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The homogeneity conjecture for supergravity backgrounds

The homogeneity conjecture for supergravity backgrounds Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 The homogeneity conjecture for supergravity backgrounds Jos e Miguel Figueroa-O'Farrill Maxwell Institute for Mathematical Sciences & School of Mathematics, The University of Edinburgh, King's Buildings, Edinburgh EH9 3JZ, United Kingdom E-mail: j.m.figueroa@ed.ac.uk Abstract. These notes record three lectures given at the workshop \Higher symmetries in Physics", held at the Universidad Complutense de Madrid in November 2008. In them we explain how to construct a Lie (super)algebra associated to a spin manifold, perhaps with extra geometric data, and a notion of privileged spinors. The typical examples are supersymmetric supergravity backgrounds; although there are more classical instances of this construction. We focus on two results: the geometric constructions of compact real forms of the simple Lie 7 8 15 algebras of type B , F and E from S , S and S , respectively; and the construction of 4 4 8 the Killing superalgebra of eleven-dimensional supergravity backgrounds. As an application of this latter construction we show that supersymmetric supergravity backgrounds with enough supersymmetry are necessarily locally homogeneous. 1. Geometric construction of exceptional Lie algebras The Killing{Cartan classi cation of complex simple Lie algebras consists of four in nite series of classical Lie algebras: A , B , C and D , and a small number of exceptional Lie n1 n2 n3 n4 algebras: E , E , E , F and G , where the subscripts denote the ranks and in the classical 6 7 8 4 2 case they have been chosen so as to avoid low rank isomorphisms. The classical Lie algebras are well-understood: they are matrix algebras and their compact real forms are the Lie algebras of (special) unitary matrices over R (B and D), C (A) and H (C ). In contrast, the exceptional Lie algebras result from \baroque" constructions involving octonions or else from constructions involving spinors, as explained by Adams in his posthumous lecture notes [1] and, for the case of E , also in [2]. It is this latter construction which we will geometrise in today's lecture, using a device well-known in supergravity and which will be subject of the next two lectures: the so-called Killing superalgebra. The basic idea of these lectures is to assign to a spin manifold a 2-graded algebra. In this rst lecture we will consider the particular example of the exceptional Hopf bration 7 15 8 (1) S ! S ! S ; n n+1 15 7 where S stands for the unit n-sphere in R . If we think of S  O O and S  O, then 8 1 S OP is the octonionic projective plane. Applying the Killing superalgebra construction to the spaces in the above bration we will obtain compact (or split) real forms of the simple Lie algebras of type B , E and F , respectively. We have been unable thus far to pinpoint 4 8 4 the relation between these Lie algebras which is suggested by the Hopf bration relating the corresponding spaces. This rst lecture is based on [3]. c 2009 IOP Publishing Ltd 1 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 1.1. Cli ord algebras, spin group and spinor representations We start with a ash review of Cli ord algebras, the spin group and the spinor representations. For more details, see the books [4] or [5]. Let E;h;i be a euclidean vector space. For instance, we could take E = R with the standard \dot" product. We de ne the Cli ord algebra C`(E) to be the associative algebra obtained by quotienting the tensor algebra T (E) by the two-sided ideal generated by elements of the form x x +hx; xi 1; in symbols, C`(E) = T (E)=fx x +hx; xi 1g : (2) Since the ideal is not homogeneous, the Cli ord algebra is not graded, but only ltered. The associated graded algebra is the exterior algebra E , whence we may think of the Cli ord algebra as a quantisation of the exterior algebra. Since the ideal has even parity, the Cli ord algebra inherits a 2-grading which agrees under the \classical limit" with the 2-grading of E into odd and even forms. When E = R relative to the usual dot product, we call the corresponding Cli ord algebra C`(n). Up to isomorphism we have the following table of euclidean Cli ord algebras: n 0 1 2 3 4 5 6 7 k + 8 (3) C`(n) R C H H H H(2) C(4) R(8) R(8) R(8) C`(k) R(16) where we use the notation K(m) for the algebra of mm matrices with entries in K, and where the last column goes by the name of Bott periodicity. The subspace  E  C`(E) is a Lie subalgebra under the Cli ord commutator isomorphic to so(E). Exponentiating inside the (associative) Cli ord algebra, gives (for n > 2) a simply- even connected Lie group Spin(E)  C`(E) , called the spin group of E. Conjugating with Spin(E) preserves E  C`(E) and de nes a two-to-one group homomorphism Spin(E) ! SO(E) : (4) When E = R with the standard dot product, we denote the spin group by Spin(n). The Cli ord algebra C`(n) is isomorphic either to a matrix algebra or to two copies of a matrix algebra, and as such has either one or two inequivalent irreducible representations. This follows from the fact that R(n) and H(n) have up to isomorphism a unique irreducible n n representation isomorphic to R and H , respectively, whereas C(n) has two non-isomorphic irreducible representations: C and its complex conjugate representation. Similarly, R(n)R(n) n n and H(n)  H(n) have two inequivalent irreducible representations, isomorphic to R or H , respectively. These are called the pinor representations of C`(E). Restricting them to Spin(E) one obtains (perhaps reducible) representations called spinor representations and denoted S(E). even The type (R, C or H) of S(E) follows from the fact that Spin(E)  C`(E) and that even C`(n) C`(n 1). For example, the spinor representations of Spin(n) for small n, have types H for n = 3; 4; 5, C for n = 2; 6 and R for n = 7; 8; 9. This is consistent with the low-dimensional isomorphisms Spin(2) = U(1), Spin(3) = Sp(1), Spin(4) = Sp(1)  Sp(1), Spin(5) = Sp(2), Spin(6) = SU(4). We will be using the fact that S(E) admits a C`(E)-invariant inner product (;) obeying (x ; ) = ( ; x ) ; (5) 1 2 1 2 for all 2 S(E) and x 2 E  C`(E). This means that (;) is also Spin(E)-invariant. For E euclidean, as we have been assuming, the spinor inner product is also positive-de nite. However if we allow for E to have arbitrary signature, then all seven types of elementary inner products appear among the spinor inner products. 2 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 The transpose of the Cli ord action E S(E) ! S(E) de nes a real bilinear map S(E) S(E) ! E which in the cases of interest in this lecture will be skewsymmetric, whence de nes a map  S(E) ! E. This will form part of a Lie bracket on a 2-graded algebra whose odd subspace will be isomorphic to (a subspace of ) S(E). 1.2. Globalisation: spin geometry Let (M ; g) be a riemannian manifold. At every point x 2 M we can consider the orthonormal frames for the tangent space T M . This is the bre at x of a principal bre bundle O(M ) called, unsurprisingly, the bundle of orthonormal frames. If M is oriented, and restricting to oriented frames, we obtain a subbundle SO(M ). The obstruction to the existence of SO(M ) is the triviality of det(TM ) which is captured by the rst Stiefel{Whitney class w (TM ) 2 H (M ;Z ). 1 2 Hence roughly speaking half the manifolds are orientable. Assuming that M is oriented, a spin structure on M is a lift Spin(M ) ! SO(M ) restricting brewise to the two-to-one homomorphism Spin(n) ! SO(n). The obstruction is now captured by the second Stiefel{ Whitney class w (TM ) 2 H (M ;Z ), whence roughly speaking one quarter of all manifolds are 2 2 spin. If w (TM ) = 0, the set of inequivalent spin structures are in one-to-one correspondence with H (M ;Z ) Hom( (M );Z ), so roughly speaking to the assignment of a sign to every 2 1 2 noncontractible loop. n n+1 n For example, if S  R is the unit sphere, then T S is the perpendicular complement n+1 n of the line in R through the origin and x. An oriented orthonormal basis for T S is then ? n an oriented orthonormal frame for x R and hence in one-to-one correspondence with the n+1 points in SO(n). Adding x itself we obtain an oriented orthonormal frame for R and hence an element of SO(n + 1): the element which takes the standard orthonormal basis to that one. Conversely, we have a map SO(n + 1) ! S sending the matrix g 2 SO(n + 1) to its rst column, n+1 say x, which is a unit vector in R . The bre of this map consists of the remaining n columns, which form an oriented frame in the n-dimensional subspace perpendicular to x. In other words, n n SO(S ) = SO(n + 1). The spin cover Spin(S ) is precisely the spin group Spin(n + 1) and since (S ) = f1g (for n > 1), there is a unique such spin structure. For n = 1 there are two spin structures, which physicists like to call Neveu{Schwarz and Ramond [2]. Let  : Spin(n) ! GL(S) be a spinor representation of Spin(n) and de ne the associated vector bundle $ = Spin(M ) S ; (6) called a bundle of spinors. Its sections are called spinor elds. (Had the lectures been given in Spanish, the spinor bundle would have been called €!) The tangent space (T M; g ) to M at x is a euclidean vector space and gives rise to a Cli ord x x algebra C`(T M ). As x varies, this globalises to a bundle C`(TM ) of Cli ord algebras. As a vector bundle, we have a natural isomorphism C`(TM ) = T M . We will always think of spinor representations as the restriction to the spin group of a pinor representation of the Cli ord algebra. Globalising, this means that the spinor bundle $ will always admit an action of the Cli ord bundle C`(TM ), making it into a bundle of Cli ord modules. In this way, di erential forms, which are sections of T M , will be able to act on spinor elds via the natural isomorphism T M = C`(TM ) and the action of C`(TM ) on $. The Levi-Civit a connection on SO(M ) lifts to a connection on Spin(M ) and hence de nes a connection on any associated vector bundle. In particular we have a covariant derivative r on sections of $: for all vector elds X 2 X(M ) and spinor elds 2 ($), r 2 ($). The covariant derivative r along X is linear and obeys the Leibniz rule r (f ) = (Xf ) + fr ; (7) X X for all functions f 2 C (M ). 3 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 1.3. Killing spinors and the cone construction The Levi-Civit a connection may be used to write down natural equations on spinors, whose solutions de ne privileged notions of spinor elds: parallel spinors: r = 0. By the holonomy principle, the holonomy group of r must be included in the stabilizer of a spinor. The determination of which manifolds admit parallel spinors was thus solved by Wang [6] using Berger's holonomy classi cation. The irreducible holonomy groups of manifolds admitting parallel spinors are SU(n), Sp(n), G and Spin(7). Killing spinors: r = X  for all X 2 X(M ) for some nonzero constant  2 C, called the Killing constant. Iterating the de nition of a Killing spinor, we nd that M 2 2 is Einstein with scalar curvature proportional to  , which means that  2 R and hence 2 R [iR . This gives rise to two separate notions of real or imaginary Killing spinors, according to whether  is real or imaginary, respectively. In this lecture we will concentrate on the case of real Killing spinors. Moreover by rescaling the metric, if necessary, we may always take  =  . Therefore such a manifold M is Einstein with positive scalar curvature and, if complete, is compact by the Bonnet{Myers theorem. The question of which complete spin manifolds admit real Killing spinors was solved by B ar [7] via his celebrated cone construction by mapping the problem to the problem of determining which manifolds admit parallel spinors. Indeed, given a spin manifold (M; g) we de ne its metric cone C (M ) = R  M , with metric 2 2 g = dr + r g ; (8) + n n+1 where r > 0 is the parameter of the R . For example, if M = S , then C (M ) = R n f0g. In this case, and in this case alone, the metric extends smoothly to the origin and it is the at n+1 metric on R written in spherical polar coordinates. In all other cases, the metric has a conical singularity at r = 0. B ar's penetrating observation was that r =  X on M becomes the e e condition r = 0 on the cone, where the tilded objects live on the cone. The sign in the Killing spinor equation is re ected either in the chirality of the parallel spinor in the case of n odd, or in the embedding C`(n)  C`(n + 1) if n is even. This says that the existence of real Killing spinors is again a holonomy problem, albeit in an auxiliary manifold one dimension higher. A theorem of Gallot [8] says that if M is complete, then its metric cone is either at | so that M is the round sphere | or irreducible. In this latter case, we may use Wang's classi cation of holonomy groups leaving a spinor invariant. In this way one arrives at Table 1 of (types of ) complete manifolds admitting real Killing spinors. Table 1. Manifolds admitting real Killing spinors n hol r C (M ) M n+1 n n f1g R S 2m 1 SU(m) Calabi{Yau Sasaki{Einstein 4m 1 Sp(m) hyperk ahler 3-Sasaki 6 G nearly K ahler (non-K ahler) 7 Spin(7) weak G holonomy Such manifolds play an important r^ ole in the AdS/CFT correspondence, as pointed out originally in [9]. 1.4. The Killing superalgebra Let (M; g) be a spin manifold and $ ! M a bundle of C`(TM )-modules. We will assume that M admits real Killing spinors and, without loss of generality (i.e., rescaling the metric and reversing 4 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 orientation, if necessary), assume that the Killing constant  = . We de ne a 2-graded vector space g = g  g , where g is the vector space of Killing vector elds on M and 0 1 0 g = 2 ($) r = X  8 X 2 X(M ) (9) is the vector space of Killing spinors. Remember we have a real bilinear map g g ! X(M ) by 1 1 transposing the Cli ord action. Explicitly, given ; 2 g , we let [ ; ] 2 X(M ) be de ned 1 2 1 1 2 by g([ ; ]; X ) = ( ; X  ) (10) 1 2 1 2 for all X 2 X(M ). The following result explains the terminology. Lemma 1. [ ; ] is a Killing vector eld. 1 2 Proof. This is a simple calculation. For all X; Y 2 X(M ), we have g(r [ ; ]; Y ) = X g([ ; ]; Y ) g([ ; ];r Y ) X 1 2 1 2 1 2 X = X ( ; Y  ) ( ;r Y  ) 1 2 1 X 2 = (r ; Y  ) + ( ; Y r ) X 1 2 1 X 2 1 1 = (X  ; Y  ) + ( ; Y  X  ) 1 2 1 2 2 2 = ( ; (Y  X X  Y ) ) ; 1 2 whence g(r [ ; ]; Y ) + g(r [ ; ]; X ) = 0 : (11) X 1 2 Y 1 2 Let K 2 g be a Killing vector eld. Then A : TM ! TM , de ned by A (Y ) = r K , is 0 K K Y a skewsymmetric endomorphism of TM . De ne the following Lie derivative on spinor elds: L = r + %(A ) ; (12) K K K where % : so(TM ) ! gl($) is the spinor representation at the level of the Lie algebra. Taking for % any other representation of so(TM ) de nes equally well a Lie derivative on sections of the corresponding associated vector bundle. For example, if we take % to be the de ning representation on TM , we have L X = r X + A X = r X r K = [K; X ] ; (13) K K K K X which is the standard Lie derivative of vector elds. Proposition 2. The Lie derivative L on ($) obeys a number of properties: (i) [L ;L ] = L , K K [K ;K ] 1 2 1 2 (ii) L (f ) = (K f ) + fL , K K (iii) L (X  ) = [K; X ] + X  L , and K K (iv) L r = r L +r , K X X K [K;X] for all Killing vector elds K; K ; K and all f 2 C (M ), X 2 X(M ) and 2 ($). 1 2 5 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 We remark that properties (1) and (2) justify calling L a Lie derivative, whereas (3) and (4) say that L leaves invariant the Cli ord action and r, respectively. The de nition of L X K goes back to Lichnerowicz and appears in the thesis of Kosmann-Schwarzbach [10]. It appeared also more recently, in a supergravity context, in [11]. The last two properties in Proposition 2 imply that if 2 g is a Killing spinor, then so is L for all Killing vector elds K . Indeed, r L = L r r by (4) in Proposition 2 X K K X [K;X] 1 1 = L (X  ) [K; X ] since 2 g K 1 2 2 = X  L by (3) in Proposition 2 whence L 2 g . This de nes a real bilinear map g  g ! g , denoted (K; ) 7! [K; ] := K 1 0 1 1 L . We now have de ned a 2-graded multiplication on g = g  g , denoted by a bracket [;] 0 1 anticipating the fact that in some cases it will be a Lie (super)algebra. To wit, we have a map g ! g , given by the Lie bracket of vector elds, the above-de ned map g g ! g given by 0 0 0 1 1 the spinorial Lie derivative, and the map g g ! g given by transposing the Cli ord action. 1 1 0 This last map is either symmetric or skewsymmetric depending on dimension. This suggests that it may de ne a Lie (super)algebra structure on g. This requires satisfying the relevant Jacobi identity. Being a 2-graded algebra, the jacobator, the element in Hom(g ; g) whose vanishing implies the Jacobi identity, breaks up into four components depending on whether we have three, two, one or no elements in g : all elements in g : this is simply the Jacobi identity of the Lie bracket of vector elds; two elements in g and one in g : 0 1 [K ; [K ; ]] [K ; [K ; ]] = L L L L 1 2 2 1 K K K K 1 2 2 1 = [L ;L ] K K 1 2 = L by (1) in Proposition 2 [K ;K ] 1 2 = [[K ; K ]; ] ; 1 2 one element in g and two in g : this is property (3) in Proposition 2; and 0 1 all elements in g : this does not follow from the formalism and has to be checked case by case. For Lie algebras, it lives in  g g , whereas for Lie superalgebras it lives in S g g . In some cases, representation theory shows that such spaces are 0, and hence this last component of the jacobator vanishes. In other cases, such spaces are not 0, but the jacobator vanishes all the same. In most cases, however, this last component of the jacobator will not vanish. Hence the generic situation is a 2-graded -Lie (super)algebra. 1.5. Some examples of 2-graded Lie algebras 7 8 8 9 15 16 Consider now the unit spheres S  R , S  R and S  R , thought of as riemannian manifolds with the canonical spin structure given by the spin groups. In all these cases, the spinor inner product is real symmetric and positive-de nite. Since Cli ord action is skewsymmetric, so is its transpose, whence the odd-odd bracket is similarly skewsymmetric, de ning a map g ! g . In other words, g is a 2-graded (possibly) Lie algebra. Notice that [g ; g ] is an 1 0 1 1 ideal of g : this does not use the vanishing of the last component of the jacobator. Since g , the 0 0 isometry Lie algebra of the above spheres, is simple, we see that [g ; g ] = g in this case. 1 1 0 To determine g as an g -module we use the cone construction and the fact, proved in [11], 1 0 that this construction is equivariant under the action of g , which is naturally a Lie subalgebra 6 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 of the isometries of the cone. This means that [K; ] = L can be lifted and calculated on the cone: e e e e L = r + %e(A ) ; (14) e e e K K K e e e but is parallel and (relative to at coordinates on the cone) and A are constant because K is a linear vector eld. Therefore this is the standard action of g = so(n + 1) (n = 7; 8; 15) on (positive-chirality, when applicable) spinors: S(8) , S(9) and S(15) , all of which are irreducible + + representations. In all cases, a roots-and-weights calculation (made less painful by using LiE [12]) shows that g = 0 ; (15) whence the Jacobi identity is satis ed and g becomes a 2-graded Lie algebra. To identify the Lie algebras in question we simply observe that g being simple and g being irreducible, implies 0 1 that g is simple. The dimensions are easy to compute and the Lie algebras are thus easy to recognise from the Killing{Cartan classi cation. The results are summarised in Table 2. Table 2. Killing Lie algebras of some spheres M g dim g g dim g dim g g 0 0 1 1 S so(8) 28 S(8) 8 36 B + 4 S so(9) 36 S(9) 16 52 F S so(16) 120 S(16) 128 248 E + 8 Since the inner products on g and g are invariant and positive-de nite, g is a compact real 0 1 form of the corresponding complex simple Lie algebra. By the usual device of taking ig instead of g , we may obtain the maximally split real forms. 2. Supergravity backgrounds Supergravity is an extension of Einstein (or Einstein{Maxwell) theory. At the level of its solutions, it is given by some geometric data (g; F; : : : ), where g is a local lorentzian metric and F; : : : stand for extra elds, all subject to partial di erential equations of the form Einstein Ric(g) Rg = T (F; : : : ) (16) \Maxwell" dF = 0 and d ? F =  (17) The details depend on the supergravity theory in question and have hence kept purposefully vague in the above description. We will consider here only so-called Poincar e supergravities. There are other supergravities: massive, gauged,... Supergravity theories are dictated by the representation theory of the Poincar e superalgebras. There are (physically interesting) supergravity theories in dimension d  11 and lorentzian signature, meaning that the local metric g is lorentzian. Supergravity theories are among the jewels of twentieth century theoretical physics and a good review of the structure of supergravity theories from the representation theory point of view can be found in [13]. 2.1. Eleven-dimensional supergravity My favourite, and to some extent the simplest yet nontrivial, supergravity theory is the unique eleven-dimensional supergravity theory. Its existence was conjectured by Nahm [14], whereas 7 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 it was constructed by Cremmer, Julia and Scherk [15]. Its eld content is a lorentzian eleven- dimensional metric g and a closed 4-form F . We can motivate this as follows. Supergravity is a theory invariant under local supersymmetry, hence the spectrum should carry a representation of the corresponding supersymmetry algebra. In the case of eleven- dimensional supergravity this is the eleven-dimensional Poincar e superalgebra (so(1; 10) 1;10 R )  S(1; 10), where S(1; 10) is the spinor representation of Spin(1; 10). It is not hard to show, using Bott periodicity, that C`(1; 10) = R(32)  R(32) and hence S(1; 10) = R . 1;10 The supertranslation ideal generated by R  S(1; 10) has as nonzero brackets the projection 2 1;10 2 1;10 S S(1; 10) ! R of the symmetric square of the spinor representation S S(1; 10) = R 2 1;10 5 1;10 R   R onto the vector representation of Spin(1; 10). Irreducible unitary representations of the Poincar e superalgebra are induced by 1;10 representations of the supertranslation ideal generated by S(1; 10)  R . This is done by 1;10 1;10 rst xing a character of the abelian translation ideal R ; that is, a momentum p 2 R . Since we are interested in a theory of gravity, which we expect even in eleven dimensions to be a long range force, we require a massless representations, whence p = 0, but p 6= 0. The little group of p, which is the maximal compact subgroup of the stabiliser of p in Spin(1; 10) is isomorphic to Spin(9). Once a momentum p has been xed, the supertranslation ideal takes the form of a Cli ord algebra [Q ; Q ] = 2 (Q ; p Q ) 1 ; (18) 1 2 1 2 where Q 2 S(1; 10) and the spinor inner product is symplectic in this signature, whence the bracket here is symmetric. The bilinear form de ning the Cli ord algebra, hQ ; Q i = 1 2 (Q ; p Q ) is degenerate because p = 0. In fact, it has rank 16. This is shown by exhibiting 1 2 a \dual" momentum q such that q = 0 and p q = 1. Then S(1; 10) = ker p ker q, where the kernel refers to the Cli ord action. The common stabiliser of p and q in Spin(1; 10) is a Spin(9)- subgroup, which we can identify with the little group of either p or q. The degenerate Cli ord algebra (18) becomes an honest Cli ord algebra on ker q  S(1; 10) isomorphic to C`(16), and in fact, as Spin(9)-module, ker q is the spinor module S(9). There is, up to isomorphism, a unique irreducible representation of C`(16) and it is real and of dimension 256. Indeed, by Bott periodicity, C`(16) C`(8) R(16) R(16) R(16) R(16 ) : (19) = = = R R As a Spin(9)-module, this is nothing but S(S(9)); that is, spinors of spinors! A roots-and-weights calculation shows that as a representation of Spin(9) we have 2 9 3 9 9 S(S(9)) S (R )  R  RS(R ) ; (20) 9 2 where R stands for the vector representation of Spin(9), S denotes traceless symmetric tensors and RS stands for the Rarita{Schwinger representation, which is the subrepresentation of 9 9 S(9) consisting of the kernel of the Cli ord action R S(9) ! S(9). Counting dimensions, we see that for the bosonic part of the representation 2 9 3 9 dim S (R ) + dim  R = 44 + 84 = 128 ; (21) whereas for the fermionic part of the representation 9 9 dim RS(R ) = dimR S(9) dim S(9) = 16 9 16 = 128 ; (22) 2 9 whence the physical degrees of freedom match, as expected. In terms of elds, S (R ) 3 9 parametrise the uctuations of a metric tensor g, whereas  R parametrises the uctuations of a (locally de ned) 3-form potential A, and RS(R ) parametrises the uctuations of a gravitino. 8 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 The supergravity action, ignoring terms involving the gravitino, consists of three terms: an Einstein{Hilbert term, a Maxwell term and a Chern{Simons term. The lagrangian density is given by 1 1 R dvol F ^ ?F + F ^ F ^ A ; (23) 4 12 where F = dA locally. Although A appears explicitly in the above lagrangian, the Euler{ Lagrange equations only involve F . The equations are of Einstein{Maxwell type with a twist provided by the Chern{Simons term; namely, the Maxwell equation is nonlinear: d ? F = F ^ F : (24) 2.2. Supersymmetric supergravity backgrounds We de ne a (bosonic) eleven-dimensional supergravity background to be an eleven- dimensional lorentzian spin manifold (M; g; $) and a closed 4-form F 2 (M ) subject to the Einstein{Maxwell equations derived from the lagrangian (23). The lagrangian (23) admits a supersymmetric completion by adding extra terms involving the gravitino 2 (M; $). The variation of the gravitino under supersymmetry de nes a connection D on the spinor bundle $: 1 1 D := r +  F  + X ^ F  ; (25) X X X 6 12 [ 1 for all 2 ($) and X 2 X(M ) and where X 2 (M ) is the one-form such that X (Y ) = g(X; Y ) for all Y 2 X(M ). The connection D is the fundamental object in this game, as it encodes virtually all the information of the theory. For example, the Einstein{Maxwell equations can be recovered by demanding the vanishing of the Cli ord-trace of its curvature. More explicitly, let e be a i i i pseudo-orthonormal frame for M and let e denote the dual frame, de ned by g(e ; e ) =  . Then, as shown in [16], the eld equations de ning the notion of a supergravity background are equivalent to i D e  R (e ; X ) = 0 8 X 2 X(M ) : (26) A nonzero spinor eld 2 ($) which is D-parallel is called a (supergravity) Killing spinor. Although this seems a priori to be a generalisation of the notion of a parallel spinor, it is in fact the original notion of a Killing spinor. The geometrical notion in the rst lecture is a special case of the supergravity Killing spinor equation for a particular Ansatz for (M; g; F ), known as a Freund{Rubin background [17]. Being a linear equation, Killing spinors form a vector space, which anticipating the construction of the Killing superalgebra, will be denoted g . Being de ned by a parallel condition, a Killing spinor is determined by its value at a point, whence the dimension of g is bounded above by the rank of the spinor bundle; that is, dim g  32. The ratio 1 1 dim g = (27) is called the supersymmetry fraction of the background (M; g; F ). If  > 0, (M; g; F ) is said to be supersymmetric. 2.3. Examples A large number of supersymmetric backgrounds are known. Maximally supersymmetric backgrounds { those with  = 1 | have been classi ed in [18, 19, 20]. For such backgrounds, D 9 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 is at and the equations of motions are automatically satis ed. These backgrounds are related as follows: KG 88 ffL r L r L r L PL PL r L r L r L r L 7 4 AdS  S AdS  S 4 7 L r L r L r L r L r L r L r %%  yyr 1;10 where AdS is the n-dimensional anti de Sitter spacetime | i.e., lorentzian hyperbolic space |, KG is a special type of plane wave [21] whose geometry is described by a lorentzian symmetric 1;10 space of Cahen{Wallach type [22], and R is Minkowski spacetime with F = 0. The arrows labelled \PL" are Penrose limits, described in this context in [23, 24], but tracing their origin to work of Guv  en [25] and, of course, Penrose [26]. The undecorated arrows are zero-curvature limits. 7 4 The AdS S and AdS S backgrounds depend on a parameter, interpreted as the scalar 4 7 curvature of the eleven-dimensional geometry. The radii of curvature of the factors are in a ratio of 2 : 1, whence these backgrounds do not describe realistic compacti cations as they once were thought to do. They are known as Freund{Rubin backgrounds. The Killing spinors of the Freund{Rubin backgrounds are of geometric Killing spinors on the two factors: real on the riemannian factor and imaginary on the lorentzian factor. One can substitute either factor by an Einstein manifold with the same scalar curvature and admitting the relevant kind of Killing spinors. In particular one can consider AdS X , where X is a riemannian manifold admitting real Killing spinors, whence its cone has holonomy contained in Spin(7). Whenever X is not a sphere, the resulting background has a smaller fraction  of supersymmetry. They can be understood as near-horizon geometries of M2-branes, to which we now turn. The M2-brane is a interesting background with  = , discovered in [27] and interpreted as an interpolating soliton in [28]. It is described as follows: 2=3 2 1;2 1=3 2 2 2 7 g = H ds (R ) + H dr + r ds (S ) 1;2 1 F = dvol(R )^ dH (28) H (r) = + ; 2 1;2 1;2 where ds (R ) and dvol(R ) are the metric and volume of 3-dimensional Minkowski spacetime, 2 7 8 ds (S ) is the metric on the unit sphere in R and H is a two-parameter harmonic function on R . If we take ! 0 while keeping 6= 0 xed, we obtain eleven-dimensional Minkowski spacetime with F = 0, but taking ! 0 while keeping 6= 0 xed, one obtains AdS  S with scalar curvature depending on . Therefore the M2-brane interpolates between these two maximally supersymmetric backgrounds. The Killing spinors are given by 1=6 = H ; (29) where is a parallel spinor in the asymptotic Minkowski spacetime obeying the projection condition 1;2 dvol(R ) = : (30) 1 1 1;2 2 1;10 Since dvol(R ) = 1 and the parallel spinors of R split into two half-dimensional eigenspaces 1;2 1 of dvol(R ). As a result the solution has  = . As observed in [9], replacing S by another seven-dimensional manifold admitting real Killing spinors | that is, weak G -holonomy, Sasaki-Einstein or 3-Sasaki manifolds | we obtain an 10 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 M2-brane at a conical singularity in an 8-dimensional manifold with Spin(7), SU(4) or Sp(2) holonomy, respectively. To this day a large class class of backgrounds with various values of  are known to exist. General local metrics with minimal supersymmetry have been written down in [16, 29]. To date, the only fraction which has been ruled out is  = [30, 31]. 3. The Killing superalgebra of supergravity backgrounds In the rst lecture we saw that from a spin manifold admitting Killing spinors one could de ne (in the good cases) a 2-graded Lie algebra and in this way we recovered the compact real forms of the simple Lie algebras of types B , F and E . At its most basic, what we have is a spin 4 4 8 manifold with a privileged subspace of spinor elds which then generates a 2-graded algebra with the spinors being the odd-subspace. In the second lecture we saw how supersymmetric (eleven-dimensional) supergravity backgrounds gave rise to precisely such a situation: an eleven-dimensional lorentzian spin manifold with a privileged notion of spinor: the supergravity Killing spinors, which are parallel with respect to a connection D on the spinor bundle. Unlike the spin connection r, the connection D is not induced from a connection on the tangent bundle: it is genuinely a spinor connection. The spinor bundle is a real rank-32 symplectic vector bundle, but D does not preserve the symplectic structure. In fact, as shown by Hull [32], the holonomy algebra of D is generically contained in sl(32;R) since only the `determinant' is preserved. In this third and last lecture we will see that to every supersymmetric background of eleven- dimensional supergravity one can assign a Lie superalgebra by the techniques in the rst lecture and using it we will show that if  is suciently large, the background is forced to be homogeneous. This lecture is based on [33]. 3.1. The Killing superalgebra Let (M; g; F ) be a supersymmetric eleven-dimensional supergravity background. Following the method in the rst lecture, let us de ne a 2-graded vector space g = g  g , where 0 1 g = fX 2 X(M )jL g = 0 = L Fg (31) 0 X X is the Lie algebra of F -preserving isometries of the background, and g = f 2 ($)jD = 0g (32) is the space of Killing spinors. Clearly g is nite-dimensional, since as mentioned above dim g  32 and dim g  66, which is the maximum dimension of the isometry algebra of 1 0 1;10 an eleven-dimensional lorentzian manifold. It is only for R with F = 0 that both of these upper bounds are realised. Given 2 ($), we may de ne [ ; ] 2 X(M ) by transposing the Cli ord action. This is nonzero because this bilinear product is now symmetric, since the inner product on $ is symplectic. In fact, it is not dicult to show that [ ; ] is always causal; that is, it has non- positive minkowskian norm. Lemma 3. If 2 g then [ ; ] 2 g . 1 0 Proof. We rst show that [ ; ] is Killing. We rst have that g(r [ ; ]; Y ) = Xg([ ; ]; Y ) g([ ; ];r Y ) since rg = 0 X X = X ( ; Y  ) ( ;r Y  ) by de nition of [ ; ] = (r ; Y  ) + ( ; Y r ) X X = 2 (r ; Y  ) since ( ; Y  ) = ( ; Y  ) X 1 2 2 1 = 2 (Y r ; ) : 11 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 Now since D = 0, 1 1 r =  F  X ^ F  ; X X 6 12 whence 1 1 [ g(r [ ; ]; Y ) = (Y   F  ; ) + Y  (X ^ F ) ; X X 3 6 1 [ = (Y ^  F   F ) ; X Y X 1 [ [ [ + (Y ^ X ^ F + g(X; Y )F X ^  F ) ; 1 1 [ [ = (  F  ; ) + Y ^ X ^ F  ; ; Y X 3 6 where we have used that for every 4-form  2 (M ), ( ; ) = 0 : It follows that g(r [ ; ]; Y ) + g(r [ ; ]; X ) = 0 ; X Y whence [ ; ] is a Killing vector. One can also prove that L F = 0. Indeed, since dF = 0, [ ; ] L F = d F and it is just a calculation to show that [ ; ] [ ; ] F = dB ; [ ; ] where B 2 (M ) is the 2-form in the square of : [ [ B(X; Y ) = ; X ^ Y  : This result explains why is a called a Killing spinor, since it is the \square root" of a Killing vector. It follows by the usual polarisation trick that if ; 2 g , then [ ; ] 2 g . 1 2 1 1 2 0 We therefore have a symmetric bilinear map g  g ! g denoted by ( ; ) 7! [ ; ]. 1 1 0 1 2 1 2 We now de ne a bilinear map g  g ! g using the spinorial Lie derivative L de ned in 0 1 1 X equation (12). Let X 2 g . It follows from Proposition 2 and the fact that X preserves F , that L preserves D; that is, [L ; D ] = D 8 2 ($) ; (33) X Y [X;Y ] whence if D = 0, also DL = 0. Therefore [X; ] = L de nes the desired bilinear map. X X Together with the Lie bracket of vector elds, under which g becomes a Lie algebra, we have on g = g  g the structure of a superalgebra. 0 1 In checking the Jacobi identity, one again sees as in the rst lecture that 3/4 of the jacobator is identically zero because of properties of the Lie derivative L . The fourth component of the jacobator vanishes if and only if for all 2 g , [[ ; ]; ] = 0 or equivalently L = 0 : (34) [ ; ] Representation theory is not useful here, since we are interested in a general result for unspeci ed g and g . An explicit calculation (made less painful with Mathematica or Maple) shows 0 1 that this is indeed the case. Therefore we have a Lie superalgebra called the symmetry superalgebra of the background. The ideal generated by g , k = [g ; g ]  g is called the 1 1 1 1 Killing superalgebra of the background. Some (but not all, see [34]) backgrounds are such that their Killing superalgebra admits an extension m = m  g which is \maximal" in the 0 1 sense that m = [g ; g ] S g ; 0 1 1 1 where the isomorphism is one of vector spaces. When it exists, it is called the maximal superalgebra of the background. 12 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 3.2. Examples Let us consider some examples. The simplest is of course the Minkowski maximally 1;10 supersymmetric background R with F = 0. The symmetry superalgebra is the Poincar e superalgebra, whereas the Killing superalgebra is the supertranslation ideal, as explained in Section 2.1. The maximal superalgebra is obtained by taking m = S g and declaring m to be 0 1 0 central. The extra elements in m not in the Killing superalgebra can be understood in terms of brane charges, as explained, for example, in [35]. Backgrounds with F = 0 are said to be purely gravitational. The Killing spinors are parallel with respect to the Levi-Civit a connection r. This means that the holonomy of r is contained in the stabiliser of a spinor in Spin(1; 10). There are two types of spinor orbits in S(1; 10) and hence two stabilisers, up to isomorphism. As shown by Bryant [36] and the author [37], the orbits are labelled by the value of a quartic polynomial q, whose value q( ) at 2 S(1; 10) is the minkowskian norm of the vector [ ; ], which as mentioned above is always non-negative. If q( ) = 0 we must distinguish between the = 0 and a 25-dimensional orbit with stabiliser isomorphic to (Spin(9)nR )R  Spin(1; 10), whereas if q( ) < 0, the stabiliser is isomorphic to SU(5). This dichotomy gives rise to two types of supersymmetric purely gravitational backgrounds: one generalising the M-wave [38], where [ ; ] is a lightlike parallel vector and thus the geometry is described by a Brinkmann metric, and another generalising the Kaluza{Klein monopole [39, 40, 41]. This latter class gives rise to reducible geometries of the form R  N , where N is a riemannian ten-dimensional manifold with holonomy contained in SU(5). The Kaluza{Klein monopole is the case N = R K , with K a 4-dimensional hyperk ahler manifold. Because is parallel with respect to r, so is [ ; ] and the resulting Killing superalgebras are of the supertranslation type, sketchily [Q; Q] = P , where P is the parallel null vector in the + + case of the waves, or else [Q; Q] = translation in the at factor, for the generalised Kaluza{Klein monopoles. For backgrounds with nonzero F , as in the M2-brane discussed in the second lecture, the Killing superalgebra is still of the supertranslation type, where now [Q; Q] = translations along the brane worldvolume. For the maximally supersymmetric Freund{Rubin backgrounds the symmetry superalgebra 7 4 is simple and isomorphic to osp(8j4) in the case of AdS  S and to osp(6; 2j2) for AdS  S . 4 7 Simplicity implies that the Killing superalgebra agrees with the symmetry superalgebra. In [34] we showed via an explicit geometric construction that the maximal superalgebra is isomorphic to osp(1j32). It was shown in [24] that the Penrose limit contracts the Killing superalgebra, whence for the maximally supersymmetric KG background, obtained via the Penrose limit from the above Freund{Rubin backgrounds, the Killing superalgebra is a contraction of either of the orthosymplectic superalgebras osp(8j4) or osp(6; 2j2). This contraction was performed explicitly in [42] obtaining the Killing superalgebra previously computed in [18]. 3.3. The homogeneity conjecture Because squaring Killing spinors one obtains Killing vectors, it stands to reason that the more supersymmetric a background is, the more isometries it has. It is therefore tempting to conjecture that there given sucient supersymmetry | that is, a suciently large value of the fraction  | the background might be homogeneous, where we say that a supergravity background (M; g; F ) is homogeneous if it admits the transitive action of a Lie group via F -preserving isometries. So the question is whether there is some critical fraction  such that if a background has a fraction  >  , then it is forced to be homogeneous. All maximally supersymmetric backgrounds discussed in the second lecture are symmetric spaces, whence in particular homogeneous. On the other hand, the M2-brane, which has  = , has cohomogeneity one: with orbits labelled by the radial coordinate r in the solution. This suggests that   . 13 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 Furthermore, inspecting the catalogue of known solution with  > , one sees that they are always homogeneous. This prompted Patrick Meessen to state the Homogeneity Conjecture. All supergravity backgrounds with  > are homogeneous. In fact, we have to be a little careful because in practice we have that g; F are only locally de ned in some open neighbourhood of R , so that a more relevant notion is that of local homogeneity, which is implied by local transitivity, by which we mean that around every point there is a local frame consisting of F -preserving Killing vectors. In [33] we proved something weaker and at the same stronger than the homogeneity conjecture. We proved that if a background has  > then it is locally homogeneous, but we proved that already [g ; g ] acts locally transitively. In other words, local homogeneity is a 1 1 direct consequence of supersymmetry. 3.4. Status of the conjecture The conjecture does not just make reference to eleven-dimensional supergravity, but in fact to any Poincar e supergravity theory. Concentrating for de niteness on the ten-dimensional supergravity theories, similar results exist for these theories as well. In [43] we showed that any background of either type IIA or IIB supergravity with  > is locally homogeneous, whereas any background of type I/heterotic supergravity with  > is locally homogeneous. This latter result bene ted from the classi cation of parallelisable backgrounds [44], which in turn was made possible by the fact that the Killing spinors are de ned by the lift to the spin bundle of a metric connection with torsion. We believe that the conjecture is true as stated, but proving this for type II and eleven-dimensional supergravities will require a better understanding of the connection D. Acknowledgments These notes are a reasonably faithful transcription of three lectures delivered at the Universidad Complutense de Madrid in November 2008 as part of the Workshop on Higher Symmetries in Physics. I would like to extend my gratitude to Marco Castrill on L opez for the invitation. The notes are an expanded version of two lectures given at the Universit at Bielefeld in May 2008 as part of the Meeting on Geometry and Supersymmetry organised by Andriy Haydys, to whom I am grateful for the invitation. The results detailed in the rst lecture had their origin in lectures delivered in May 2007 at the Dipartamento di Matematica \U. Dini" of the Universit a degli Studi di Firenze. It is a pleasure to take this opportunity to thank Dmitri Alekseevsky and Andrea Spiro for arranging that visit, Luigi Mangiarotti for support, Cristina Giannotti for the many delicious meals and Suor Tarcisia of le Suore \Stabilite nella Carit a" della Villa Agape for keeping me o the streets at night. Most of the work described here is based on collaborations with a number of people, whom it is my pleasure to thank: Bobby Acharya, Matthias Blau, Emily Hackett-Jones, Chris Hull, Patrick Meessen, George Moutsopoulos, George Papadopoulos, Simon Philip, Hannu Rajaniemi, Joan Sim on and Bill Spence. Finally, these notes were prepared while on sabbatical at the Universitat de Val encia, supported under research grant FIS2008-01980. I am grateful to Jos e de Azc arraga for making this visit possible. 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(IV) 91 317{395 [11] Figueroa-O'Farrill J M 1999 Class. Quant. Grav. 16 2043{2055 (Preprint hep-th/9902066) [12] van Leeuwen M A A 1994 Euromath Bull. 1 83{94 ISSN 1359-4346 [13] Van Proeyen A (Preprint hep-th/0301005) [14] Nahm W 1978 Nucl. Phys. B135 149{166 [15] Cremmer E, Julia B and Scherk J 1978 Phys. Lett. 76B 409{412 [16] Gauntlett J P and Pakis S 2003 J. High Energy Phys. 04 039 (Preprint hep-th/0212008) [17] Freund P and Rubin M 1980 Phys. Lett. B97 233{235 [18] Figueroa-O'Farrill J M and Papadopoulos G 2001 J. High Energy Phys. 06 036 (Preprint hep-th/0105308) [19] Figueroa-O'Farrill J M 2001 Special geometric structures in String Theory ed Alekseevsky D V, Cort es V, Devchand C and Proeyen A V (Bonn: EMIS) (Preprint math.DG/0109162) [20] Figueroa-O'Farrill J M and Papadopoulos G 2003 J. High Energy Phys. 03 048 (Preprint hep-th/0211089) [21] Kowalski-Glikman J 1984 Phys. Lett. 134B 194{196 [22] Cahen M and Wallach N 1970 Bull. Am. Math. 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Grav. 22 207{226 (Preprint hep-th/0409170) [34] Figueroa-O'Farrill J M, Hackett-Jones E, Moutsopoulos G and Sim on J 2008 (Preprint 0809.5034) [35] Townsend P K (Preprint hep-th/9712004) [36] Bryant R L 2000 Global analysis and harmonic analysis (Marseille-Luminy, 1999) (S emin. Congr. vol 4) ed Bourgignon J P, Branson T and Hijazi O (Paris: Soc. Math. France) pp 53{94 (Preprint math.DG/0004073) [37] Figueroa-O'Farrill J M 2000 Class. Quant. Grav. 17 2925{2947 (Preprint hep-th/9904124) [38] Hull C M 1984 Phys. Lett. 139B 39{41 [39] Sorkin R 1983 Phys. Rev. Lett. 51 87{90 [40] Gross D and Perry M 1983 Nucl. Phys. B226 29 [41] Han S and Koh I 1985 Phys. Rev. D31 2503 [42] Hatsuda M, Kamimura K and Sakaguchi M 2002 Nucl. Phys. B637 168{176 (Preprint hep-th/0204002) [43] Figueroa-O'Farrill J M, Hackett-Jones E and Moutsopoulos G 2007 Class. Quant. Grav. 24 3291{3308 (Preprint hep-th/0703192) [44] Figueroa-O'Farrill J M, Kawano T and Yamaguchi S 2003 J. High Energy Phys. 10 012 (Preprint hep-th/0308141) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Physics: Conference Series IOP Publishing

The homogeneity conjecture for supergravity backgrounds

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Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 The homogeneity conjecture for supergravity backgrounds Jos e Miguel Figueroa-O'Farrill Maxwell Institute for Mathematical Sciences & School of Mathematics, The University of Edinburgh, King's Buildings, Edinburgh EH9 3JZ, United Kingdom E-mail: j.m.figueroa@ed.ac.uk Abstract. These notes record three lectures given at the workshop \Higher symmetries in Physics", held at the Universidad Complutense de Madrid in November 2008. In them we explain how to construct a Lie (super)algebra associated to a spin manifold, perhaps with extra geometric data, and a notion of privileged spinors. The typical examples are supersymmetric supergravity backgrounds; although there are more classical instances of this construction. We focus on two results: the geometric constructions of compact real forms of the simple Lie 7 8 15 algebras of type B , F and E from S , S and S , respectively; and the construction of 4 4 8 the Killing superalgebra of eleven-dimensional supergravity backgrounds. As an application of this latter construction we show that supersymmetric supergravity backgrounds with enough supersymmetry are necessarily locally homogeneous. 1. Geometric construction of exceptional Lie algebras The Killing{Cartan classi cation of complex simple Lie algebras consists of four in nite series of classical Lie algebras: A , B , C and D , and a small number of exceptional Lie n1 n2 n3 n4 algebras: E , E , E , F and G , where the subscripts denote the ranks and in the classical 6 7 8 4 2 case they have been chosen so as to avoid low rank isomorphisms. The classical Lie algebras are well-understood: they are matrix algebras and their compact real forms are the Lie algebras of (special) unitary matrices over R (B and D), C (A) and H (C ). In contrast, the exceptional Lie algebras result from \baroque" constructions involving octonions or else from constructions involving spinors, as explained by Adams in his posthumous lecture notes [1] and, for the case of E , also in [2]. It is this latter construction which we will geometrise in today's lecture, using a device well-known in supergravity and which will be subject of the next two lectures: the so-called Killing superalgebra. The basic idea of these lectures is to assign to a spin manifold a 2-graded algebra. In this rst lecture we will consider the particular example of the exceptional Hopf bration 7 15 8 (1) S ! S ! S ; n n+1 15 7 where S stands for the unit n-sphere in R . If we think of S  O O and S  O, then 8 1 S OP is the octonionic projective plane. Applying the Killing superalgebra construction to the spaces in the above bration we will obtain compact (or split) real forms of the simple Lie algebras of type B , E and F , respectively. We have been unable thus far to pinpoint 4 8 4 the relation between these Lie algebras which is suggested by the Hopf bration relating the corresponding spaces. This rst lecture is based on [3]. c 2009 IOP Publishing Ltd 1 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 1.1. Cli ord algebras, spin group and spinor representations We start with a ash review of Cli ord algebras, the spin group and the spinor representations. For more details, see the books [4] or [5]. Let E;h;i be a euclidean vector space. For instance, we could take E = R with the standard \dot" product. We de ne the Cli ord algebra C`(E) to be the associative algebra obtained by quotienting the tensor algebra T (E) by the two-sided ideal generated by elements of the form x x +hx; xi 1; in symbols, C`(E) = T (E)=fx x +hx; xi 1g : (2) Since the ideal is not homogeneous, the Cli ord algebra is not graded, but only ltered. The associated graded algebra is the exterior algebra E , whence we may think of the Cli ord algebra as a quantisation of the exterior algebra. Since the ideal has even parity, the Cli ord algebra inherits a 2-grading which agrees under the \classical limit" with the 2-grading of E into odd and even forms. When E = R relative to the usual dot product, we call the corresponding Cli ord algebra C`(n). Up to isomorphism we have the following table of euclidean Cli ord algebras: n 0 1 2 3 4 5 6 7 k + 8 (3) C`(n) R C H H H H(2) C(4) R(8) R(8) R(8) C`(k) R(16) where we use the notation K(m) for the algebra of mm matrices with entries in K, and where the last column goes by the name of Bott periodicity. The subspace  E  C`(E) is a Lie subalgebra under the Cli ord commutator isomorphic to so(E). Exponentiating inside the (associative) Cli ord algebra, gives (for n > 2) a simply- even connected Lie group Spin(E)  C`(E) , called the spin group of E. Conjugating with Spin(E) preserves E  C`(E) and de nes a two-to-one group homomorphism Spin(E) ! SO(E) : (4) When E = R with the standard dot product, we denote the spin group by Spin(n). The Cli ord algebra C`(n) is isomorphic either to a matrix algebra or to two copies of a matrix algebra, and as such has either one or two inequivalent irreducible representations. This follows from the fact that R(n) and H(n) have up to isomorphism a unique irreducible n n representation isomorphic to R and H , respectively, whereas C(n) has two non-isomorphic irreducible representations: C and its complex conjugate representation. Similarly, R(n)R(n) n n and H(n)  H(n) have two inequivalent irreducible representations, isomorphic to R or H , respectively. These are called the pinor representations of C`(E). Restricting them to Spin(E) one obtains (perhaps reducible) representations called spinor representations and denoted S(E). even The type (R, C or H) of S(E) follows from the fact that Spin(E)  C`(E) and that even C`(n) C`(n 1). For example, the spinor representations of Spin(n) for small n, have types H for n = 3; 4; 5, C for n = 2; 6 and R for n = 7; 8; 9. This is consistent with the low-dimensional isomorphisms Spin(2) = U(1), Spin(3) = Sp(1), Spin(4) = Sp(1)  Sp(1), Spin(5) = Sp(2), Spin(6) = SU(4). We will be using the fact that S(E) admits a C`(E)-invariant inner product (;) obeying (x ; ) = ( ; x ) ; (5) 1 2 1 2 for all 2 S(E) and x 2 E  C`(E). This means that (;) is also Spin(E)-invariant. For E euclidean, as we have been assuming, the spinor inner product is also positive-de nite. However if we allow for E to have arbitrary signature, then all seven types of elementary inner products appear among the spinor inner products. 2 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 The transpose of the Cli ord action E S(E) ! S(E) de nes a real bilinear map S(E) S(E) ! E which in the cases of interest in this lecture will be skewsymmetric, whence de nes a map  S(E) ! E. This will form part of a Lie bracket on a 2-graded algebra whose odd subspace will be isomorphic to (a subspace of ) S(E). 1.2. Globalisation: spin geometry Let (M ; g) be a riemannian manifold. At every point x 2 M we can consider the orthonormal frames for the tangent space T M . This is the bre at x of a principal bre bundle O(M ) called, unsurprisingly, the bundle of orthonormal frames. If M is oriented, and restricting to oriented frames, we obtain a subbundle SO(M ). The obstruction to the existence of SO(M ) is the triviality of det(TM ) which is captured by the rst Stiefel{Whitney class w (TM ) 2 H (M ;Z ). 1 2 Hence roughly speaking half the manifolds are orientable. Assuming that M is oriented, a spin structure on M is a lift Spin(M ) ! SO(M ) restricting brewise to the two-to-one homomorphism Spin(n) ! SO(n). The obstruction is now captured by the second Stiefel{ Whitney class w (TM ) 2 H (M ;Z ), whence roughly speaking one quarter of all manifolds are 2 2 spin. If w (TM ) = 0, the set of inequivalent spin structures are in one-to-one correspondence with H (M ;Z ) Hom( (M );Z ), so roughly speaking to the assignment of a sign to every 2 1 2 noncontractible loop. n n+1 n For example, if S  R is the unit sphere, then T S is the perpendicular complement n+1 n of the line in R through the origin and x. An oriented orthonormal basis for T S is then ? n an oriented orthonormal frame for x R and hence in one-to-one correspondence with the n+1 points in SO(n). Adding x itself we obtain an oriented orthonormal frame for R and hence an element of SO(n + 1): the element which takes the standard orthonormal basis to that one. Conversely, we have a map SO(n + 1) ! S sending the matrix g 2 SO(n + 1) to its rst column, n+1 say x, which is a unit vector in R . The bre of this map consists of the remaining n columns, which form an oriented frame in the n-dimensional subspace perpendicular to x. In other words, n n SO(S ) = SO(n + 1). The spin cover Spin(S ) is precisely the spin group Spin(n + 1) and since (S ) = f1g (for n > 1), there is a unique such spin structure. For n = 1 there are two spin structures, which physicists like to call Neveu{Schwarz and Ramond [2]. Let  : Spin(n) ! GL(S) be a spinor representation of Spin(n) and de ne the associated vector bundle $ = Spin(M ) S ; (6) called a bundle of spinors. Its sections are called spinor elds. (Had the lectures been given in Spanish, the spinor bundle would have been called €!) The tangent space (T M; g ) to M at x is a euclidean vector space and gives rise to a Cli ord x x algebra C`(T M ). As x varies, this globalises to a bundle C`(TM ) of Cli ord algebras. As a vector bundle, we have a natural isomorphism C`(TM ) = T M . We will always think of spinor representations as the restriction to the spin group of a pinor representation of the Cli ord algebra. Globalising, this means that the spinor bundle $ will always admit an action of the Cli ord bundle C`(TM ), making it into a bundle of Cli ord modules. In this way, di erential forms, which are sections of T M , will be able to act on spinor elds via the natural isomorphism T M = C`(TM ) and the action of C`(TM ) on $. The Levi-Civit a connection on SO(M ) lifts to a connection on Spin(M ) and hence de nes a connection on any associated vector bundle. In particular we have a covariant derivative r on sections of $: for all vector elds X 2 X(M ) and spinor elds 2 ($), r 2 ($). The covariant derivative r along X is linear and obeys the Leibniz rule r (f ) = (Xf ) + fr ; (7) X X for all functions f 2 C (M ). 3 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 1.3. Killing spinors and the cone construction The Levi-Civit a connection may be used to write down natural equations on spinors, whose solutions de ne privileged notions of spinor elds: parallel spinors: r = 0. By the holonomy principle, the holonomy group of r must be included in the stabilizer of a spinor. The determination of which manifolds admit parallel spinors was thus solved by Wang [6] using Berger's holonomy classi cation. The irreducible holonomy groups of manifolds admitting parallel spinors are SU(n), Sp(n), G and Spin(7). Killing spinors: r = X  for all X 2 X(M ) for some nonzero constant  2 C, called the Killing constant. Iterating the de nition of a Killing spinor, we nd that M 2 2 is Einstein with scalar curvature proportional to  , which means that  2 R and hence 2 R [iR . This gives rise to two separate notions of real or imaginary Killing spinors, according to whether  is real or imaginary, respectively. In this lecture we will concentrate on the case of real Killing spinors. Moreover by rescaling the metric, if necessary, we may always take  =  . Therefore such a manifold M is Einstein with positive scalar curvature and, if complete, is compact by the Bonnet{Myers theorem. The question of which complete spin manifolds admit real Killing spinors was solved by B ar [7] via his celebrated cone construction by mapping the problem to the problem of determining which manifolds admit parallel spinors. Indeed, given a spin manifold (M; g) we de ne its metric cone C (M ) = R  M , with metric 2 2 g = dr + r g ; (8) + n n+1 where r > 0 is the parameter of the R . For example, if M = S , then C (M ) = R n f0g. In this case, and in this case alone, the metric extends smoothly to the origin and it is the at n+1 metric on R written in spherical polar coordinates. In all other cases, the metric has a conical singularity at r = 0. B ar's penetrating observation was that r =  X on M becomes the e e condition r = 0 on the cone, where the tilded objects live on the cone. The sign in the Killing spinor equation is re ected either in the chirality of the parallel spinor in the case of n odd, or in the embedding C`(n)  C`(n + 1) if n is even. This says that the existence of real Killing spinors is again a holonomy problem, albeit in an auxiliary manifold one dimension higher. A theorem of Gallot [8] says that if M is complete, then its metric cone is either at | so that M is the round sphere | or irreducible. In this latter case, we may use Wang's classi cation of holonomy groups leaving a spinor invariant. In this way one arrives at Table 1 of (types of ) complete manifolds admitting real Killing spinors. Table 1. Manifolds admitting real Killing spinors n hol r C (M ) M n+1 n n f1g R S 2m 1 SU(m) Calabi{Yau Sasaki{Einstein 4m 1 Sp(m) hyperk ahler 3-Sasaki 6 G nearly K ahler (non-K ahler) 7 Spin(7) weak G holonomy Such manifolds play an important r^ ole in the AdS/CFT correspondence, as pointed out originally in [9]. 1.4. The Killing superalgebra Let (M; g) be a spin manifold and $ ! M a bundle of C`(TM )-modules. We will assume that M admits real Killing spinors and, without loss of generality (i.e., rescaling the metric and reversing 4 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 orientation, if necessary), assume that the Killing constant  = . We de ne a 2-graded vector space g = g  g , where g is the vector space of Killing vector elds on M and 0 1 0 g = 2 ($) r = X  8 X 2 X(M ) (9) is the vector space of Killing spinors. Remember we have a real bilinear map g g ! X(M ) by 1 1 transposing the Cli ord action. Explicitly, given ; 2 g , we let [ ; ] 2 X(M ) be de ned 1 2 1 1 2 by g([ ; ]; X ) = ( ; X  ) (10) 1 2 1 2 for all X 2 X(M ). The following result explains the terminology. Lemma 1. [ ; ] is a Killing vector eld. 1 2 Proof. This is a simple calculation. For all X; Y 2 X(M ), we have g(r [ ; ]; Y ) = X g([ ; ]; Y ) g([ ; ];r Y ) X 1 2 1 2 1 2 X = X ( ; Y  ) ( ;r Y  ) 1 2 1 X 2 = (r ; Y  ) + ( ; Y r ) X 1 2 1 X 2 1 1 = (X  ; Y  ) + ( ; Y  X  ) 1 2 1 2 2 2 = ( ; (Y  X X  Y ) ) ; 1 2 whence g(r [ ; ]; Y ) + g(r [ ; ]; X ) = 0 : (11) X 1 2 Y 1 2 Let K 2 g be a Killing vector eld. Then A : TM ! TM , de ned by A (Y ) = r K , is 0 K K Y a skewsymmetric endomorphism of TM . De ne the following Lie derivative on spinor elds: L = r + %(A ) ; (12) K K K where % : so(TM ) ! gl($) is the spinor representation at the level of the Lie algebra. Taking for % any other representation of so(TM ) de nes equally well a Lie derivative on sections of the corresponding associated vector bundle. For example, if we take % to be the de ning representation on TM , we have L X = r X + A X = r X r K = [K; X ] ; (13) K K K K X which is the standard Lie derivative of vector elds. Proposition 2. The Lie derivative L on ($) obeys a number of properties: (i) [L ;L ] = L , K K [K ;K ] 1 2 1 2 (ii) L (f ) = (K f ) + fL , K K (iii) L (X  ) = [K; X ] + X  L , and K K (iv) L r = r L +r , K X X K [K;X] for all Killing vector elds K; K ; K and all f 2 C (M ), X 2 X(M ) and 2 ($). 1 2 5 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 We remark that properties (1) and (2) justify calling L a Lie derivative, whereas (3) and (4) say that L leaves invariant the Cli ord action and r, respectively. The de nition of L X K goes back to Lichnerowicz and appears in the thesis of Kosmann-Schwarzbach [10]. It appeared also more recently, in a supergravity context, in [11]. The last two properties in Proposition 2 imply that if 2 g is a Killing spinor, then so is L for all Killing vector elds K . Indeed, r L = L r r by (4) in Proposition 2 X K K X [K;X] 1 1 = L (X  ) [K; X ] since 2 g K 1 2 2 = X  L by (3) in Proposition 2 whence L 2 g . This de nes a real bilinear map g  g ! g , denoted (K; ) 7! [K; ] := K 1 0 1 1 L . We now have de ned a 2-graded multiplication on g = g  g , denoted by a bracket [;] 0 1 anticipating the fact that in some cases it will be a Lie (super)algebra. To wit, we have a map g ! g , given by the Lie bracket of vector elds, the above-de ned map g g ! g given by 0 0 0 1 1 the spinorial Lie derivative, and the map g g ! g given by transposing the Cli ord action. 1 1 0 This last map is either symmetric or skewsymmetric depending on dimension. This suggests that it may de ne a Lie (super)algebra structure on g. This requires satisfying the relevant Jacobi identity. Being a 2-graded algebra, the jacobator, the element in Hom(g ; g) whose vanishing implies the Jacobi identity, breaks up into four components depending on whether we have three, two, one or no elements in g : all elements in g : this is simply the Jacobi identity of the Lie bracket of vector elds; two elements in g and one in g : 0 1 [K ; [K ; ]] [K ; [K ; ]] = L L L L 1 2 2 1 K K K K 1 2 2 1 = [L ;L ] K K 1 2 = L by (1) in Proposition 2 [K ;K ] 1 2 = [[K ; K ]; ] ; 1 2 one element in g and two in g : this is property (3) in Proposition 2; and 0 1 all elements in g : this does not follow from the formalism and has to be checked case by case. For Lie algebras, it lives in  g g , whereas for Lie superalgebras it lives in S g g . In some cases, representation theory shows that such spaces are 0, and hence this last component of the jacobator vanishes. In other cases, such spaces are not 0, but the jacobator vanishes all the same. In most cases, however, this last component of the jacobator will not vanish. Hence the generic situation is a 2-graded -Lie (super)algebra. 1.5. Some examples of 2-graded Lie algebras 7 8 8 9 15 16 Consider now the unit spheres S  R , S  R and S  R , thought of as riemannian manifolds with the canonical spin structure given by the spin groups. In all these cases, the spinor inner product is real symmetric and positive-de nite. Since Cli ord action is skewsymmetric, so is its transpose, whence the odd-odd bracket is similarly skewsymmetric, de ning a map g ! g . In other words, g is a 2-graded (possibly) Lie algebra. Notice that [g ; g ] is an 1 0 1 1 ideal of g : this does not use the vanishing of the last component of the jacobator. Since g , the 0 0 isometry Lie algebra of the above spheres, is simple, we see that [g ; g ] = g in this case. 1 1 0 To determine g as an g -module we use the cone construction and the fact, proved in [11], 1 0 that this construction is equivariant under the action of g , which is naturally a Lie subalgebra 6 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 of the isometries of the cone. This means that [K; ] = L can be lifted and calculated on the cone: e e e e L = r + %e(A ) ; (14) e e e K K K e e e but is parallel and (relative to at coordinates on the cone) and A are constant because K is a linear vector eld. Therefore this is the standard action of g = so(n + 1) (n = 7; 8; 15) on (positive-chirality, when applicable) spinors: S(8) , S(9) and S(15) , all of which are irreducible + + representations. In all cases, a roots-and-weights calculation (made less painful by using LiE [12]) shows that g = 0 ; (15) whence the Jacobi identity is satis ed and g becomes a 2-graded Lie algebra. To identify the Lie algebras in question we simply observe that g being simple and g being irreducible, implies 0 1 that g is simple. The dimensions are easy to compute and the Lie algebras are thus easy to recognise from the Killing{Cartan classi cation. The results are summarised in Table 2. Table 2. Killing Lie algebras of some spheres M g dim g g dim g dim g g 0 0 1 1 S so(8) 28 S(8) 8 36 B + 4 S so(9) 36 S(9) 16 52 F S so(16) 120 S(16) 128 248 E + 8 Since the inner products on g and g are invariant and positive-de nite, g is a compact real 0 1 form of the corresponding complex simple Lie algebra. By the usual device of taking ig instead of g , we may obtain the maximally split real forms. 2. Supergravity backgrounds Supergravity is an extension of Einstein (or Einstein{Maxwell) theory. At the level of its solutions, it is given by some geometric data (g; F; : : : ), where g is a local lorentzian metric and F; : : : stand for extra elds, all subject to partial di erential equations of the form Einstein Ric(g) Rg = T (F; : : : ) (16) \Maxwell" dF = 0 and d ? F =  (17) The details depend on the supergravity theory in question and have hence kept purposefully vague in the above description. We will consider here only so-called Poincar e supergravities. There are other supergravities: massive, gauged,... Supergravity theories are dictated by the representation theory of the Poincar e superalgebras. There are (physically interesting) supergravity theories in dimension d  11 and lorentzian signature, meaning that the local metric g is lorentzian. Supergravity theories are among the jewels of twentieth century theoretical physics and a good review of the structure of supergravity theories from the representation theory point of view can be found in [13]. 2.1. Eleven-dimensional supergravity My favourite, and to some extent the simplest yet nontrivial, supergravity theory is the unique eleven-dimensional supergravity theory. Its existence was conjectured by Nahm [14], whereas 7 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 it was constructed by Cremmer, Julia and Scherk [15]. Its eld content is a lorentzian eleven- dimensional metric g and a closed 4-form F . We can motivate this as follows. Supergravity is a theory invariant under local supersymmetry, hence the spectrum should carry a representation of the corresponding supersymmetry algebra. In the case of eleven- dimensional supergravity this is the eleven-dimensional Poincar e superalgebra (so(1; 10) 1;10 R )  S(1; 10), where S(1; 10) is the spinor representation of Spin(1; 10). It is not hard to show, using Bott periodicity, that C`(1; 10) = R(32)  R(32) and hence S(1; 10) = R . 1;10 The supertranslation ideal generated by R  S(1; 10) has as nonzero brackets the projection 2 1;10 2 1;10 S S(1; 10) ! R of the symmetric square of the spinor representation S S(1; 10) = R 2 1;10 5 1;10 R   R onto the vector representation of Spin(1; 10). Irreducible unitary representations of the Poincar e superalgebra are induced by 1;10 representations of the supertranslation ideal generated by S(1; 10)  R . This is done by 1;10 1;10 rst xing a character of the abelian translation ideal R ; that is, a momentum p 2 R . Since we are interested in a theory of gravity, which we expect even in eleven dimensions to be a long range force, we require a massless representations, whence p = 0, but p 6= 0. The little group of p, which is the maximal compact subgroup of the stabiliser of p in Spin(1; 10) is isomorphic to Spin(9). Once a momentum p has been xed, the supertranslation ideal takes the form of a Cli ord algebra [Q ; Q ] = 2 (Q ; p Q ) 1 ; (18) 1 2 1 2 where Q 2 S(1; 10) and the spinor inner product is symplectic in this signature, whence the bracket here is symmetric. The bilinear form de ning the Cli ord algebra, hQ ; Q i = 1 2 (Q ; p Q ) is degenerate because p = 0. In fact, it has rank 16. This is shown by exhibiting 1 2 a \dual" momentum q such that q = 0 and p q = 1. Then S(1; 10) = ker p ker q, where the kernel refers to the Cli ord action. The common stabiliser of p and q in Spin(1; 10) is a Spin(9)- subgroup, which we can identify with the little group of either p or q. The degenerate Cli ord algebra (18) becomes an honest Cli ord algebra on ker q  S(1; 10) isomorphic to C`(16), and in fact, as Spin(9)-module, ker q is the spinor module S(9). There is, up to isomorphism, a unique irreducible representation of C`(16) and it is real and of dimension 256. Indeed, by Bott periodicity, C`(16) C`(8) R(16) R(16) R(16) R(16 ) : (19) = = = R R As a Spin(9)-module, this is nothing but S(S(9)); that is, spinors of spinors! A roots-and-weights calculation shows that as a representation of Spin(9) we have 2 9 3 9 9 S(S(9)) S (R )  R  RS(R ) ; (20) 9 2 where R stands for the vector representation of Spin(9), S denotes traceless symmetric tensors and RS stands for the Rarita{Schwinger representation, which is the subrepresentation of 9 9 S(9) consisting of the kernel of the Cli ord action R S(9) ! S(9). Counting dimensions, we see that for the bosonic part of the representation 2 9 3 9 dim S (R ) + dim  R = 44 + 84 = 128 ; (21) whereas for the fermionic part of the representation 9 9 dim RS(R ) = dimR S(9) dim S(9) = 16 9 16 = 128 ; (22) 2 9 whence the physical degrees of freedom match, as expected. In terms of elds, S (R ) 3 9 parametrise the uctuations of a metric tensor g, whereas  R parametrises the uctuations of a (locally de ned) 3-form potential A, and RS(R ) parametrises the uctuations of a gravitino. 8 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 The supergravity action, ignoring terms involving the gravitino, consists of three terms: an Einstein{Hilbert term, a Maxwell term and a Chern{Simons term. The lagrangian density is given by 1 1 R dvol F ^ ?F + F ^ F ^ A ; (23) 4 12 where F = dA locally. Although A appears explicitly in the above lagrangian, the Euler{ Lagrange equations only involve F . The equations are of Einstein{Maxwell type with a twist provided by the Chern{Simons term; namely, the Maxwell equation is nonlinear: d ? F = F ^ F : (24) 2.2. Supersymmetric supergravity backgrounds We de ne a (bosonic) eleven-dimensional supergravity background to be an eleven- dimensional lorentzian spin manifold (M; g; $) and a closed 4-form F 2 (M ) subject to the Einstein{Maxwell equations derived from the lagrangian (23). The lagrangian (23) admits a supersymmetric completion by adding extra terms involving the gravitino 2 (M; $). The variation of the gravitino under supersymmetry de nes a connection D on the spinor bundle $: 1 1 D := r +  F  + X ^ F  ; (25) X X X 6 12 [ 1 for all 2 ($) and X 2 X(M ) and where X 2 (M ) is the one-form such that X (Y ) = g(X; Y ) for all Y 2 X(M ). The connection D is the fundamental object in this game, as it encodes virtually all the information of the theory. For example, the Einstein{Maxwell equations can be recovered by demanding the vanishing of the Cli ord-trace of its curvature. More explicitly, let e be a i i i pseudo-orthonormal frame for M and let e denote the dual frame, de ned by g(e ; e ) =  . Then, as shown in [16], the eld equations de ning the notion of a supergravity background are equivalent to i D e  R (e ; X ) = 0 8 X 2 X(M ) : (26) A nonzero spinor eld 2 ($) which is D-parallel is called a (supergravity) Killing spinor. Although this seems a priori to be a generalisation of the notion of a parallel spinor, it is in fact the original notion of a Killing spinor. The geometrical notion in the rst lecture is a special case of the supergravity Killing spinor equation for a particular Ansatz for (M; g; F ), known as a Freund{Rubin background [17]. Being a linear equation, Killing spinors form a vector space, which anticipating the construction of the Killing superalgebra, will be denoted g . Being de ned by a parallel condition, a Killing spinor is determined by its value at a point, whence the dimension of g is bounded above by the rank of the spinor bundle; that is, dim g  32. The ratio 1 1 dim g = (27) is called the supersymmetry fraction of the background (M; g; F ). If  > 0, (M; g; F ) is said to be supersymmetric. 2.3. Examples A large number of supersymmetric backgrounds are known. Maximally supersymmetric backgrounds { those with  = 1 | have been classi ed in [18, 19, 20]. For such backgrounds, D 9 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 is at and the equations of motions are automatically satis ed. These backgrounds are related as follows: KG 88 ffL r L r L r L PL PL r L r L r L r L 7 4 AdS  S AdS  S 4 7 L r L r L r L r L r L r L r %%  yyr 1;10 where AdS is the n-dimensional anti de Sitter spacetime | i.e., lorentzian hyperbolic space |, KG is a special type of plane wave [21] whose geometry is described by a lorentzian symmetric 1;10 space of Cahen{Wallach type [22], and R is Minkowski spacetime with F = 0. The arrows labelled \PL" are Penrose limits, described in this context in [23, 24], but tracing their origin to work of Guv  en [25] and, of course, Penrose [26]. The undecorated arrows are zero-curvature limits. 7 4 The AdS S and AdS S backgrounds depend on a parameter, interpreted as the scalar 4 7 curvature of the eleven-dimensional geometry. The radii of curvature of the factors are in a ratio of 2 : 1, whence these backgrounds do not describe realistic compacti cations as they once were thought to do. They are known as Freund{Rubin backgrounds. The Killing spinors of the Freund{Rubin backgrounds are of geometric Killing spinors on the two factors: real on the riemannian factor and imaginary on the lorentzian factor. One can substitute either factor by an Einstein manifold with the same scalar curvature and admitting the relevant kind of Killing spinors. In particular one can consider AdS X , where X is a riemannian manifold admitting real Killing spinors, whence its cone has holonomy contained in Spin(7). Whenever X is not a sphere, the resulting background has a smaller fraction  of supersymmetry. They can be understood as near-horizon geometries of M2-branes, to which we now turn. The M2-brane is a interesting background with  = , discovered in [27] and interpreted as an interpolating soliton in [28]. It is described as follows: 2=3 2 1;2 1=3 2 2 2 7 g = H ds (R ) + H dr + r ds (S ) 1;2 1 F = dvol(R )^ dH (28) H (r) = + ; 2 1;2 1;2 where ds (R ) and dvol(R ) are the metric and volume of 3-dimensional Minkowski spacetime, 2 7 8 ds (S ) is the metric on the unit sphere in R and H is a two-parameter harmonic function on R . If we take ! 0 while keeping 6= 0 xed, we obtain eleven-dimensional Minkowski spacetime with F = 0, but taking ! 0 while keeping 6= 0 xed, one obtains AdS  S with scalar curvature depending on . Therefore the M2-brane interpolates between these two maximally supersymmetric backgrounds. The Killing spinors are given by 1=6 = H ; (29) where is a parallel spinor in the asymptotic Minkowski spacetime obeying the projection condition 1;2 dvol(R ) = : (30) 1 1 1;2 2 1;10 Since dvol(R ) = 1 and the parallel spinors of R split into two half-dimensional eigenspaces 1;2 1 of dvol(R ). As a result the solution has  = . As observed in [9], replacing S by another seven-dimensional manifold admitting real Killing spinors | that is, weak G -holonomy, Sasaki-Einstein or 3-Sasaki manifolds | we obtain an 10 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 M2-brane at a conical singularity in an 8-dimensional manifold with Spin(7), SU(4) or Sp(2) holonomy, respectively. To this day a large class class of backgrounds with various values of  are known to exist. General local metrics with minimal supersymmetry have been written down in [16, 29]. To date, the only fraction which has been ruled out is  = [30, 31]. 3. The Killing superalgebra of supergravity backgrounds In the rst lecture we saw that from a spin manifold admitting Killing spinors one could de ne (in the good cases) a 2-graded Lie algebra and in this way we recovered the compact real forms of the simple Lie algebras of types B , F and E . At its most basic, what we have is a spin 4 4 8 manifold with a privileged subspace of spinor elds which then generates a 2-graded algebra with the spinors being the odd-subspace. In the second lecture we saw how supersymmetric (eleven-dimensional) supergravity backgrounds gave rise to precisely such a situation: an eleven-dimensional lorentzian spin manifold with a privileged notion of spinor: the supergravity Killing spinors, which are parallel with respect to a connection D on the spinor bundle. Unlike the spin connection r, the connection D is not induced from a connection on the tangent bundle: it is genuinely a spinor connection. The spinor bundle is a real rank-32 symplectic vector bundle, but D does not preserve the symplectic structure. In fact, as shown by Hull [32], the holonomy algebra of D is generically contained in sl(32;R) since only the `determinant' is preserved. In this third and last lecture we will see that to every supersymmetric background of eleven- dimensional supergravity one can assign a Lie superalgebra by the techniques in the rst lecture and using it we will show that if  is suciently large, the background is forced to be homogeneous. This lecture is based on [33]. 3.1. The Killing superalgebra Let (M; g; F ) be a supersymmetric eleven-dimensional supergravity background. Following the method in the rst lecture, let us de ne a 2-graded vector space g = g  g , where 0 1 g = fX 2 X(M )jL g = 0 = L Fg (31) 0 X X is the Lie algebra of F -preserving isometries of the background, and g = f 2 ($)jD = 0g (32) is the space of Killing spinors. Clearly g is nite-dimensional, since as mentioned above dim g  32 and dim g  66, which is the maximum dimension of the isometry algebra of 1 0 1;10 an eleven-dimensional lorentzian manifold. It is only for R with F = 0 that both of these upper bounds are realised. Given 2 ($), we may de ne [ ; ] 2 X(M ) by transposing the Cli ord action. This is nonzero because this bilinear product is now symmetric, since the inner product on $ is symplectic. In fact, it is not dicult to show that [ ; ] is always causal; that is, it has non- positive minkowskian norm. Lemma 3. If 2 g then [ ; ] 2 g . 1 0 Proof. We rst show that [ ; ] is Killing. We rst have that g(r [ ; ]; Y ) = Xg([ ; ]; Y ) g([ ; ];r Y ) since rg = 0 X X = X ( ; Y  ) ( ;r Y  ) by de nition of [ ; ] = (r ; Y  ) + ( ; Y r ) X X = 2 (r ; Y  ) since ( ; Y  ) = ( ; Y  ) X 1 2 2 1 = 2 (Y r ; ) : 11 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 Now since D = 0, 1 1 r =  F  X ^ F  ; X X 6 12 whence 1 1 [ g(r [ ; ]; Y ) = (Y   F  ; ) + Y  (X ^ F ) ; X X 3 6 1 [ = (Y ^  F   F ) ; X Y X 1 [ [ [ + (Y ^ X ^ F + g(X; Y )F X ^  F ) ; 1 1 [ [ = (  F  ; ) + Y ^ X ^ F  ; ; Y X 3 6 where we have used that for every 4-form  2 (M ), ( ; ) = 0 : It follows that g(r [ ; ]; Y ) + g(r [ ; ]; X ) = 0 ; X Y whence [ ; ] is a Killing vector. One can also prove that L F = 0. Indeed, since dF = 0, [ ; ] L F = d F and it is just a calculation to show that [ ; ] [ ; ] F = dB ; [ ; ] where B 2 (M ) is the 2-form in the square of : [ [ B(X; Y ) = ; X ^ Y  : This result explains why is a called a Killing spinor, since it is the \square root" of a Killing vector. It follows by the usual polarisation trick that if ; 2 g , then [ ; ] 2 g . 1 2 1 1 2 0 We therefore have a symmetric bilinear map g  g ! g denoted by ( ; ) 7! [ ; ]. 1 1 0 1 2 1 2 We now de ne a bilinear map g  g ! g using the spinorial Lie derivative L de ned in 0 1 1 X equation (12). Let X 2 g . It follows from Proposition 2 and the fact that X preserves F , that L preserves D; that is, [L ; D ] = D 8 2 ($) ; (33) X Y [X;Y ] whence if D = 0, also DL = 0. Therefore [X; ] = L de nes the desired bilinear map. X X Together with the Lie bracket of vector elds, under which g becomes a Lie algebra, we have on g = g  g the structure of a superalgebra. 0 1 In checking the Jacobi identity, one again sees as in the rst lecture that 3/4 of the jacobator is identically zero because of properties of the Lie derivative L . The fourth component of the jacobator vanishes if and only if for all 2 g , [[ ; ]; ] = 0 or equivalently L = 0 : (34) [ ; ] Representation theory is not useful here, since we are interested in a general result for unspeci ed g and g . An explicit calculation (made less painful with Mathematica or Maple) shows 0 1 that this is indeed the case. Therefore we have a Lie superalgebra called the symmetry superalgebra of the background. The ideal generated by g , k = [g ; g ]  g is called the 1 1 1 1 Killing superalgebra of the background. Some (but not all, see [34]) backgrounds are such that their Killing superalgebra admits an extension m = m  g which is \maximal" in the 0 1 sense that m = [g ; g ] S g ; 0 1 1 1 where the isomorphism is one of vector spaces. When it exists, it is called the maximal superalgebra of the background. 12 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 3.2. Examples Let us consider some examples. The simplest is of course the Minkowski maximally 1;10 supersymmetric background R with F = 0. The symmetry superalgebra is the Poincar e superalgebra, whereas the Killing superalgebra is the supertranslation ideal, as explained in Section 2.1. The maximal superalgebra is obtained by taking m = S g and declaring m to be 0 1 0 central. The extra elements in m not in the Killing superalgebra can be understood in terms of brane charges, as explained, for example, in [35]. Backgrounds with F = 0 are said to be purely gravitational. The Killing spinors are parallel with respect to the Levi-Civit a connection r. This means that the holonomy of r is contained in the stabiliser of a spinor in Spin(1; 10). There are two types of spinor orbits in S(1; 10) and hence two stabilisers, up to isomorphism. As shown by Bryant [36] and the author [37], the orbits are labelled by the value of a quartic polynomial q, whose value q( ) at 2 S(1; 10) is the minkowskian norm of the vector [ ; ], which as mentioned above is always non-negative. If q( ) = 0 we must distinguish between the = 0 and a 25-dimensional orbit with stabiliser isomorphic to (Spin(9)nR )R  Spin(1; 10), whereas if q( ) < 0, the stabiliser is isomorphic to SU(5). This dichotomy gives rise to two types of supersymmetric purely gravitational backgrounds: one generalising the M-wave [38], where [ ; ] is a lightlike parallel vector and thus the geometry is described by a Brinkmann metric, and another generalising the Kaluza{Klein monopole [39, 40, 41]. This latter class gives rise to reducible geometries of the form R  N , where N is a riemannian ten-dimensional manifold with holonomy contained in SU(5). The Kaluza{Klein monopole is the case N = R K , with K a 4-dimensional hyperk ahler manifold. Because is parallel with respect to r, so is [ ; ] and the resulting Killing superalgebras are of the supertranslation type, sketchily [Q; Q] = P , where P is the parallel null vector in the + + case of the waves, or else [Q; Q] = translation in the at factor, for the generalised Kaluza{Klein monopoles. For backgrounds with nonzero F , as in the M2-brane discussed in the second lecture, the Killing superalgebra is still of the supertranslation type, where now [Q; Q] = translations along the brane worldvolume. For the maximally supersymmetric Freund{Rubin backgrounds the symmetry superalgebra 7 4 is simple and isomorphic to osp(8j4) in the case of AdS  S and to osp(6; 2j2) for AdS  S . 4 7 Simplicity implies that the Killing superalgebra agrees with the symmetry superalgebra. In [34] we showed via an explicit geometric construction that the maximal superalgebra is isomorphic to osp(1j32). It was shown in [24] that the Penrose limit contracts the Killing superalgebra, whence for the maximally supersymmetric KG background, obtained via the Penrose limit from the above Freund{Rubin backgrounds, the Killing superalgebra is a contraction of either of the orthosymplectic superalgebras osp(8j4) or osp(6; 2j2). This contraction was performed explicitly in [42] obtaining the Killing superalgebra previously computed in [18]. 3.3. The homogeneity conjecture Because squaring Killing spinors one obtains Killing vectors, it stands to reason that the more supersymmetric a background is, the more isometries it has. It is therefore tempting to conjecture that there given sucient supersymmetry | that is, a suciently large value of the fraction  | the background might be homogeneous, where we say that a supergravity background (M; g; F ) is homogeneous if it admits the transitive action of a Lie group via F -preserving isometries. So the question is whether there is some critical fraction  such that if a background has a fraction  >  , then it is forced to be homogeneous. All maximally supersymmetric backgrounds discussed in the second lecture are symmetric spaces, whence in particular homogeneous. On the other hand, the M2-brane, which has  = , has cohomogeneity one: with orbits labelled by the radial coordinate r in the solution. This suggests that   . 13 Workshop on Higher Symmetries in Physics IOP Publishing Journal of Physics: Conference Series 175 (2009) 012002 doi:10.1088/1742-6596/175/1/012002 Furthermore, inspecting the catalogue of known solution with  > , one sees that they are always homogeneous. This prompted Patrick Meessen to state the Homogeneity Conjecture. All supergravity backgrounds with  > are homogeneous. In fact, we have to be a little careful because in practice we have that g; F are only locally de ned in some open neighbourhood of R , so that a more relevant notion is that of local homogeneity, which is implied by local transitivity, by which we mean that around every point there is a local frame consisting of F -preserving Killing vectors. In [33] we proved something weaker and at the same stronger than the homogeneity conjecture. We proved that if a background has  > then it is locally homogeneous, but we proved that already [g ; g ] acts locally transitively. In other words, local homogeneity is a 1 1 direct consequence of supersymmetry. 3.4. Status of the conjecture The conjecture does not just make reference to eleven-dimensional supergravity, but in fact to any Poincar e supergravity theory. Concentrating for de niteness on the ten-dimensional supergravity theories, similar results exist for these theories as well. In [43] we showed that any background of either type IIA or IIB supergravity with  > is locally homogeneous, whereas any background of type I/heterotic supergravity with  > is locally homogeneous. This latter result bene ted from the classi cation of parallelisable backgrounds [44], which in turn was made possible by the fact that the Killing spinors are de ned by the lift to the spin bundle of a metric connection with torsion. We believe that the conjecture is true as stated, but proving this for type II and eleven-dimensional supergravities will require a better understanding of the connection D. Acknowledgments These notes are a reasonably faithful transcription of three lectures delivered at the Universidad Complutense de Madrid in November 2008 as part of the Workshop on Higher Symmetries in Physics. I would like to extend my gratitude to Marco Castrill on L opez for the invitation. The notes are an expanded version of two lectures given at the Universit at Bielefeld in May 2008 as part of the Meeting on Geometry and Supersymmetry organised by Andriy Haydys, to whom I am grateful for the invitation. The results detailed in the rst lecture had their origin in lectures delivered in May 2007 at the Dipartamento di Matematica \U. Dini" of the Universit a degli Studi di Firenze. It is a pleasure to take this opportunity to thank Dmitri Alekseevsky and Andrea Spiro for arranging that visit, Luigi Mangiarotti for support, Cristina Giannotti for the many delicious meals and Suor Tarcisia of le Suore \Stabilite nella Carit a" della Villa Agape for keeping me o the streets at night. Most of the work described here is based on collaborations with a number of people, whom it is my pleasure to thank: Bobby Acharya, Matthias Blau, Emily Hackett-Jones, Chris Hull, Patrick Meessen, George Moutsopoulos, George Papadopoulos, Simon Philip, Hannu Rajaniemi, Joan Sim on and Bill Spence. Finally, these notes were prepared while on sabbatical at the Universitat de Val encia, supported under research grant FIS2008-01980. I am grateful to Jos e de Azc arraga for making this visit possible. 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