal holonomy group is
Sim ( d − H = H ( u, x ), i.e. not depending on v , eq. E.5 becomes a Brinkmann wave[170 - 172]. This means that ∇ µ n ν = 0 and hence n is a (null) Killing vector field. In this casethe maximal holonomy group is not Sim ( d −
1) but rather a subgroup of it, the Euclideangroup of R d − , where one further generator is no longer needed, since now R µν + − = 0. ppendix F The Lorentz and the Spin groups
The group of diffeomorphisms, paramount to Einstein’s theory of gravity, does not allowfor half-spin representations , and hence a different recipe is needed to include fermionsin a formulation of gravity. This was first done by Weyl, who realised that the connectionexistent between the Lorentz and the Spin group could be exploited for this purpose [173]. Byexponentiating the generators of the Lorentz algebra so (1 , d −
1) in a spinorial representation,one obtains the simply-connected
Spin (1 , d −
1) group. This is the famous 2-1 correspondencebetween
Spin (1 , d −
1) and SO (1 , d − M ab with M ab = − M ba (where a, b =0 , , . . . , d −
1) such that [ M ab , M cd ] = − η ac M bd − η bd M ac + η ad M bc + η bc M ad . (F.1)This relation guarantees that the exponential map on the algebra gives the group of Lorentz-Fitzgerald transformations Λ cd = e σ ab ( M ab ) cd , which respects the metric η = (+ , − , ..., − ) inaccordance with the special principle of relativity V (cid:48) a η ab V (cid:48) b = V a η ab V b , (F.2)where V (cid:48) a = Λ ab V b . This implies a generalised constraint for orthogonalityΛ ∈ SO (1 , d − → η ab Λ bc η cd = (Λ − ) da , det (Λ) = 1 , (F.3)where η cd is the inverse of η , and the flat metric can be used to raise/ lower Lorentz indices.This condition translates at the level of the Lie algebra into( M ab ) T = − M ab → tr ( M ab ) = 0 , (F.4)which implies that the generators are antisymmetric in their Lorentz-indices, i.e. M ab = − M ba . (F.5)The generators of the Lorentz algebra in the vector representation are thus given byΓ v ( M ab ) cd = 2 η [ ac η b ] d , (F.6) At least not ones that fall naturally within the framework of the gravity theories we shall be discussing inthis thesis. Notice that the following rule coincides (modulo conventions) with the first equation in the system (E.1).This is so because the Poincar´e algebra is the Lorentz one plus translations.
APPENDIX F. THE LORENTZ AND THE SPIN GROUPS which obviously satisfy eq. (F.1).The
Spin (1 , d −
1) group, on the other hand, is generated by the product of an evennumber of elements of the Clifford algebra with an inverse, where the defining relation for theClifford algebra is given by { γ a , γ b } ≡ γ a γ b + γ b γ a = 2 η ab (cid:98) d/ (cid:99) × (cid:98) d/ (cid:99) . (F.7)The connection between Clifford and Lorentz algebras becomes obvious with the followingchoice for the spinorial representation of the Lorentz algebraΓ s ( M ab ) αβ = 12 ( γ [ a γ b ] ) αβ = 14 ( γ a γ b − γ b γ a ) αβ , (F.8)where the α , β indices run from 1 to 2 (cid:98) d/ (cid:99) , and label the components of a general complexspinor ψ α , which by definition is an irreducible representation of the Spin group. In otherwords, one can use the Clifford algebra for η ab to construct a spinorial representation for so (1 , d − Spin (1 , d − i.e. at every point on our curvedmanifold, on which one considers a Minkowskian tangent space with SO (1 , d −
1) as thestructure group for changes of frame.The experienced reader will have noticed that the exponential map e σ ab M ab = + σ ab M ab + . . . (F.9)will only give the elements of the Lorentz group connected to the identity, what is usuallycalled the proper orthochronous, or restricted, Lorentz group SO + (1 , d − , one has to act with (discrete) parity P = diag (1 , − , − , −
1) and time-reversal T = diag ( − , , ,
1) transformations. However, as commonly done in the Physicsliterature, throughout this thesis -and with the exception of the following paragraph- we shallrefer to the restricted Lorentz group simply as the Lorentz group, and we will genericallylabel it by SO (1 , d − Spin + (1 , d − so (1 , d −
1) algebra in the spinorial representation Γ s ( M ab )will give the simply-connected group, Spin + (1 , d − SO + (1 , d − i.e. the algebras are the same. At the level of the group, however, Spin + (1 , d − SO + (1 , d − d > of the Lorentz group. Furthermore, this cover is a doublecover, in that there are two elements of Spin mapping to each element of Lorentz. There are four orbits in total, relative to the possible orientations of space and time. Proper/ inproper( det (Λ) = ±
1) determine whether the subgroup respects the orientation of space, and orthochronicity is ameasure of the group respecting the direction of time (Λ > A group manifold G is simply-connected, or 1-connected, if it is path-connected, and has trivial funda-mental group, i.e. π ( G ) = 1. The latter is defined as the set of loops defined on the space modulo continuousdeformations (homotopies). This means that a simply-connected manifold presents no obstructions (holes) todeforming any loop homeomorphically into another loop with the same base point. This means that the Spin group will cover any other possible connected cover for the Lorentz group. ρ : SL (2 , C ) (cid:39) Spin + (1 , → SO + (1 , , (F.10)characterised by ρ ( P ) : X → P XP † , where X is a generic Hermitian matrixwith the extended Pauli matrices { σ = , σ , σ , σ } as basis, which isidentified with Minkowski space -where a generic vector v T = ( t, x, y, z ) lives-through the determinant function, i.e. det ( X ) = t − x − y − z = v T ηv ,for X = (cid:18) t + z x − iyx + iy t − z (cid:19) , (F.11)and P ∈ SL (2 , C ). The endomorphism on the space of Hermitian matricespreserves the determinat, and hence its action on M (1 , is isometric (vectorlength-preserving), which was of course expected since ρ ( P ) is a Lorentztransformation. Furthermore, one can easily see that both P = K and P = − K , for K any element of SL (2 , C ), give the same element of M (1 , ,and hence the 2-1 mapping.This is why it is common to hear that the latter is doubly-connected, namely that its funda-mental group is isomorphic to Z . Despite these differences, it is also quite common to hearthem referred interchangeably.52 APPENDIX F. THE LORENTZ AND THE SPIN GROUPS ibliography [1] J. P. Gauntlett, J. B. Gutowski, C. M. Hull, S. Pakis, and H. S. Reall, “Allsupersymmetric solutions of minimal supergravity in five- dimensions”,
Class. Quant.Grav. (2003) 4587–4634, [hep-th/0209114] .[2] J. Gillard, U. Gran, and G. Papadopoulos, “The Spinorial geometry of supersymmetricbackgrounds”, Class. Quant. Grav. (2005) 1033–1076, [hep-th/0410155] .[3] H. Baum, T. Friedrich, R. Grunewald, and I. Kath, “Twistors and Killing Spinors onRiemannian Manifolds”, in Teubner-Texte zur Mathematik , vol. 24. Teubner-Verlag,Stuttgart/ Leipzig, 1991.[4] N. Hitchin, “Harmonic spinors”,
Advances in Mathematics (1974) 1–55.[5] E. Witten, “A Simple Proof of the Positive Energy Theorem”, Commun.Math.Phys. (1981) 381.[6] R. Schoen and S.-T. Yau, “On the Proof of the positive mass conjecture in generalrelativity”, Commun.Math.Phys. (1979) 45–76.[7] R. Schoen and S.-T. Yau, “Proof of the positive mass theorem. 2.”, Commun.Math.Phys. (1981) 231–260.[8] M. Duff, B. Nilsson, and C. Pope, “Kaluza-Klein Supergravity”, Phys.Rept. (1986) 1–142.[9] R. Kallosh and T. Ort´ın, “Killing spinor identities”, [hep-th/9306085] .[10] T. Banks and M. B. Green, “Nonperturbative effects in AdS in five-dimensions x S**5string theory and d = 4 SUSY Yang-Mills”,
JHEP (1998) 002, [hep-th/9804170] .[11] R. Kallosh and A. Rajaraman, “Vacua of M theory and string theory”,
Phys.Rev.
D58 (1998) 125003, [hep-th/9805041] .[12] A. Coley, A. Fuster, S. Hervik, and N. Pelavas, “Vanishing Scalar InvariantSpacetimes in Supergravity”,
JHEP (2007) 032, [hep-th/0703256] .[13] P. Meessen, “All-order consistency of 5d sugra vacua”,
Phys.Rev.
D76 (2007) 046006, [hep-th/0705.1966] .[14] A. Strominger and C. Vafa, “Microscopic origin of the Bekenstein-Hawking entropy”,
Phys. Lett.
B379 (1996) 99–104, [hep-th/9601029] .15354
BIBLIOGRAPHY [15] S. Ferrara, R. Kallosh, and A. Strominger, “N=2 extremal black holes”,
Phys.Rev.
D52 (1995) 5412–5416, [hep-th/9508072] .[16] A. Strominger, “Macroscopic entropy of N=2 extremal black holes”,
Phys.Lett.
B383 (1996) 39–43, [hep-th/9602111] .[17] S. Ferrara and R. Kallosh, “Supersymmetry and attractors”,
Phys.Rev.
D54 (1996)1514–1524, [hep-th/9602136] .[18] J. M. Maldacena, “The Large N limit of superconformal field theories andsupergravity”,
Adv.Theor.Math.Phys. (1998) 231–252, [hep-th/9711200] .[19] P. Fayet and J. Iliopoulos, “Spontaneously Broken Supergauge Symmetries andGoldstone Spinors”, Phys.Lett.
B51 (1974) 461–464.[20] D. Kastor and J. H. Traschen, “Cosmological multi-black hole solutions”,
Phys. Rev.
D47 (1993) 5370–5375, [hep-th/9212035] .[21] K. Tod, “All Metrics Admitting Supercovariantly Constant Spinors”,
Phys. Lett.
B121 (1983) 241–244.[22] D. Kastor and J. H. Traschen, “Particle production and positive energy theorems forcharged black holes in De Sitter”,
Class. Quant. Grav. (1996) 2753–2762, [gr-qc/9311025] .[23] D. Freedman, C. Nu˜nez, M. Schnabl, and K. Skenderis, “Fake supergravity anddomain wall stability”, Phys.Rev.
D69 (2004) 104027, [hep-th/0312055] .[24] L. London, “Arbitrary dimensional cosmological multi - black holes”,
Nucl. Phys.
B434 (1995) 709–735.[25] T. Shiromizu, “Cosmological spinning multi - ’black hole’ solution in string theory”,
Prog. Theor. Phys. (1999) 1207–1211, [hep-th/9910176] .[26] D. Klemm and W. Sabra, “Charged rotating black holes in 5-D Einstein-Maxwell(A)dS gravity”,
Phys. Lett.
B503 (2001) 147–153, [hep-th/0010200] .[27] D. Klemm and W. Sabra, “General (anti-)de Sitter black holes in five-dimensions”,
JHEP (2001) 031, [hep-th/0011016] .[28] K. Behrndt and M. Cvetiˇc, “Time dependent backgrounds from supergravity withgauged noncompact R symmetry”,
Class. Quant. Grav. (2003) 4177–4194, [hep-th/0303266] .[29] L. Andrianopoli, M. Bertolini, A. Ceresole, R. D’Auria, S. Ferrara, P. Fr´e, andT. Magri, “N=2 supergravity and N=2 superYang-Mills theory on general scalarmanifolds: Symplectic covariance, gaugings and the momentum map”, J.Geom.Phys. (1997) 111–189, [hep-th/9605032] .[30] J. Grover, J. B. Gutowski, C. A. Herdeiro, and W. Sabra, “HKT Geometry and deSitter Supergravity”, Nucl. Phys.
B809 (2009) 406–425, [hep-th/0806.2626] . IBLIOGRAPHY
Phys.Rev.
D68 (2003) 105009, [hep-th/0304064] .[32] K. Pilch, P. van Nieuwenhuizen, and M. F. Sohnius, “De S itter superalgebras andsupergravity”, Comm. Math. Phys. (1985) 105–117.[33] J. Lukierski and A. Nowicki, “All possible de Sitter superalgebras and the presence ofghosts”, Phys. Lett.
B151 (1985) 382.[34] A. Celi, A. Ceresole, G. Dall’Agata, A. V. Proeyen, and M. Zagermann, “On thefakeness of fake supergravity”,
Phys. Rev.
D71 (2005) 045009, [hep-th/0410126] .[35] K. Skenderis, P. K. Townsend, and A. Van Proeyen, “Domain-wall/cosmologycorrespondence in adS/dS supergravity”,
JHEP (2007) 036, [hep-th/0704.3918] .[36] M. Cvetiˇc and H. H. Soleng, “Naked singularities in dilatonic domain wall spacetimes”,
Phys.Rev.
D51 (1995) 5768–5784, [hep-th/9411170] .[37] K. Skenderis and P. K. Townsend, “Hidden supersymmetry of domain walls andcosmologies”,
Phys. Rev. Lett. (2006) 191301, [hep-th/0602260] .[38] K. Skenderis and P. K. Townsend, “Hamilton-Jacobi method for curved domain wallsand cosmologies”, Phys.Rev.
D74 (2006) 125008, [hep-th/0609056] .[39] K. Skenderis and P. Townsend, “Pseudo-Supersymmetry and theDomain-Wall/Cosmology Correspondence”,
J. Phys. A
A40 (2007) 6733–6742, [hep-th/0610253] .[40] E. A. Bergshoeff, J. Hartong, A. Ploegh, J. Rosseel, and D. V. den Bleeken,“Pseudo-supersymmetry and a tale of alternate realities”,
JHEP (2007) 067, [hep-th/0704.3559] .[41] G. Gibbons, “Supersymmetric soliton states in extended supergravity theories” (talk),in
Unified Theories of Elementary Particles. Lecture Notes in Physics ,P. Breitenlohner and H. D¨urr, eds., vol. 160, pp. 145–151. Springer, 1982.[42] R. Kallosh, A. D. Linde, T. Ort´ın, A. W. Peet, and A. Van Proeyen, “Supersymmetryas a cosmic censor”,
Phys.Rev.
D46 (1992) 5278–5302, [hep-th/9205027] .[43] R. Kallosh and T. Ort´ın, “Charge quantization of axion - dilaton black holes”,
Phys.Rev.
D48 (1993) 742–747, [hep-th/9302109] .[44] G. Gibbons, D. Kastor, L. London, P. Townsend, and J. H. Traschen,“Supersymmetric selfgravitating solitons”,
Nucl. Phys.
B416 (1994) 850–880, [hep-th/9310118] .[45] K. Tod, “More on supercovariantly constant spinors”,
Class. Quant. Grav. (1995)1801–1820.[46] E. Bergshoeff, R. Kallosh, and T. Ort´ın, “Stationary axion/ dilaton solutions andsupersymmetry”, Nucl. Phys.
B478 (1996) 156–180, [hep-th/9605059] .56
BIBLIOGRAPHY [47] K. Behrndt, D. L¨ust, and W. A. Sabra, “Stationary solutions of N=2 supergravity”,
Nucl. Phys.
B510 (1998) 264–288, [hep-th/9705169] .[48] W. Sabra, “General BPS black holes in five-dimensions”,
Mod. Phys. Lett.
A13 (1998)239–251, [hep-th/9708103] .[49] G. Lopes Cardoso, B. de Wit, J. Kappeli, and T. Mohaupt, “Stationary BPS solutionsin N=2 supergravity with R**2 interactions”,
JHEP (2000) 019, [hep-th/0009234] .[50] S. Hawking, “The conservation of matter in general relativity”,
Commun. Math. Phys. (1970) 301–306.[51] G. Gibbons and C. Hull, “A Bogomolny Bound for General Relativity and Solitons inN=2 Supergravity”, Phys. Lett.
B109 (1982) 190.[52] E. Newman and R. Penrose, “An Approach to gravitational radiation by a method ofspin coefficients”,
J.Math.Phys. (1962) 566–578.[53] B. de Wit, D. Smit, and N. Hari Dass, “Residual Supersymmetry of CompactifiedD=10 Supergravity”, Nucl.Phys.
B283 (1987) 165.[54] J. Bellor´ın and T. Ort´ın, “A Note on simple applications of the Killing SpinorIdentities”,
Phys. Lett.
B616 (2005) 118–124, [hep-th/0501246] .[55] E. Cremmer, “Supergravities in five dimensions”, in
Superspace and Supergravity ,S. W. Hawking and M. Rocek, eds., pp. 267–282. Cambridge University Press, 1981.[56] J. Bellor´ın and T. Ort´ın, “Characterization of all the supersymmetric solutions ofgauged N=1, d=5 supergravity”,
JHEP (2007) 096, [hep-th/0705.2567] .[57] D. Joyce,
Compact Manifolds with Special Holonomy . Oxford University Press, 2000.[58] J. P. Gauntlett, D. Martelli, S. Pakis, and D. Waldram, “G structures and wrappedNS5-branes”,
Commun. Math. Phys. (2004) 421–445, [hep-th/0205050] .[59] S. Salamon, “Riemannian Geometry and Holonomy Groups”, in
Vol. 201 of PitmanResearch Notes in Mathematics . Longman, Harlow, 1989.[60] U. Gran, J. Gutowski, and G. Papadopoulos, “The Spinorial geometry ofsupersymmetric IIB backgrounds”,
Class. Quant. Grav. (2005) 2453–2492, [hep-th/0501177] .[61] U. Gran, G. Papadopoulos, and D. Roest, “Systematics of M-theory spinorialgeometry”, Class. Quant. Grav. (2005) 2701–2744, [hep-th/0503046] .[62] U. Gran, J. Gutowski, G. Papadopoulos, and D. Roest, “Systematics of IIB spinorialgeometry”, Class.Quant.Grav. (2006) 1617–1678, [hep-th/0507087] .[63] U. Gran, P. Lohrmann, and G. Papadopoulos, “The Spinorial geometry ofsupersymmetric heterotic string backgrounds”, JHEP (2006) 063, [hep-th/0510176] . IBLIOGRAPHY
JHEP (2006) 049, [hep-th/0602250] .[65] U. Gran, G. Papadopoulos, D. Roest, and P. Sloane, “Geometry of all supersymmetrictype I backgrounds”,
JHEP (2007) 074, [hep-th/0703143] .[66] S. L. Cacciatori, M. M. Caldarelli, D. Klemm, D. S. Mansi, and D. Roest, “Geometryof four-dimensional Killing spinors”,
JHEP (2007) 046, [hep-th/0704.0247] .[67] U. Gran, J. Gutowski, and G. Papadopoulos, “Geometry of all supersymmetricfour-dimensional N = 1 supergravity backgrounds”,
JHEP (2008) 102, [hep-th/0802.1779] .[68] M. Elvin, “Spinorial geometry for supersymmetric type IIB backgrounds”, Master’sthesis, Chalmers Tekniska H¨ogksola, Sweden, 2009.[69] M. Akyol and G. Papadopoulos, “(1,0) superconformal theories in six dimensions andKilling spinor equations”, [hep-th/1204.2167] .[70] M. Wang, “Parallel spinors and parallel forms”,
Ann. Glob. Anal. Geom. (1989) 59.[71] F. R. Harvey, Spinors and calibrations . Academic Press, 1990.[72] H. Lawson and M. Michelsohn,
Spin geometry . Princeton University Press, 1998.[73] J. M. Figueroa-O’Farrill, “Majorana spinors”. Extracted from ∼ jmf/Teaching/Lectures/Majorana.pdf .[74] T. Ort´ın, Gravity and strings . Cambridge University Press, 2004.[75] J. Grover, J. B. Gutowski, and W. Sabra, “Null Half-Supersymmetric Solutions inFive-Dimensional Supergravity”,
JHEP (2008) 103, [hep-th/0802.0231] .[76] R. Slansky, “Group Theory for Unified Model Building”,
Phys.Rept. (1981) 1–128.[77] J. Grover, J. B. Gutowski, C. A. Herdeiro, P. Meessen, A. Palomo-Lozano, andW. Sabra, “Gauduchon-Tod structures, Sim holonomy and De Sitter supergravity”, JHEP (2009) 069, [hep-th/0905.3047] .[78] G. Gibbons, “Aspects of supergravity theories”, in
Supersymmetry, Supergravity andRelated Topics , F. del Aguila, J. de Azc´arraga, and L. Ib´a˜nez, eds., p. 147. WorldScientific, Singapore, 1985.[79] P. Meessen and A. Palomo-Lozano, “Cosmological solutions from fake N=2 EYMsupergravity”,
JHEP (2009) 042, [hep-th/0902.4814] .[80] E. Cremmer, C. Kounnas, A. Van Proeyen, J. Derendinger, S. Ferrara, B. de Wit, andL. Girardello, “Vector Multiplets Coupled to N=2 Supergravity: SuperHiggs Effect,Flat Potentials and Geometric Structure”,
Nucl. Phys.
B250 (1985) 385.[81] M. H¨ubscher, P. Meessen, T. Ort´ın, and S. Vaul`a, “N=2 Einstein-Yang-Mills’s BPSsolutions”,
JHEP (2008) 099, [hep-th/0806.1477] .58
BIBLIOGRAPHY [82] P. Meessen and T. Ort´ın, “The Supersymmetric configurations of N=2, D=4supergravity coupled to vector supermultiplets”,
Nucl. Phys.
B749 (2006) 291–324, [hep-th/0603099] .[83] J. Bellor´ın, P. Meessen, and T. Ort´ın, “Supersymmetry, attractors and cosmiccensorship”,
Nucl. Phys.
B762 (2007) 229–255, [hep-th/0606201] .[84] P. Gauduchon and P. K. Tod, “Hyper-hermitian metrics with symmetry”,
J. Geom.Phys. (1998) 291–304.[85] P. Meessen, “Unfrozen hyperscalars and supersymmetry”, Fortsch.Phys. (2007)777–780.[86] V. Buchholz, “Spinor equations in W eyl geometry”, Suppl.Rend.Circ.Mat.di Palermo (2000) 63, [math.dg/9901125] .[87] V. Buchholz, “A note on real K illing spinors in W eyl geometry”, J.Geom.Phys. (2000) 93–98, [math.dg/9912115] .[88] M. M. Caldarelli and D. Klemm, “All supersymmetric solutions of N=2, D = 4gauged supergravity”, JHEP (2003) 019, [hep-th/0307022] .[89] M. Dunajski and P. Tod, “Einstein-Weyl structures from hyperKahler metrics withconformal Killing vectors”,
Differ. Geom. Appl. (2001) 39–55, [math.dg/9907146] .[90] D. M. Calderbank and P. Tod, “Einstein metrics, hypercomplex structures and theToda field equation”, Differ.Geom.Appl. (2001) 199–208, [math.dg/9911121] .[91] P. Meessen, “Supersymmetric coloured/hairy black holes”, Phys. Lett.
B665 (2008)388–391, [hep-th/0803.0684] .[92] D. Kastor and J. H. Traschen, “A Positive energy theorem for asymptotically de Sitterspace-times”,
Class. Quant. Grav. (2002) 5901–5920, [hep-th/0206105] .[93] A. Ashtekar and B. Krishnan, “Isolated and dynamical horizons and theirapplications”, Living Rev.Rel. (2004) 10, [gr-qc/0407042] .[94] T. Liko and I. Booth, “Supersymmetric isolated horizons”, Class. Quant. Grav. (2008) 105020, [gr-qc/0712.3308] .[95] I. Booth and T. Liko, “Supersymmetric isolated horizons in ADS spacetime”, Phys.Lett.
B670 (2008) 61–66, [gr-qc/0808.0905] .[96] G. Gibbons and C. Pope, “Time-dependent multi-centre solutions from new metricswith holonomy SIM(n-2)”,
Class. Quant. Grav. (2008) 125015, [hep-th/0709.2440] .[97] A. Walker, “Canonical form for a riemannian space with a parallel field of nullplanes”, Quart. J. Math. Oxford (1950) 69–79.[98] H. Nariai, “On some static solutions of Einstein’s gravitational field equations in aspherically symmetric case”, Sci. Rept. T¯ohoku Univ. (1950) 160. IBLIOGRAPHY
Sci. Rept. T¯ohoku Univ. (1951) 62.[100] J. P. Gauntlett, R. C. Myers, and P. Townsend, “Supersymmetry of rotating branes”, Phys.Rev.
D59 (1999) 025001, [hep-th/9809065] .[101] J. M. Figueroa-O’Farrill, “On the supersymmetries of Anti-de Sitter vacua”,
Class.Quant. Grav. (1999) 2043–2055, [hep-th/9902066] .[102] Y. Kosmann-Schwarzbach, “D´eriv´ees de Lie des spineurs”, Annali di Mat. Pura Appl. (1972) 317–395.[103] T. Ort´ın, “A Note on Lie-Lorentz derivatives”, Class. Quant. Grav. (2002)L143–L150, [hep-th/0206159] .[104] E. Lozano-Tellechea, P. Meessen, and T. Ort´ın, “On d = 4, d = 5, d = 6 vacua witheight supercharges”, Class. Quant. Grav. (2002) 5921–5934, [hep-th/0206200] .[105] A. Coley, G. Gibbons, S. Hervik, and C. Pope, “Metrics With Vanishing QuantumCorrections”, Class. Quant. Grav. (2008) 145017, [hep-th/0803.2438] .[106] J. Gutowski and W. Sabra, “Solutions of Minimal Four Dimensional de SitterSupergravity”, Class. Quant. Grav. (2010) 235017, [hep-th/0903.0179] .[107] P. Jones and K. Tod, “Minitwistor space and Einstein-Weyl spaces”, Class. Quant.Grav. (1985) 565–577.[108] J. Bellor´ın, P. Meessen, and T. Ort´ın, “All the supersymmetric solutions of N=1,d=5ungauged supergravity”, JHEP (2007) 020, [hep-th/0610196] .[109] V. Cardoso, O. J. Dias, and J. P. Lemos, “Nariai, Bertotti-Robinson and anti-Nariaisolutions in higher dimensions”,
Phys.Rev.
D70 (2004) 024002, [hep-th/0401192] .[110] W. Kundt, “The plane-fronted gravitational waves”,
Zeitshrift f¨ur Physik (1961)77.[111] A. Coley, S. Hervik, and N. Pelavas, “On spacetimes with constant scalar invariants”,
Class. Quant. Grav. (2006) 3053–3074, [gr-qc/0509113] .[112] J. Brannlund, A. Coley, and S. Hervik, “Supersymmetry, holonomy and Kundtspacetimes”, Class. Quant. Grav. (2008) 195007, [gr-qc/0807.4542] .[113] J. Podolsky and M. Zofka, “General Kundt spacetimes in higher dimensions”, Class.Quant. Grav. (2009) 105008, [gr-qc/0812.4928] .[114] A. Coley, S. Hervik, G. Papadopoulos, and N. Pelavas, “Kundt Spacetimes”, Class.Quant. Grav. (2009) 105016, [gr-qc/0901.0394] .[115] A. G. Cohen and S. L. Glashow, “Very special relativity”, Phys. Rev. Lett. (2006)021601, [hep-ph/0601236] .[116] G. Gibbons, J. Gomis, and C. Pope, “General very special relativity is Finslergeometry”, Phys.Rev.
D76 (2007) 081701, [hep-th/0707.2174] .60
BIBLIOGRAPHY [117] J. B. Gutowski, A. Palomo-Lozano, and W. Sabra, “Einstein Weyl Structures and deSitter Supergravity”,
Class. Quantum Grav. (2012) 105006, [hep-th/1109.5257] .[118] M. G¨unaydin, G. Sierra, and P. Townsend, “Gauging the d = 5 Maxwell-EinsteinSupergravity Theories: More on Jordan Algebras”, Nucl. Phys.
B253 (1985) 573.[119] J. Gutowski and W. Sabra, “Towards Cosmological Black Rings”,
JHEP (2011)020, [hep-th/1012.2120] .[120] J. B. Gutowski, D. Martelli, and H. S. Reall, “All Supersymmetric solutions ofminimal supergravity in six-dimensions”,
Class. Quant. Grav. (2003) 5049–5078, [hep-th/0306235] .[121] M. Berger, “Les vari´et´es riemanniennes homog`enes normales simplement connexe `acourbure strictement positive”, Ann. Scoula. Norm. Sup. Pisa (1961) 179.[122] P. Meessen, T. Ortin, and A. Palomo-Lozano, “On supersymmetric Einstein-Weylspaces”, J.Geom.Phys. (2012) 301, [gr-qc/1107.0937] .[123] R. Bryant, “Pseudo-Riemannian metrics with parallel spinor fields and vanishing Riccitensor”, S´emin. Congr. Soc. Math. France (2000) 53–94, [math.dg/0004073] .[124] A. Galaev and T. Leistner, “Holonomy groups of lorentzian manifolds: classification,examples, and applications”, in Recent Developments in pseudo-RiemannianGeometry , D. Alekseevski and H. Baum, eds. European Mathematical Society, 2007.[125] T. Ort´ın, “The Supersymmetric solutions and extensions of ungauged matter-coupledN=1, d=4 supergravity”,
JHEP (2008) 034, [hep-th/0802.1799] .[126] R. Penrose and W. Rindler,
Spinors and space-time 1. Two spinor calculus andrelativistic fields . Cambridge University Press, 1985.[127] J. Bellor´ın and T. Ort´ın, “All the supersymmetric configurations of N=4, d=4supergravity”,
Nucl. Phys.
B726 (2005) 171–209, [hep-th/0506056] .[128] D. Calderbank and M. Dunajski, “Scalar-flat Lorentzian Einstein-Weyl spaces”,
Class.Quantum Grav. (2001) L77–L80.[129] M. Cariglia and O. A. Mac Conamhna, “The General form of supersymmetricsolutions of N=(1,0) U(1) and SU(2) gauged supergravities in six-dimensions”, Class.Quant.Grav. (2004) 3171–3196, [hep-th/0402055] .[130] P. S. Howe and G. Papadopoulos, “Twistor spaces for HKT manifolds”, Phys. Lett.
B379 (1996) 80–86, [hep-th/9602108] .[131] J. Gates, S.J., C. Hull, and M. Roˇcek, “Twisted Multiplets and New SupersymmetricNonlinear Sigma Models”,
Nucl. Phys.
B248 (1984) 157.[132] A. Moroianu, “Structures de Weyl admettant des spineurs parall`eles”,
Bull. Soc. math. France (1996) 685’695.[133] G. Gibbons, G. Papadopoulos, and K. Stelle, “HKT and OKT geometries on solitonblack hole moduli spaces”,
Nucl. Phys.
B508 (1997) 623–658, [hep-th/9706207] . IBLIOGRAPHY
Solitons, instantons and twistors . Oxford University Press, 2010.[135] B. B. Kadomtsev and V. I. Petviashvili, “On the Stability of Solitary Waves inWeakly Dispersing Media”,
Soviet Physics Doklady (1970) 539.[136] Y. Kodama, “A method for solving the dispersionless KP equation and its exactsolutions”, Physics Letters A , issue 4 (1988) 223–226.[137] Z. Bern, J. Carrasco, L. J. Dixon, H. Johansson, D. Kosower, et al. , “Three-LoopSuperfiniteness of N=8 Supergravity”,
Phys.Rev.Lett. (2007) 161303, [hep-th/0702112] .[138] J. P. Gauntlett and S. Pakis, “The Geometry of D = 11 killing spinors”, JHEP (2003) 039, [hep-th/0212008] .[139] J. P. Gauntlett, J. B. Gutowski, and S. Pakis, “The Geometry of D = 11 null Killingspinors”,
JHEP (2003) 049, [hep-th/0311112] .[140] M. H¨ubscher, P. Meessen, and T. Ort´ın, “Supersymmetric solutions of N=2 D=4sugra: The Whole ungauged shebang”,
Nucl. Phys.
B759 (2006) 228–248, [hep-th/0606281] .[141] E. Bergshoeff, S. Cucu, T. de Wit, J. Gheerardyn, S. Vandoren, and A. Van Proeyen,“N = 2 supergravity in five-dimensions revisited”,
Class. Quant. Grav. (2004)3015–3042, [hep-th/0403045] .[142] S. Hawking and G. Ellis, The Large scale structure of space-time . CambridgeUniversity Press, 1973.[143] B. Zumino, “Supersymmetry and Kahler Manifolds”,
Phys. Lett.
B87 (1979) 203.[144] J. Bagger, “Supersymmetric Sigma Models”, in
Lectures at the NATO SummerInstitute on Supersymmetry and supergravity (Bonn) . 1984.[145] L. Alvarez-Gaum´e and D. Z. Freedman, “Geometrical Structure and UltravioletFiniteness in the Supersymmetric Sigma Model”,
Commun. Math. Phys. (1981)443.[146] J. Bagger and E. Witten, “Matter Couplings in N=2 Supergravity”, Nucl. Phys.
B222 (1983) 1.[147] B. de Wit, P. Lauwers, R. Philippe, S. Su, and A. Van Proeyen, “Gauge and MatterFields Coupled to N=2 Supergravity”,
Phys. Lett.
B134 (1984) 37.[148] B. de Wit, P. Lauwers, and A. Van Proeyen, “Lagrangians of N=2 Supergravity -Matter Systems”,
Nucl. Phys.
B255 (1985) 569.[149] B. de Wit and A. Van Proeyen, “Potentials and Symmetries of General Gauged N=2Supergravity: Yang-Mills Models”,
Nucl. Phys.
B245 (1984) 89.[150] B. de Wit and A. Van Proeyen, “Broken sigma model isometries in very specialgeometry”,
Phys.Lett.
B293 (1992) 94–99, [hep-th/9207091] .62
BIBLIOGRAPHY [151] A. Strominger, “Special geometry”,
Commun. Math. Phys. (1990) 163–180.[152] L. Castellani, R. D’Auria, and S. Ferrara, “Special K¨ahler geometry: An intrinsicformulation from N=2 space-time Supersymmetry”,
Phys. Lett.
B241 (1990) 57.[153] B. Craps, F. Roose, W. Troost, and A. Van Proeyen, “What is special Kahlergeometry?”,
Nucl. Phys.
B503 (1997) 565–613, [hep-th/9703082] .[154] A. Ceresole, R. D’Auria, and S. Ferrara, “The Symplectic Structure of N=2Supergravity and its Central Extension”,
Nucl. Phys. Proc. Suppl. (1996) 67–74, [hep-th/9509160] .[155] P. Fr´e and P. Soriani, The N=2 wonderland: From Calabi-Yau manifolds to topologicalfield theories . World Scientific, 1995.[156] A. Ceresole, R. D’Auria, S. Ferrara, and A. Van Proeyen, “Duality transformations insupersymmetric Yang-Mills theories coupled to supergravity”,
Nucl. Phys.
B444 (1995) 92–124, [hep-th/9502072] .[157] P. Fr´e, “The Complete form of N=2 supergravity and its place in the generalframework of D = 4 N extended supergravities”,
Nucl. Phys.Proc.Suppl. (1997)229–239, [hep-th/9611182] .[158] H. Weyl, “Gravitation and electricity”,
Sitzungsber. Pr¨ess. Akad. Wiss. Berlin (Math.Phys.) (1918) 465.[159] P. Gauduchon, “La 1-forme de torsion d’une vari´et´e hermitienne compacte”,
Math.Ann. (1984) 495–518.[160] K. Tod, “Compact 3-dimensional Einstein-Weyl structures”,
J. London Math. Soc. (1992) 341–351.[161] G. Gibbons, “Sim(n-2): Very special relativity and its deformations, holonomy andquantum corrections”, AIP Conf.Proc. (2009) 63–71.[162] H. Bacry and J. Nuyts, “Classification of Ten-dimensional Kinematical Groups withSpace Isotropy”,
J.Math.Phys. (1986) 2455.[163] C. Duval, G. Burdet, H. Kunzle, and M. Perrin, “Bargmann Structures andNewton-Cartan Theory”, Phys.Rev.
D31 (1985) 1841.[164] J. L´evy-Leblond, “Une nouvelle limite non-relativiste du groupe de Poincar´e”,
Ann.Inst. H. Poincar´e (1965) 1–12.[165] J. B. Kogut and D. E. Soper, “Quantum electrodynamics in the infinite-momentumframe”, Phys. Rev. D (1970) 2901–2914.[166] M. Berger, “Sur les groupes d‘holonomie homog`ene des vari´et`es `a connexion affine etdes vari´et`es riemanniennes”, Bull. Soc. Math. France (1955) 279–330.[167] T. Friedrich, “Zur Existenz paralleler Spinorfelder ¨uber RiemannschenMannigfaltigkeiten”, Czechoslavakian-GDR-Polish scientific school on differentialgeometry, Boszkowo/ Poland 1978, Sci. Comm., Part 1,2 (1979) 104–124. IBLIOGRAPHY
Actes dela Table Ronde de G´eom´etrie Diff´erentielle (Luminy, 1992), S´emin. Congr. Soc.Math. France (1996) 93–165.[169] C. Bohle, “Killing and Twistor Spinors on Lorentzian Manifolds”, Master’s thesis,Freien Universit¨at Berlin, Germany, 1998.[170] M. Brinkmann, “On riemann spaces conformal to euclidean space”, Proc. Natl. Acad.Sci. U.S. (1923) 1.[171] M. Brinkmann, “On riemann spaces conformal to einstein spaces”, Proc. Natl. Acad.Sci. U.S. (1923) 172.[172] M. Brinkmann, “Einstein spaces which are mapped conformally on each other”, Math.Ann. (1925) 119.[173] H. Weyl, “Elektron und Gravitation I. ( transl. Gravitation and the electron)”,
Z.Phys.56