Hidden Symmetries and Black Holes in Supergravity

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Hidden Symmetries and Black Holes in

Supergravity

Ella Jamsin

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Hidden Symmetries and Black Holes in

Supergravity

Ella Jamsin

Boursi`ere FRIA-FNRS

Thèse présentée en vue de l’obtention du grade de Docteur en Sciences Année académique 2009-2010

Facult´e des Sciences

Service de Physique Th´eorique et Math´ematique

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I would like to thank first my advisor, Marc Henneaux. Every time I walked out of his office, I had new motivation, my head filled with ideas and advice that helped me progress in my work.

He managed to give me a large independence while making sure I always had a project to keep me busy.

I would also like to thank Roberto Emparan, Laurent Houart, Axel Kleinschmidt and Philippe Spindel for accepting to be members of my thesis jury.

During my PhD, I worked very closely with Daniel Persson and Amitabh Virmani. Their infectious enthousiasm and their wide knowledge of their fields got me through many obstacles. I also thank them for the careful reading and useful comments on parts of this thesis. And Daniel, reference[64]is just for you!

It was a blessing to have someone as wise and friendly as Axel Kleinschmidt around. My regular visits to his office helped me to understand many aspects of this thesis.

I would also like to thank Sophie de Buyl and Nassiba Tabti, my ‘Kac-Moody sistahs’. Not only was it very interesting to work and discuss with each of you but moreover, it is oh so refreshing to talk to a girl sometimes! And thank you Sophie for proofreading and encouragements.

I had four more collaborators around the world: Geoffrey Comp`ere, Pau Figueras, Jakob Palmkvist and Jorge Rocha. It has been a pleasure to interact with you, both on scientific and human levels. And thanks Jakob for the comments on the thesis.

I would also like to thank the whole group of Fundamental Interactions for the nice and informal atmosphere they create, not to mention the parties and other nights out. In particular, I thank Fran¸cois Dehouck and Josef Lindmand H¨ornlund for having been easy-going office mates.

Last but not least, I want to thank Alex Wijns for proofreading many parts of this thesis and for, within the last three years, improving my English, filling my shelves with physics and math books, calmly answering my traditional last minute panick questions and repeatedly providing encouragements. I owe you a lot!

Of course, more people deserve my acknowledgments. They did not have a direct influence on the content of this thesis, I think most of them don’t even care about it, but they certainly had a positive influence of my mental health, without which the last four years would have been much less fun.

This first concerns Yassin Chaffi, Cyril Closset, Pierre de Buyl, Fran¸cois Dehouck, Julie Delvax, Jon Demaeyer, Nathan Goldman, Nassiba Tabti and Vincent Wens for four years of lunchtimes spent sharing all kinds of stories, worries and advice in the dreamy setting of the cafeteria... I especially owe one to Nathan for getting me out of printing drama.

Because the road to the PhD really started not four but eight years ago, I would like to thank my other fellow students for making physics more fun, until they emigrated to far away campuses. Among them, a special thank to Ariane Razavi (and her mother) for letting me borrow their house for a night and to J´erˆome Loreau for having greatly helped my early understanding of string theory.

I’m also grateful to my old friend Pauline Danhaive for regularly providing me with totally physics-free fun and for keeping inviting me to join parties despite my extreme lazyness. You should notice, though, that the I-have-a-thesis-to-finish excuse is gonna be over soon...

Finally, these acknowledgments would be incomplete without thanking Marie-Claire and Michel Jamsin for being such supportive, helpful and fun parents. Thank you so much!

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Introduction

The main topic of this thesis is the study of hidden symmetries of supergravity theories of particular interest and how these help constructing black hole solutions in dimensions D >4. We here introduce the different ingredients that motivate this work and play an important role in understanding it.

1.1 String theory and supergravity

One often feels better comprehension is achieved when one is able to describe a concept in simpler terms, in a more concise fashion. In a large interpretation, this can be seen as the way science progresses. In certain fields, in particular in physics, this simplification often arises when one becomes able to describe various phenomena that seemed unrelated using just one set of principles. This process of unification appears at key steps of the history of physics. A striking example dates back to the 19^th century when Maxwell described electricity and magnetism in a unique theory, electromagnetism. Not only is the unified theory esthetically attractive, but separate theories of electricity and magnetism would be inconsistent. As another example, the General Relativity of Einstein unified in 1915 special relativity and gravitation.

It is believed that physics is dictated by four fundamental forces: gravitation, electromagnetism, the weak force and the strong - or color - force. A partial unification of these forces was done in the early 1970’s in the Standard Model, which provides a unified quantum theory of the electromagnetic, weak and strong interactions. This theory contains a large number of fundamental particles: bosons, which are the mediator of forces, and fermions, which constitute the matter. In a certain extension of the Standard Model, bosons and fermions are related by a symmetry, called supersymmetry. It predicts that each particle must have a superpartner of same mass. As current observations imply that this is not the case, supersymmetry must be broken in the evolution of the early universe.

The standard model, combined with the classical theory of gravitation provides a good description of most situations. Nevertheless, a theory of gravitation at the quantum level is absolutely essential in certain situations, in particular for cosmological models of

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the early universe and for certain aspects of black holes. However, quantum field theory, the quantization approach that allowed to construct quantum versions of the other three interactions, leads, when applied to general relativity, to inconsistencies.

A candidate to the unification of all interactions and quantization of gravity came up from a completely different approach in the 1970’s, giving rise to string theory. In this theory, the fundamental objects are not pointlike as in quantum field theory, but they are extended in one dimension: they are strings. All elementary particles then correspond to different vibrational modes of the strings. One of them is the graviton, which is the mediator of gravity. This theory might thus provide a deep unification of all interactions. Moreover, as it includes gravity and can be consistently quantized, in principle string theory yields a quantum theory of gravity. Unlike the Standard Model, it does not contain any adjustable dimensionless parameter, which makes it a unique theory in the sense that it cannot be continuously deformed. Moreover, its consistency at the quantum level requires a fixed value of the spacetime dimension. In superstring theory, the supersymmetric version of string theory, this dimension is D= 10. As a consequence, a mechanism is needed to explain why according to all current observations, the spacetime has four dimensions. Compactification of unobserved dimensions can provide such a mechanism.

It was showed in the mid 1980’s that one can construct five different superstring theories in ten dimensions, that are related by dualities: type I, type IIA, type IIB, heterotic SO(32) and heterotic E₈ ×E₈. It was later discovered that in the strong coupling limit of type IIA and heterotic E8 ×E8, there was a sixth theory, which was eleven-dimensional. These six theories happen to be different facets of one single, unique theory [1], yet unknown and called M-theory.

It is interesting to study these superstring theories in the low energy limit, where they reduce to supergravity theories, which can be seen as supersymmetric generalizations of gravity. Eleven-dimensional supergravity is the field theory with the largest possible spacetime supersymmetry and Poincar´e invariance. Taking it as a starting point, one can find the low energy limits of all five superstring theories in ten dimensions through compactifications and dualities. For example, dimensional reduction on a circle gives rise to type IIA supergravity. Eleven-dimensional supergravity is in this sense a fundamental theory, and thus an important field of investigation. Its bosonic sector consists of a metric g11 and a three-form potentialA₍₃₎, which enter the Lagrangian as follows:

L11=R11?1−1

2 ? F₍₄₎∧F₍₄₎+1

6F₍₄₎∧F₍₄₎∧A₍₃₎, (1.1) whereF₍₄₎=dA₍₃₎.

One property of eleven-dimensional supergravity is that in the presence of eight commuting Killing vectors, it exhibits a global symmetry under the largest exceptional simple finite-dimensional Lie group E8.

Upon a certain compactification of eleven-dimensional supergravity to five dimensions followed by a truncation, one obtains a theory which exhibits a structure very similar to D= 11 supergravity. This theory, D= 5 minimal supergravity, contains on the bosonic

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1.2. Symmetries and Kac-Moody algebras 11

side a metric g₅ and a one-form potentialA₍₁₎. Its Lagrangian reads L₅ =R₅?1−1

2 ? F₍₂₎∧F₍₂₎+ 1 3√

3F₍₂₎∧F₍₂₎∧A₍₁₎, (1.2) whereF₍₂₎=dA₍₁₎.

In the presence of two commuting Killing vectors, D = 5 minimal supergravity is symmetric under the smallest exceptional simple finite-dimensional Lie group G2. This theory thus has properties similar to the ones of D = 11 supergravity but in a simpler setting. Moreover, it admits a large variety of black holes. A certain number of these solutions are known but the most general ones are still to be constructed. This theory is our center of focus in chapters 5 and 6 of this thesis.

Another important type of supergravity theory is deformed supergravity. By this, we mean a theory that is different from standard Kaluza–Klein reductions of maximal D = 11 supergravity. Often, these theories cannot even be obtained by other types of reductions or compactifications from D = 11 and are therefore to be interpreted as genuine new low energy actions for M-theory. One of the pertinent features of deformed supergravity theories is that they support domain wall solutions. These are required to maintain invariance under the duality symmetries expected from an underlying string theory description. The most prominent example of this phenomenon is the D8-brane of type IIA string theory that can be reached from lower-dimensional branes by sequences of T-dualities [2]. However, there is no corresponding supergravity solution in undeformed type IIA supergravity in D= 10. Only when considering Romans’ massive deformation of type IIA supergravity [3] can one accommodate the D8-brane [2]. Therefore any attempt at describing M-theory, at least at low energies, should be able to reproduce these deformed supergravity theories. The analysis of chapter 7 is applied to Romans’

massive type IIA supergravity.

1.2 Symmetries and Kac-Moody algebras

Whether it is in the search for elegance or comprehension, symmetry is a key concept in science. It is moreover deeply related to the notion of unification – both induce a (relative) simplification and a deeper comprehension. Moreover, several theories can be considered unified when they appear to be related by symmetries. In physics, we often seek for symmetries of a theory, defined as the transformations that leave the action or the equations of motion invariant. In particular, much can be learnt about string theory from the study of its symmetries. In this thesis, we investigate symmetries of certain supergravity theories and their applications.

Symmetries are described by a large field of mathematics which is group theory. The symmetry groups of continuous transformations are Lie groups. In the context of this thesis, we construct them by exponential mapping of the corresponding Lie algebras. The two notions can be thus considered equivalent.

In full generality, Lie algebras are algebras whose product operation is the Lie bracket.

In most physics applications, the relevant Lie algebras are direct sums of abelian and

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simple¹ Lie algebras of finite dimension. Abelian Lie algebras can always be written as a direct sum of u(1) algebras. The non trivial part is thus simple Lie algebras. Their classification was completed by Cartan and Killing using a description of each simple Lie algebra by a so-called Cartan matrix, or equivalently a Dynkin diagram. A certain number of rules indicate whether a given matrix can be the Cartan matrix of a simple finite-dimensional Lie algebra.

The description of simple finite Lie algebras in terms of the Cartan matrix has the double advantage of simplifying the classification and of being easily generalized. Indeed, the ‘generalized Cartan matrix’ is defined with a slight modification of the rules defining Cartan matrices [4]. The corresponding Lie algebras are the Kac-Moody algebras. These contain three classes, depending on the properties of the generalized Cartan matrix: finite, affine and indefinite Kac-Moody algebras. The finite ones are simple finite-dimensional Lie algebras. The two other types have infinite dimension. Affine algebras are well known and they can be completely classified as extensions of the finite algebras. Not much is known about indefinite algebras, though. However, for the purpose of the present thesis, only the subclass of indefinite algebras known as hyperbolic algebras plays a role, and this is partly understood. Each hyperbolic algebra can be seen as an extension of an affine algebra, or a double extension of a finite one.

These three types of Kac-Moody algebras, finite, affine and hyperbolic, correspond to symmetries of (super)gravity theories. We are here especially interested in ‘hidden symmetries’. Let us consider a (super)gravity theory inDdimensions. Upon dimensional reduction on ann-dimensional torusTⁿ, (n≤D−3), this (super)gravity theory reduces to an effectived-dimensional theory (d=D−n) including a certain number of scalar fields.

These scalar fields couple to form a non-linear sigma model on the coset space G/K(G) where G is the group obtained by exponentiation of a finite Kac-Moody algebra g and K(G) is a certain subgroup of G. The lower-dimensional theory thus possesses a global symmetry underg. As the reduction is performed on ann-torus, which is equivalent to assuming the existence ofncommuting Killing vectors, a global symmetry undergl(n) is to be expected. However, in certain cases, that is, for certain supergravity theories and reduction of enough dimensions, one finds the symmetrygto be bigger thangl(n), hence the concept of hidden symmetry.

In order to see the full hidden symmetry, one has to take into account the scalar fields that arise from dualization of certain p-forms. An extreme situation happens in dimensional reductions down tod= 3, as allp-forms can then be dualized to scalars. The reduced theory then consists only of a three-dimensional metric and a number of scalar fields. Since finite Kac-Moody algebras are classified, it is possible to also classify theories in dimensions D >3 such that the scalar fields obtained after dimensional reduction to d= 3 form a non-linear sigma model with symmetry under a finite Kac-Moody algebra [5].

If one performs the reduction one step further, down tod= 2, the hidden symmetry becomes infinite-dimensional and is described by the affine extension of the finite symme-

1Simple Lie algebras are non abelian and do not possess any non trivial ideal. This implies for example that it cannot be written as a direct sum of subalgebras.

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1.3. Black holes in higher dimensions 13 try algebra that appears ind= 3 [6, 7]. Moreover, the system is then integrable, which in the context of vacuum gravity has been known for long to provide a very efficient method to generate solutions [8–11].

The appearance of successive extensions of algebras when one reduces on more dimensions is the first hint of a symmetry under a hyperbolic Kac-Moody algebra upon reduction tod= 1. Another motivation comes from the study of certain gravity theories close to a spacelike singularity. In this limit, the dynamics of gravity is controlled by the Weyl group of a hyperbolic algebra, the hyperbolic extension of the finite symmetry that appears in the reduction to d= 3 [12]. These considerations led to the conjecture of the symmetry of M-theory under the hyperbolic group E10. A partial correspondence was established between eleven-dimensional supergravity and a non-linear sigma model on the coset E₁₀/K(E₁₀), whereK(E₁₀) is the maximal compact subgroup of E₁₀ [13].

Finding unexpected symmetries is of course very interesting in its own right. However, from a more pragmatic point of view, symmetries can also be used to construct new solutions of a theory starting from known ones. They provide powerful techniques to generate black hole solutions.

1.3 Black holes in higher dimensions

A fascinating consequence of General Relativity is the existence of black holes, these regions in spacetime from which nothing, not even light, can escape. As they are also the most fundamental objects of General Relativity, it is no surprise that they have been an important topic of research in mathematical physics since the 1930’s. The classical study of four-dimensional black holes of Einstein-Maxwell theory reached a peak (and an end) when the joint effort of a number of physicists led to the no-hair theorem: Four- dimensional black holes are completely determined by their three conserved charges (the mass M, the angular momentum J and the electric charge Q) [14, 15]. Moreover, their horizon must be topologically spherical [16].

In recent years, there has been a growing interest for black holes in dimensions higher than four [17]. The motivation for this can be first attributed to the appearance of higher dimensions in string theory, which might provide a microscopic description of gravity and should thus in particular be able to predict the entropy of a black hole. The first successful calculation of such a quantity was applied to a five-dimensional black hole [18].

Moreover, supergravity theories can be seen as supersymmetric extensions of gravity. It is therefore natural to study their most basic objects: black holes.

From a more intrinsic point of view, one can hope to discover new insights about gravity by considering the spacetime dimension D as a tunable parameter and investigate which properties generalize to all dimensions and which ones are specific to D= 4.

It turns out that the physics of higher-dimensional black holes is richer and more diffi- cult. For example, non-spherical topologies become possible [19] and the corresponding solutions are not completely determined by their conserved charges.

An origin to the richness of black hole solutions in higher dimensions is the possibility for several independent rotations. In D = 4, through a change of coordinates, any

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rotation can be written as a rotation around one of the axes, or equivalently in one plane.

InD >4, one can construct perpendicular spatial planes, each of them corresponding to an independent rotation.

Another important novel feature in dimensions D >4 is the appearance of extended black objects. The simplest example is the black string which is obtained inDdimensions by adding one flat direction to a (D−1)-dimensional spherical black hole. One finds p- branes by adding p flat directions. Because some directions can be much larger than the other ones, these objects can present dynamical horizon instablilities [20]. They are moreover not asymptotically flat, but they can be used to give intuition and insight for certain asymptotically flat objects, for example black rings. Indeed, these can be seen as black strings whose extended direction is bent into a circle.

In the presence of gauge fields, black rings present another new feature which is the possibility of a dipole [21]. This introduces a new parameter in the solution although it is not a conserved charge. This implies that for some given conserved charges, there is not only a discrete degeneracy due to the existence of several possible topologies of the horizon, but one moreover has to take into account a continuous degeneracy due to the existence of non conserved dipoles in black rings.

1.4 Solution generating techniques

As a consequence of the greater richness of black holes in higher dimensions, the development of efficient generating techniques is very important. This is not easy as the techniques used in four dimensions do not all generalize to any higher dimension. In this thesis, we consider two generating techniques based on the existence of hidden symmetries. In both cases, they apply to D-dimensional (super)gravity theories which exhibit hidden symmetry under a finite Kac-Moody algebra g upon reduction to d= 3 [5]. We assume this is the case in the following.

The first generating technique makes direct use of theg symmetry. It thus applies to solutions possessingD−3 commuting Killing vectors. Such solutions can be equivalently given as a three-dimensional metric and a set of scalars, obtained by dimensional reduction of the higher-dimensional description. These scalar fields couple to form a non-linear sigma model symmetric under g. More precisely, one can construct out of the scalar fields a spacetime dependent matrix M(x) that is, at each point x of the spacetime, an element of the group G obtained by exponentiation of g. Conversely, given a d = 3 metric and a set of scalars satisfying the equations of motion of the effectived= 3 theory, one can uplift, or ‘oxidise’, it back to a D-dimensional solutions. This process however requires certain gauge choices.

The generating process then proceeds as follows. LetS be a known solution of theD- dimensional theory possessingD−3 Killing vectors. Thus,Sis given by aD-dimensional metric and possiblyp-forms and scalar fields. The first step consists in re-expressing the solution as a metric and scalar fields in three dimensions. Then, one constructs the matrix M(x) using the scalar fields. The third step consists in acting with an appropriate element ofg on M(x). The new solution is thus described by the same three-dimensional metric

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1.5. Overview 15 and a new, transformed, matrix M⁰(x). The last step is to deduce the D-dimensional solution S⁰ fromM⁰(x).

From a technical point of view, the last part is the most tricky one because of the dualizations and the gauge choices involved. However, in order for the final solution S⁰ to be physically interesting, that is, different from the seed solution and regular, the non-trivial step is choosing the right element of G to act on M(x). It turns out that this question is already partly answered by making use of a well-known decomposition of G, namely the Iwasawa decomposition. This says that one can write any element of G as the product of three elements, lying in three different subgroups of G: the nilpotent subgroup, the Cartan subgroup and K(G)². When acting on a solution, these three subgroups correspond respectively to gauge transformations (which modify the scalars in such a way that the D-dimensional solution stays unchanged), scaling transformations (which can possibly modify the original charges but not bring any new ones) and actual charging transformations (which can generate new charges). One can thus focus on the last type of transformation.

The second generating technique we consider requires the existence ofD−2 commuting Killing vectors. Note that asymptotically flat solutions can only fulfill this requirement inD= 4,5. Not only does the two-dimensional effective theory possess a symmetry under an infinite-dimensional affine algebra but moreover it is completely integrable. This implies the possibility to use techniques available for integrable systems. More precisely, Belinsky and Zakharov [8–10] applied the inverse scattering technique as a generating technique for black holes of four-dimensional vacuum gravity. This was later generalized to five dimensions by Pomeransky [11]. The construction of Belinsky, Zakharov and Pomeransky has showed very useful in obtaining new solutions. However, it cannot be generalized directly to theories including p-forms. Nevertheless, from a different per- spective, Breitenlohner and Maison [6] and Nicolai [7] showed the integrability (as well as the affine symmetry) of certain supergravity theories reduced to two dimensions. A combination of these two approaches allows the generalization of inverse scattering to all (super)gravity theories with hidden symmetry, that is, to all theories of [5].

1.5 Overview

This thesis contains two parts. Part I introduces purely background material: a review of higher-dimensional black holes and an introduction to Kac-Moody algebras. In part II, we get into the subject of this thesis, that is, the presence of hidden symmetries in (super)gravity theories and their use in constructing black hole solutions. Chapters 2 to 4 contain essentially review material. The original work is presented in chapters 5, 6 and 7, which are based respectively on the three papers [22], [23] and [24].

2The Iwasawa decomposition is strictly valid whenK(G) is the maximal compact subgroup ofG. This is the relevant decomposition when the reduction is performed only on spacelike directions. When one direction is timelike,K(G) is non compact and some subtleties must be taken into account.

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Part I. Chapter 2 is a review of black holes in higher dimensions. We consider separately vacuum and supergravity solutions. We first review known solutions of vacuum gravity.

We then provide a more detailed description of stationary axisymmetric solutions of vacuum gravity, that is, vacuum solutions with three commuting Killing vectors including time translations. We review the different approaches to describe the structure of these solutions and present a solution generating technique: the inverse scattering method.

In chapter 3, we review the essential mathematical tools used in this thesis, that is, Kac-Moody algebras. We start by defining the generalized Cartan matrix and Kac-Moody algebras. Follow a series of general definitions and properties. We then focus separately on the three subclasses that play a role in this thesis: finite, affine and hyperbolic Kac- Moody algebras. For finite algebras, we first give some general considerations and then study in further detail three algebras relevant for this thesis. Concerning affine algebras, we present two equivalent descriptions, one in terms of the generalized Cartan matrix and one as an extension of a loop algebra. A simple example of affine algebra is discussed a bit more explicitly. Finally we consider hyperbolic Kac-Moody algebras. After discussing some generalities, we present a method that allows to describe the infinite-dimensional hyperbolic algebras in terms of finite subparts. We then briefly introduce the hyperbolic algebrae₁₀.

Part II. To begin with, we review in chapter 4 the appearance of hidden symmetries in (super)gravity theories. First we summarize the main aspects of dimensional reduction on ann-dimensional torus and the corresponding gl(n) symmetry. We then explain how to construct non-linear sigma models on coset spaces. This allows us to understand how hidden symmetries arise in various situations, including infinite-dimensional symmetries.

Finally we briefly discuss the inclusion of fermions in the theories. We then proceed with applications in three situations.

In chapter 5, we focus on the hidden symmetries under finite Kac-Moody algebras appearing upon reduction to three dimensions. As a warm up, we first study how the hidden sl(2,R) symmetry of four-dimensional vacuum gravity with one Killing vector can be used to generate solutions. In the rest of the chapter, we focus on the case of five-dimensional minimal supergravity. We first make explicit the hidden symmetry of the theory under the algebra g₂ upon dimensional reduction tod= 3. In particular, we collect all definitions in our formalism. We then give the general result of the action of the Cartan and the nilpotent subalgebras ofg₂ and show in particular that these do not give rise to interesting new solutions. We then study the action of the third subalgebra of g₂. We focus on stationary axisymmetric solutions. In that case, one has to use a slightly modified version of the Iwasawa decomposition in which the third subalgebra is not compact. Since this subalgebra conserves Kaluza-Klein asymptotics, we study its action on the precise case of black strings. We present the results of acting with each of the generators. In particular, we give a new construction of electrically and magnetically charged black strings. We discuss the thermodynamics of these solutions. The study of the pressureless conditions of the black strings can be put in relation with the balance

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1.5. Overview 17 condition of the corresponding black rings, although they are yet unknown.

We go one step further in chapter 6, devoted to the affine symmetry and integrability of gravity theories reduced to two dimensions. After a brief discussion of the possibility to use the affine symmetry algebra to generate new solutions, we focus on five-dimensional minimal supergravity and exploit the other solution generating technique at hand, the inverse scattering method. To start with, we generalize the construction, previously available for vacuum gravity, to five-dimensional minimal supergravity. We then discuss some general issues related to the generating technique. Finally, we apply the method in two cases. The first one is simply a generalization of a known vacuum gravity result to our formalism. The second example consists in adding two rotation parameters to a charged solution. It is the first application of the inverse scattering technique to a charged solution.

Finally, in chapter 7, we consider symmetries under hyperbolic algebras. More precisely we investigate the conjectured symmetry of M-theory under E₁₀ by focussing on massive type IIA supergravity. We give a detailed account of the correspondence between the algebraic and the physical theory including the dynamics. A consequence of that work is that, although they are not related by dimensional reduction, eleven dimensional supergravity and massive IIA supergravity have the sameE10 origin. Moreover, unlike the rest of this thesis, the analysis of that chapter also considers the fermionic sector of the theory. The content is structured as follows: we first give all necessary details on massive type IIA supergravity; second we present the non-linear sigma model for E10/K(E10);

third we give the correspondence between the two sides.

Chapter 8 contains the general conclusions of this thesis.

Appendices A, B and C gather conventions and the more lengthy or technical aspects of chapters 5, 6 and 7 respectively.

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Part I

Preliminaries

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Chapter 2

Black holes in higher dimensions

In this section, we review the theory of black holes in dimensions higher than four, with a focus on aspects that play a role in part II of this thesis. The basic notions of the theory of four-dimensional black holes is assumed to be known. In section 2.1, we present briefly the known solutions in vacuum. In section 2.2, we explain a structure theory and a generating technique that have shown very useful in the study of stationary axisymmetric solutions of vacuum gravity. We then switch to supergravity theories. Section 2.3 reviews the current knowledge of black hole solutions in maximal supergravity theories and certain truncations of them. Throughout these sections, we focus essentially on asymptotically flat solutions. In five dimensions though, one can learn about black rings by studying objects with Kaluza-Klein asymptotics, in particular black strings. Correspondences between the parameters of each of these objects and four-dimensional black holes are described in section 2.4. Finally, the development of uniqueness theorems in higher dimensions is discussed in section 2.5.

2.1 Vacuum solutions

2.1.1 Spherical topology

In four-dimensional vacuum gravity, the most general stationary black hole is described by the Kerr metric, which is determined by a mass M and an angular momentum J and whose horizon has spherical topology. A first generalization of black holes to higher dimensions is achieved by constructing higher-dimensional spherical black holes.

Tangherlini solution

The first black hole constructed in dimension D > 4 is naturally the generalization of the Schwarzschild black hole, found by Tangherlini in 1963 [25]. The Tangherlini metric reads

ds² =−(1− µ

r^D⁻³)dt²+ (1− µ

r^D⁻³)⁻¹+r²dΩ²_D−2, (2.1) 21

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wheredΩ²_D₋₂ is the unit (D−2)-sphere line element. The parameterµis related to the massM as

µ= 16πGM

(D−2)A_D−2 , (2.2)

whereGis the gravitational constant and A_D−2 is the area of the unit (D−2)-sphere A_D−2 = 2π^(D⁻^1)/2

Γ(^D⁻₂¹) .

Essentially, the D-dimensional metric is obtained by demanding that the deviation from the flat metric falls off asr^D⁻³, which is the way the Newtonian potential generalizes inDdimensions.

Myers-Perry solution

The generalization of the Kerr metric to arbitrary dimensions was done by Myers and Perry [26] using the fact that it can be written in cartesian coordinates in the Kerr-Schild form

g_µν =η_µν+ 2H(x^ρ)k_µk_ν (2.3)

whereη_µν is the flat Minkowski metric, H(x^ρ) is some function of spacetime andk_µ is a null vector with respect to bothgµν and ηµν.

In four spacetime dimensions, through a change of coordinates, any rotation can be written as a rotation around one of the axes, or equivalently in a (x, y) plane. Therefore, only one rotation parameter is needed in the Kerr solution. In higher dimensions, more complicated rotations are possible. Indeed, there is an independent rotation in all perpendicular spatial planes (x_i, x_j) one can construct. If the number of spatial dimensions D−1 is even, there are (D−1)/2 perpendicular planes (x1, x2), . . . ,(xD−2, xD−1), and therefore as many independent rotations. IfD−1 is odd, there are (D−2)/2 of them. In summary, the numbern of independent angular momenta of aD-dimensional spacetime is

n≡

D−1 2

(2.4) where bac is the biggest integer smaller than or equal to a. One can also see n as the dimension of a maximal abelian subalgebra of the rotation algebraso(D−1). As we will see in chapter 3, this defines the Cartan subalgebra ofso(D−1), whose dimension is the rank ofso(D−1). The Myers-Perry metric reads nicely in terms of polar coordinates on the planes of rotation.

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2.1. Vacuum solutions 23

2.1.2 New topologies in D = 5

Black strings

A novelty in dimensions higher than 4 is the existence of extended black objects. In general, given a vacuum black hole solutionBinddimensions, one can construct another solution in D=d+p dimensions by adding pflat extra dimensions. The corresponding metric

ds²_d+p=ds²_d(B) +

p

X

i=1

(dxⁱ)². (2.5)

describes a blackp-brane.

In particular, and most simply, starting from four-dimensional solutions, thus taking d = 4, one can construct black one-branes, i.e., black strings, in five dimensions. The horizon of a black string is topologically S²×R. Besides the famous though very formal proof that all black holes have spherical topology in four dimensions [16], one can more intuitively understand that black strings do not exist in four dimensions because there is no vacuum asymptotically flat black hole in three dimensions. This can be attributed to the absence of propagating degrees of freedom or equivalently to the absence of a length scale inD= 3. Indeed the quantityMGis then dimensionless.

Although black strings are not asymptotically flat, but rather asympotically Kaluza- Klein R^1,3×S¹, they are useful in the construction and the study of asymptotically flat black objects in D= 5, as black rings.

Black rings

Consider a black string in five dimensions, constructed out of a Schwarzschild or Kerr black hole in four dimensions. One can identify periodically points along the extended direction z of the string, z ∼ z+L, and bend it into a circle of circumference L. The final object has a horizon of topology S²×S¹. The S¹ circle is contractible and hence the gravitational attraction must be compensated by the centrifugal repulsion due to a rotation in theS¹. Fixing the parameters in order to have the right balance corresponds to the removal of a conical singularity in the metric.

One therefore has two possibilities, a singly spinning black ring, with rotation only in the S¹, and a doubly rotating black ring, with one rotation in the S¹ and one in the S². The first one was constructed in [19]. It was the first asymptotically flat vacuum solution that was completely regular on and outside the event horizon and of non-spherical topology; it provided therefore the first proof that uniqueness theorems did not trivially generalize to higher dimensions. Actually, one can even see that in a certain range of the two parameters, which can be taken to be the mass M and the angular momentum J, there exists three different solutions: a singly spinning Myers-Perry black hole (which can be obtained as a certain limit of the non equilibrium black ring solution), a thin black ring and a thick one.

The doubly spinning black ring was constructed in [27] using inverse scattering techniques, which are reviewed in section 2.2. The equilibrium solution depends on three

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parameters: the mass M, the angular momentum J_φ in theS² and the angular momen- tumJ_ψ in the S¹.

A very different type of black ring, the so-called helical black ring, is constructed in [28] in all dimensions using the recently proposed blackfold theory [29, 30]. One of its most striking features is that it is the first example of an asymptotically flat black object in dimensionD >4 possessing only one spatial U(1) Killing vector.

Multi-black holes

While it is believed that there is no stationary vacuum black hole solution with dis- connected horizons in four dimensions, several such solutions were constructed in five dimensions.

One can first construct black Saturns [31], composed of a Myers-Perry black hole surrounded by a concentric rotating black ring. The two pieces can be co or counter- rotating. These solutions present interesting features, such as rotational dragging. One can construct a solution where the S³ horizon is static while the angular momentum associated to the black hole in the center does not vanish: the proper rotation of the black hole is exactly compensated at the horizon by the dragging of the ring.

There are also multi-ring solutions. The di-ring [32] is composed of two concentric and coplanar rotating black rings. Bicycling black rings [33, 34] on the other hand lie and rotate in two orthogonal planes.

Note that these two types of multi-black holes can be constructed with the inverse scattering method [31, 33–35].

2.1.3 Some words on D > 5

Due to the lack of available techniques to construct asymptotically flat exact solutions in six dimensions and higher, much less is known in those regions. Indeed, the inverse scattering method, which led to many discoveries in five dimensions and is explained in section 2.2 applies to stationary solutions possessingD−3 spacelike commuting Killing vectors. This requirement is compatible with asympotic flatness only forD= 4,5. Indeed, a general D-dimensional asymptotically flat solution cannot possess more than b^D⁻₂¹c commuting spacelike Killing vectors as its geometry must asymptotically be the one of a D−2-sphere.

Note however that another generating technique is provided by the so-called hidden symmetries, as developed in section 5. The technique we present there assumes the existence of D−3 commuting Killing vectors in total, thus D−4 spacelike ones if one focuses on stationary black holes. This technique could therefore apply also toD= 6,7.

This direction has not been explored yet though.

However, strong indications lead to believe that the situation in D >5 is even richer and more complicated. For example, it was conjectured [36] that it is possible to construct black holes with spherical horizon topology but with axially symmetric ‘ripples’. The existence of black rings in any higher dimension was also argued [37, 38]. Some aproximate solutions of that type have been constructed [39].

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2.2. Stationary axisymmetric solutions in vacuum 25 A new strategy for an approximate study of higher-dimensional black holes was recently proposed; it is the blackfold approach [29, 30]. It is based on the existence of ultra-spinning regimes in dimensions D > 4. Indeed, for a neutral vacuum black hole, one can construct two length scales associated respectively to the mass and to the angular momentum:

l_M ∼(GM)^D−3¹ , l_J ∼ J

M . (2.6)

In D = 4, these are always parametrically similar, due to the Kerr bound lJ ≤ lM. In higher dimensions though, one can have ultra spinning regimel_J >> l_M, exhibiting novel features compared to D= 4 and allowing the developpement of approximate analytical methods in terms of the small parameterlM/lJ. This leads indeed to an effective description of a black hole as a blackbrane whose worldvolume spans a curved submanifold of a background spacetime - hence the term blackfold. A systematic scan of the landscape of black holes in any spacetime dimension using this theory was recently initiated [28].

2.2 Stationary axisymmetric solutions in vacuum

In this section, we discuss the structure of stationary axisymmetric black hole solutions, that is, stationary metrics in the presence of D−3 commuting spacelike Killing vectors.

We moreover present a generating technique that is used to construct several of the black holes mentioned earlier. Note that this symmetry requirement allows asymptotically flat spacetimes only in dimensions D= 4 and D= 5.

In the presence ofD−2 commuting Killing vector fields ξ_(a) = _∂x^∂a,a= 0, . . . , D−3, one can show [40, 41] that under weak conditions¹ a metric admits the form

ds²=G_abdx^adx^b+e^2ν(dρ²+dz²) , (2.7) where a = 0, . . . , D −3. Here we choose x⁰ = t so that ξ₍₀₎ is the time translation generator. Note thatρ, z cover the space outside the horizon.

Furthermore, without loss of generality we can choose coordinates so that

detG=−ρ². (2.8)

Under this decomposition, the vacuum Einstein equations divide into two groups, one for the metric components along the Killing directions,

∂_ρU¯ +∂_zV¯ = 0, (2.9)

with

U¯ =ρ(∂_ρG)G⁻¹, V¯ =ρ(∂_zG)G⁻¹ , (2.10)

1As an example, a sufficient condition is that one of the Killing directions be an angle.

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and the second group of equations forν

∂ρν =− 1 2ρ + 1

8ρTr( ¯U²−V¯²) , ∂zν = 1

4ρTr( ¯UV¯) . (2.11) These equations (2.9-2.11) are the starting point of the rod structure analysis. Moreover, as they form an integrable system, they allow the development of an inverse scattering method.

2.2.1 Rod structure

There are at least three equivalent ways to describe the so-called rod structure of a solution:

• as intervals where the Killing metric Ghas a one-dimensional kernel [38, 42],

• as sources of a Poisson equation for G[41, 43],

• as the boundary of the orbit space [44].

In this section, we review the two first approaches and give examples. The orbit space approach, which is more global, is sumarized and illustrated in section 2.2.2.

From equation (2.8), it is clear that

detG(ρ= 0, z) = 0. (2.12)

As a consequence, the kernel of G(0, z) is at least one-dimensional. Actually, one can show [38] that demanding the regularity of the solution inρ= 0 requires that the kernel is exactly one-dimensional except for a finite number of isolatedzvaluesa₁, . . . , a_N. This defines N + 1 intervals (−∞, a1],[a1, a2], . . . ,[aN,∞) on the z axis, the so-called rods of the solution. There are thus semi-infinite rods, (−∞, a₁] and [a_N,∞), and finite rods [a_k, a_k+1]. One can also have the case when there is only one infinite rod (−∞,+∞).

Given a rodR (thusR is an interval on thezaxis), there exists a linear combination of the Killing vectors

v=

D−3

X

a=0

v^aξ_(a) (2.13)

such that

G(0, z)abv^b = 0, z∈R , a= 0, . . . , D−3. (2.14) This vectorv is called thedirection of the rodR. In other words, the directionv of a rod is a Killing vector whose norm|v|² =Gabv^av^b vanishes one the rod

|v|² −−−→^ρ^→⁰ 0, z∈R . (2.15)

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2.2. Stationary axisymmetric solutions in vacuum 27 The position of the rods together with the corresponding directions is called the rod structure of a solution.

The rodR is said to be timelike or spacelike depending on the sign of|v|²/ρ² on the rod (ρ= 0, z∈R):

|v|²

ρ² (0, z)<0 ⇒ the rod is timelike, (2.16)

|v|²

ρ² (0, z)>0 ⇒ the rod is spacelike. (2.17) If the rod is spacelike, the condition (2.15) translates into the presence of fixed points of the orbits ofvon the rod. These correspond to potential conical singularities. They are avoided ifvcorresponds to a compact direction, with a certain periodicity [38, 42]. More- over, to prevent closed timelike curves, one must require that the directions of spacelike rods do not contain a time component.

For a timelike rod, one says that v becomes null on the rod. Moreover, if v is nor- malized so that its coefficient along the asymptotic time-translation generatorξ₍₀₎ is one, v⁰ = 1, the other components correspond to the angular velocities at the horizon. For example, at a rod corresponding to a rotating horizon, the direction v is typically of the form

v=ξ₍₀₎+ Ωiξ_(i) (2.18)

where Ω_i is the angular velocity along φ_i at the horizon.

Rods can also be seen as sources of a Poisson equation. This is most easily seen in the particular case when the Killing metric G is diagonal, that is, when the Killing vectors not only commute but are also perpendicular. These correspond to static axisymmetric or Weyl solutions. In that case, one can write the metric (2.7) in the following diagonal form

ds² =−e^2U⁰dt²+^D

−3

X

i=1

e^2Uⁱ(dxⁱ)²+e^2ν(dρ²+dz²). (2.19) The equation of motion (2.9) for Gcan then be rewritten [41] as

(∂_ρ²+1

ρ∂_ρ+∂_z²)U_a= 0, a= 0, . . . , D−3. (2.20) This is a Laplacian equation on the auxiliary three-dimensional flat space

ds²=dz²+dρ²+ρ²dγ². (2.21) where γ is some auxiliary, non physical, coordinate. In ρ = 0 we can have sources.

Moreover, equation (2.8) is equivalent to

D−3

X

a=0

U_a= logρ . (2.22)

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Note that logρ is the solution of a Poisson equation whose source is an infinite rod of zero thickness and linear mass density 1/2 along the axis ρ = 0. Each U_a is actually the potential corresponding to a rod source, with the condition that they add up to the potential of an infinite rod. In the particular case of Weyl solutions, these rod sources coincide with the rods defined as intervals above.

Three types of rods are possible: semi-infinite with positive infinity, semi-infinite with negative infinity and finite. In the first case of a semi infinite rod [a_k,+∞) with linear densityδ, the potential is

U =δlogµ_k (2.23)

where

µ_k =p

ρ²+ (z−a_k)²−(z−a_k). (2.24) This potential corresponds to a rod lying whereµ_k→0, thus forρ→0 and z≥a_k.

In the second case, when the rod extends along (−∞, ak], the corresponding potential is

U =δlog ¯µ_k (2.25)

where

¯ µ_k=p

ρ²+ (z−a_k)²+ (z−a_k) = ρ²

µ_k. (2.26)

The potential for a finite rod of density δ along the finite rod [ak−1, ak] is U =δlog

µ_k−1

µ_k

. (2.27)

The regularity requirement as to which the kernel of G(ρ = 0, z) must be one- dimensional implies in this case that the linear density of the rods must be δ = 1/2.

Indeed, the rods then do not overlap and therefore only one of the eigenvalues ofG(0, z) vanishes for a given z.

Using these considerations, one sees that the components of the diagonal Killing metric G are given, up to a sign by µ_k, ¯µ_k or ^µ_µ^k−1_k for a certain k. For each index a∈ {0, . . . , D−3}, the component G_aa thus vanishes on a certain rod (or a collection of rods). The corresponding rod directionv, defined by

X

c

G_bcv^c=G_bbv^b= 0, b= 0, . . . , D−3, (2.28) can thus be simply taken to bev^a= 1,v^b= 0 if b6=a. In summary, for a static solution, the direction of the rod whereG_aa vanishes isv=ξ_(a).

Note that the rod structure can also be defined for a metric in a theory with p-forms but it then only describes the gravity part of that theory.

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2.2. Stationary axisymmetric solutions in vacuum 29

Examples

The difficulty in uncovering the rod structure of a known solution consists essentially in finding the right change of coordinates that will bring the solution into the canonical form (2.7). These are called the canonical coordinates. We do not go into the details of these changes of coordinates here² but simply give the rod structure of rotating black holes in four and five dimensions. The details can be found in [38]. The flat and static analogs are directly obtained by taking the limit where the mass or the rotation parameters vanish.

The Kerr black hole. In canonical coordinates (t, φ, ρ, z), where ∂/∂t and ∂/∂φ are the two commuting Killing vectors, there are three rods that in a certain choice of the z coordinate can be described as (−∞,−α], [−α, α] and [α,∞). The parameter α is constructed out of the mass M and the rotation parameteraof the Kerr black hole:

α=p

M²−a². (2.29)

The Kerr black hole is represented on picture 2.1 where the points at ρ = 0 are in red.

The rods are the two semi-infinite pieces of the z axis going up and down and the finite piece along the horizon.

Figure 2.1: Kerr black hole.

The semi-infinite rods are spacelike and correspond both to the direction∂/∂φas the orbits of the rotation generator indeed has fixed points along the axis. The finite rod is timelike with direction∂/∂t+Ω∂/∂φwhere Ω is the angular velocity at the horizon. This combination of the Killing vectors indeed becomes null at the horizon. The rod structure of the Kerr black hole can therefore be represented as figure 2.2.

2The canonical coordinates are discussed in detail for the charged rotating black hole in five dimensions in appendix B.1.

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−α +α

∂φ∂

∂t∂ + Ω_∂φ^∂

Figure 2.2: Rod structure of the Kerr black hole.

Note that the static limit (a= 0) has schematically the same rod structure but with Ω = 0 andα =M. For four-dimensional Minkowski space, (M = 0 = a), the finite rod reduces to a point and one is therefore left with one infinite rod with direction∂/∂φ.

In the extremal limitM =a, the diagram degenerates and reproduces the rod diagram of Minkowski space although the solution is physically non trivial. This illustrates the fact that the rod structure alone is not sufficient to describe extremal objects.

The 5D Myers-Perry black hole. In the canonical coordinates (t, φ, ψ, ρ, z), the commuting Killing vectors are∂/∂t,∂/∂φ and ∂/∂ψ. There are again three rods delimited by points±β whereβ depends on the mass M and the two rotation parametersl1, l2 of the black hole

β = 1 4

q(M −l²₁−l₂²)²−4l²₁l₂². (2.30) This situation is represented in figure 2.3. Thezaxis is not represented as a straight line in order to emphasize the fact that the positive and the negative values ofz correspond to the fixed points of different rotations, generated respectively by∂/∂φ and ∂/∂ψ.

Figure 2.3: Myers-Perry black hole.

In other words, the semi-infinite rods [+β,+∞) and (−∞,−β] are spacelike, they have the directions∂/∂φand ∂/∂ψ respectively, while the finite rod [−β,+β] is timelike

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2.2. Stationary axisymmetric solutions in vacuum 31

−β +β

∂φ∂

∂ψ∂

v_{T L}

Figure 2.4: Rod structure of the Myers-Perry black hole.

with the direction v_{T L} ≡ ∂/∂t+ Ω₁∂/∂φ+ Ω₂∂/∂ψ where Ω₁ and Ω₂ are the angular velocities along the angles φand ψrespectively. This is summarized in picture 2.4.

As in the four-dimensional case, in the static limit, the rod structure stays schematically the same but with different rod positions and directions. TheD= 5 asymptotically flat space consits of two semi-infinite rods that interesect at z= 0.

2.2.2 Orbit space

Another definition of rod structure is introduced in the context of orbit space. Following [44], let (M, gµν) describe the spacetime of aD-dimensional stationary asympotically flat black hole in vacuum withD−3 axial symmetries,D= 4,5. For simplicity,M is assumed to be the exterior of the black hole, with the horizon as boundary∂M =H. The isotropy group G of this spacetime is generated by theD−2 commuting Killing fields -ξ₍₀₎ = _∂t^∂ for time translations and ξ_(i)= _∂φ^∂_i,i= 1, . . . , D−3, for rotations. One thus has

G=R×U(1) if D= 4,

G=R×U(1)×U(1) if D= 5. (2.31)

The orbits of the points ofM under the action ofGgive rise to an equivalence relation.

Two points x, y ∈ M are in the same equivalence class if there exists g ∈ G such that y=gx. The set of these equivalence classes, given by the factor space ˆM =M/G, is the orbit space of the spacetimeM.

To fix the ideas, consider the Kerr spacetime inD= 4. After factoring out the time translations and the rotation, one is left with the two dimensional space on the right of the red line in figure 2.1. This region is the orbit space of the Kerr spacetime.

It is shown in [44] that the orbit space ˆM is a simply connected two-dimensional manifold with one-dimensional boundaries and corners where two boundaries intersect.

This means that locally, the manifold ˆM can be modelled over R×R (interior points), R₊×R(boundary segments) andR₊×R₊ (corners).

Moreover, ˆMcan be analytically mapped to the (ρ, z) half plane. The one-dimensional boundaries then lie along the z axis with the corners corresponding to pointsa₁, . . . , a_N of z. The intervals thus defined

(−∞, a₁],[a₁, a₂], . . . ,[a_N,+∞) (2.32)

Hidden Symmetries and Black Holes in Supergravity

Hidden Symmetries and Black Holes in

Supergravity

Ella Jamsin

Hidden Symmetries and Black Holes in

Supergravity

Ella Jamsin

Contents

Chapter 1

Introduction

1.1 String theory and supergravity

1.2 Symmetries and Kac-Moody algebras

1.3 Black holes in higher dimensions

1.4 Solution generating techniques

1.5 Overview

Part I

Preliminaries

Chapter 2

Black holes in higher dimensions

2.1 Vacuum solutions

2.1.1 Spherical topology

2.1.2 New topologies in D = 5

2.1.3 Some words on D > 5

2.2 Stationary axisymmetric solutions in vacuum

2.2.1 Rod structure

2.2.2 Orbit space