CTP TAMU-22/98 LPTENS-98/22 SU-ITP-98/3 hep-th/98055 May 998 A Construction of Killing Spinors on S n H. Lu y, C.N. Pope z and J. Rahmfeld 2 y Laboratoire de Physique Theorique de l' Ecole Normale Superieure 3 24 Rue Lhomond - 7523 Paris CEDEX 05 z Center for Theoretical Physics, Texas A&M University, College Station, Texas 77843 Department of Physics, Stanford University Stanford, CA 94305-4060 ABSTRACT We derive simple general expressions for the explicit Killing spinors on the n-sphere, for arbitrary n. Using these results we also construct the Killing spinors on various AdSSphere supergravity backgrounds, including AdS 5 S 5, AdS 4 S 7 and AdS 7 S 4. In addition, we extend previous results to obtain the Killing spinors on the hyperbolic spaces H n. Research supported in part by DOE grant DE-FG03-95ER4097. 2 Research supported by NSF Grant PHY-929345. 3 Unite Propre du Centre National de la Recherche Scientique, associee a l' Ecole Normale Superieure et a l'universite de Paris-Sud.
Introduction Finding the explicit form of Killing spinors on curved spaces can be an involved task. Often, one merely uses integrability conditions to establish their existence and to determine their multiplicities. In this way it is easy to show that spheres and anti-de Sitter spacetimes preserve all supersymmetries, i.e. they admit the maximum number of Killing spinors. However, one does not obtain explicit solutions by this method. Although establishing their existence is often sucient, there are situations where it is necessary to know their explicit forms. There exists a very simple explicit construction of the Killing spinors on n-dimensional anti-de Sitter spacetime AdS n, for arbitrary n []. This exploits the fact that AdS n can be written in horospherical coordinates, in terms of which the metric takes the simple form ds 2 = dr 2 + e 2r dx dx ; () where is the Minkowski metric in (n ) dimensions, and the Ricci tensor satises R = (n ) g. It was shown in [] that the Killing spinors, satisfying D = 2, are then expressible as = e 2 r + 0 ; or = e 2 r + e 2 r x 0 ; (2) where are arbitrary constant spinors satisfying r = 0 0. One can alternatively write the two kinds of Killing spinor together in one equation, as = e 2 r r + 2 x ( r) 0 ; (3) where 0 is an arbitrary constant spinor. It is therefore manifest that the number of independent Killing spinors is equal to the number of components in the spinors. (The Killing spinors for AdS 4, written in the standard AdS coordinate system, were obtained in [2].) It is worth remarking that the horospherical metric () can equally well have other spacetime signatures (p; n p), by taking other signatures (p; n p ) for the metric. The isometry group is SO(p +;n p). The case p =gives AdS n, with SO(2;n ), while p = 0 gives the positive-denite hyperbolic metric on H n, with SO(;n). (Expressions for the Killing spinors on H 2 and H 3, which are special cases of (3), were given in [3].) Thus equation (3) gives the Killing spinors on all of the AdS n spacetimes, hyperbolic spaces H n, and the other maximally-symmetric spacetimes with (p; n p) signature. There is an alternative Killing spinor equation that one can consider when n is even, namely D = i 2, where is the chirality operator, expressed as an appropriate
product over the written as, with 2 =. Weeasily see that the solutions of this equation can be = e i 2 + r r i 2 x ( i r ) 0 : (4) Note that in all the cases above, we considered a \unit radius" AdS n,orh n,etc., given by (). It is trivial to extend the results to an arbitrary scale size, by replacing () by ds 2 = 2 (dr 2 + e 2r dx dx ), which has the Ricci tensor R = (n ) 2 g. The Killing spinor equations then become D = 2, etc. It is easily seen that the solutions are given by precisely the same expressions (3), etc., with no modications whatsoever. (In [] a dierent coordinatisation of the general-radius AdS n was used, in which the expressions for the Killing spinors do depend upon the scale-setting parameter.) In this paper, we nd an explicit construction of the Killing spinors on S n. (Explicit results for n =2and n = 3 were obtained in [3].) One might think that since AdS n can be related to S n by appropriate complexications of coordinates, it should be possible to obtain expressions for the Killing spinors on S n that are analogous to those given above. However, things are not quite so simple, because the ability to write the metric on AdS n in the simple form () depends rather crucially on the fact that its isometry group SO(2;n ) is non-compact. (One can easily see that () has (n ) commuting Killing vectors @, which exceeds the rank [(n +)=2] of the isometry group when n > 3. This is not possible for compact groups.) We shall thus present a dierent construction for the Killing spinors of S n, which, although more complicated, is still explicit, and of an essentially simple structure. Our main result is contained in equation (7) in section 2, which also contains a detailed proof. In section 3 we combine the results for AdS and spheres, to give the explicit expressions for Killing spinors in various AdS m S n supergravity backgrounds, with (m; n) = (4,7), (7,4), (5,5), (3,3), (3,2), (2,3), (2,2). In appendix A, we collect some useful expressions for the representation and decomposition of Dirac matrices. 2 Killing spinors on S n 2. Results We begin by writing the metric on a unit S n in terms of that for a unit S n as ds 2 n = d 2 n + sin 2 n ds 2 n ; (5) 2
with ds 2 = d 2. This has Ricci tensor given by R ij = (n Killing spinor equation on the unit n-sphere, for arbitrary n, namely ) g ij. We then consider the D j = i 2 j : (6) We shall rst present our results for the solutions to this equation, and then present the proof later. We nd that the Killing spinors can be written as = e i ny 2 n n j= e 2 j j;j+ 0 ; (7) where 0 is an arbitrary constant spinor, and the indices on the Dirac matrices are vielbein indices. We use the convention that the algebra f i ; matrices are Hermitean, satisfying the Cliord jg =2 ij. Note that here, and in all other analogous formulae in the paper, the factors in the product in (7) are ordered anti-lexigraphically, i.e. starting with the n term at the left. Note also that the exponential factors in (7) can be written as e i 2 n n =l cos 2 n + i n sin 2 n ; e 2 j j;j+ =l cos 2 j j;j+ sin 2 j : (8) One can also consider the Killing spinor equation with the opposite sign for the namely j term, D j = i 2 j : (9) The previous solution (7) is easily modied to give solutions of this equation. One nds = e i ny 2 n n e 2 j j;j+ 0 : (0) j= This is immediately veried by noting that (9) is obtained from (6) by changing the sign of the gamma matrices. The Killing spinors discussed above exist on S n for any n. When n is even, there is an alternative equation that can also be considered, namely D j = 2 j ; () where is the chirality operator formed from the product of the j= matrices, satisfying 2 =. In this case, we nd that the corresponding Killing spinors can be written as = e ny 2 n n e 2 j j;j+ 0 ; (2) We may again also consider the Killing spinors satisfying () with the sign of the reversed, namely j term D j = 2 j : (3) 3
The solutions are again obtained by sending n! = e ny 2 n n j= n, giving e 2 j j;j+ 0 ; (4) As in the AdS and H m cases discussed in section, we mayagain trivially extend the results to an n-sphere of arbitrary radius, with metric ds 2 n = 2 (d 2 n + sin 2 n ds 2 n ) and Ricci tensor R ij =(n ) 2 g ij. The Killing spinor equations are modied to D j = i 2 j, etc., but again the expressions (7), etc. for the Killing spinors receive no modication whatsoever. 2.2 Proofs The proofs of these results proceed by substituting our expressions into the corresponding Killing spinor equations. We begin by showing that in the orthonormal basis e n = d n, e a = sin n e a (n, the spin connection for the metric (5) is given by )! ab =! ab (n ) ;!a;n = cos n e a (n ) ; (5) where a n, and e a (n and!ab ) (n are the vielbein and spin connection for ) Sn. (Note that the index n always denotes the specic value n of the dimension of the n-sphere.) Thus we can write the vielbein and spin connection on S n as e j = n Y k=j+! jk = cos k k sin k d j ; Y `=j+ sin ` The Killing spinor equation (6) can be written as d j ; j<kn: (6) @ j + 4! j k` k` = i 2 e j k k ; (7) where! j k` and e j k are the coordinate-index components of! k` and e k, i.e.! k` =! j k` d j and e k = e j k d j. These can be read o from (6). Note that the indices on the matrices in (7) are vielbein indices. We now make the following two denitions: U j k k Y e 2 ` `;`+ j;j+ k Y e 2 ` `;`+ ; k j (8) `=j+ `=j+ V j e i 2 n n U j n e i 2 n n ; (9) 4
where as usual, the factors with the larger ` values in the product sit to the left of those with smaller ` values. (Note that if the upper limit on the product is less than the lower limit, then it is dened to be.) It is now evident that verifying that the expression (7) gives a solution to the Killing spinor equation (6) amounts to proving that j V j = i e j j + nx k>j! j jk jk : (20) We prove this by rst establishing two lemmata. The rst, whose proof is elementary, states that if X and Y are matrices such that [X; Y ]=2Z, and [X; Z] = 2Y, then e 2 X Y e 2 X = cos Y + sin Z : (2) The second lemma states that U j k = sec k+! j j;k+ j;k+ + kx `>j! j j` j` ; k j : (22) We prove this by induction. From the denition (8), we know that U j j = j;j+, which j;j+ clearly satises (22) since! j = cos j+. Assuming then that (22) holds for a specic k j, we will have that U j k+ e 2 k+ k+;k+2 U j k e + 2 k+ k+;k+2 ; = sec k+! j j;k+ e 2 k+ k+;k+2 j;k+ e 2 k+ k+;k+2 + kx `>j! j j` j` ; (23) where we have made use of the fact that the j` in the last term all commute with k+;k+2, since ` k. The rst term can be evaluated using lemma, giving k+ j;k+ U j = sec k+! j cos k+ j;k+ + sin k+ j;k+2 + = tan k+! j j;k+ X k+ j` j;k+2 +! j `>j kx `>j! j j` j` ; j` : (24) Now, it follows from (6) that! j j;k+2 = cos k+2 tan k+! j j;k+. Using this, we then obtain (22) with k replaced by k +, completing the inductive proof. Having established the lemmata, we can substitute the expression U j n the denition of V j given in (8), giving V j = sec n! jn j e i 2 n n jn e i n 2 n n + = i tan n! j jn j + 5 nx `>j! j j` X `>j! j j` j` ; from (22) into j` ; (25)
where we have used lemma to derive the second line. Since e j j = tan n! jn j, as can be seen from (6), it follows that (25) gives (20). This completes the proof that (7) satises the Killing spinor equation (6). An essentially identical proof shows that (2) satises the alternative Killing spinor equation () in even dimensions. 3 Killing spinors on AdSSphere An application of the formulae obtained in this paper is to construct the explicit forms of the Killing spinors in the full D-dimensional spacetime of a supergravity theory that admits an AdS m S n solution, where D = m + n. Consider, for example, the AdS 4 S 7 solution of D = supergravity. This is obtained by taking F = 6m with = 0; 8; 9; 0, implying that the Ricci tensors on AdS 4 and S 7 satisfy R = 2m 2 g and R mn =6m 2 g mn respectively [4]. The Killing spinors must satisfy 0= M = D M 288 (^MNPQR F NPQR 8F ^NPQ MNPQ ): (26) Using the appropriate decomposition of Dirac matrices given in appendix A, this implies that on AdS 4 and S 7 wemust have AdS 4 : D AdS =im AdS ; S 7 : D j = i 2 m j (27) with j =;:::;7. From the results obtained in this paper we nd that the Killing spinors on AdS 4 S 7 can be written as AdS 4 S 7 : = e i 2 r ^ ^r 6Y + 2 x (i ^ ^ + ^r ^) e i 2 7 ^ ^7 j= e 2 j ^j;j+ 0 ; (28) where ^ i 24 ^ = l is a \pseudo chirality operator," and 0 is an arbitrary 32- component constant spinor in D =. Note that the explicit numerically-assigned indices refer to the seven directions on the 7-sphere. In D = supergravity there is also a solution AdS 7 S 4. An analogous calculation gives the result that the Killing spinors in this background can be written as AdS 7 S 4 : = e 2 r ^ ^r 3Y + 2 x (^ ^ + ^r ^) e 2 4 ^ ^4 j= e 2 j ^j;j+ 0 ; (29) Note that as implied by (27), the AdS 4 and S 7 have dierent radii, which are related by the elevendimensional eld equations. However, as noted before, in our coordinatisation the Killing spinors are independent of the scale sizes. 6
where ^ ^234 =l,and all numerically-assigned indices refer to the four directions on S 4. Again, 0 is an arbitrary 32-component constant spinor in D =. As another explicit example let us look at Type IIB supergravity on AdS 5 S 5. The gravitino transformation rules are 0= M = D M + i 920 ^NP QRS ^MF NP QRS ; (30) where is a ten dimensional spinor of positive chirality, satisfying ^0:::^9 = : (3) Choosing now F =4m and F ijk`m =4m ijk`m, equation (30) reduces to D M m ( 2 l l) ^M =0; (32) where we are using the (odd,odd) decomposition of Dirac matrices given in appendix A. With the ansatz! = AdS (33) 0 for a spinor of positive chirality, we obtain the equations for the AdS 5 and S 5 subspaces: AdS 5 : D AdS = 2 m AdS ; S 5 : D j = i 2 m j ; (34) which are the standard Killing spinor equations. Putting the AdS and S n results together, we obtain the explicit expression for the Killing spinors on AdS 5 S 5 AdS 5 S 5 : = e i r ^ 2 ^r + x 2 i^^+^r^ e i 2 5^^5 4Y j= e 2 j ^j;j+ 0 ; (35) where 0 is an arbitrary 32-component constant spinor of positivechirality, and ^ ^2345 = 2 l l, where the numerical indices lie in S 5. Four further analogous examples that arise in lower-dimensional supergravities are AdS 3 S 3 : = e i r ^~ 2 ^r AdS 3 S 2 : AdS 2 S 3 : AdS 2 S 2 : or = e 2 = e i 2 = e i 2 = e 2 r ^ ^r r ^ ^r r ^ ^r r ^ ^r + x 2 i^~^+^r^ e i 2 3 ^ ^3 2Y j= + 2 x (^ ^ + ^r ^) e 2 2 ^ ^2 e 2 ^2 0 ; 2Y + 2 x(i ^ ^x + ^r ^x) e i 2 3 ^ ^3 j= e e 2 j ^j;j+ 2 j ^j;j+ 0 ; 0 ; + x(i ^ 2 ^x + ^r ^x) e i 2 2 ^ ^2 e 2 ^2 0 ; + x(^ 2 ^x + ^r ^x) e 2 2 ^ ^2 e 2 ^2 0 ; (36) 7
where the Dirac matrices are the ones appropriate to the total spacetime dimension in each case. In the case where one or other space in the factored product is even dimensional, ^ is the pseudo chirality operator given by the appropriate product of the hatted Dirac matrices in the even-dimensional factor. For this reason, there are two possibilities in the AdS 2 S 2 example, reecting the two possibilities for the Dirac matrix decomposition given in appendix A. The rst corresponds to taking ^ to be the pseudo chirality operator in AdS 2, and the second to taking it instead to be in S 2. In the case of AdS 3 S 3, ^ ^23 = i 2 ll, where the numerical indices lie in S 3, while ^~ 6 ^ = ll. In all the examples, 0 is an arbitrary constant spinor in the total space. It will be subject toachirality condition in the AdS 3 S 3 example, if the D = 6 supergravity ischosen to be the minimal chiral theory, and ^~ can then be replaced by ^ in the expression for the Killing spinors. 4 Discussion In this paper, we have obtained explicit expressions for the Killing spinors on S n for all n. We then used the results to obtain the full Killing spinors on various AdS m S n spacetimes that arise as solutions in supergravity theories. These are of considerable interest owing to the conjectured duality relation to conformal theories on the AdS boundaries. One further application of these results is to construct the Killing vectors, and conformal Killing vectors, from appropriate bilinear products 0 y i of Killing spinors. As discussed in [3], products where the Killing spinors 0 and on S n either both satisfy (6) or both satisfy (9) give Killing vectors, while products where one satises (6) and the other satises (9) give conformal Killing vectors. In general, it is necessary to use both of the Killing-vector constructions in order to obtain all the Killing vectors on S n. At large n there is a considerable redundancy in the construction, since the number of Killing spinors grows exponentially with n, while the number of Killing vectors grows only quadratically with n. In certain low dimensions, there is a more elegant exact spanning of the Killing vectors using this construction, such as for S 7 where the antisymmetric products i of the eight Killing spinors give the 28 Killing vectors of SO(8) [4]. We shall present just one simple example here, for the case of S 2. From the matrix a expression (4) in the appendix, we nd that from the Killing spinors = 2 and b! 8
0 = 2 a 0 b 0!,we obtain the Killing vectors K = K i @ i = E i j 0 y j = (b b 0 aa 0 ) @ @ +i(a b 0 e i a 0 be i ) @ +(a b 0 e i +a 0 be i @ ) cot 2 ; (37) @ 2 @ where E i j are the components of the inverse vielbein E j = E i j @ i. Choosing dierent values for the constants a; b; a 0 ;b 0 spans the complete set of three Killing vectors of SO(3). Acknowledgment We are grateful to Renata Kallosh for posing the question of whether simple explicit expressions for the Killing spinors on S n can be obtained. J.R. thanks Arvind Rajaraman for useful discussions. A Dirac matrices and their decomposition on product spaces It is useful in general to represent the Dirac matrices in terms of the 2 2 Pauli matrices f ; 2 ; 3 g as follows. In even dimensions D =2n,wehave = lll ; 2 = 2 lll ; 3 = 3 ll ; 4 = 3 2 ll ; 5 = 3 3 l ; ; 2n = 3 3 3 ; 2n = 3 3 3 2 ; (38) In odd dimensions D =2n+,we use the above construction for the Dirac matrices of 2n dimensions, and take 2n+ = 3 3 3 3 : (39) When performing Kaluza-Klein reductions, it is necessary to decompose the Dirac matrices of D dimensions in terms of those of the lower-dimensional spacetime M m, and the internal space K n, whose respective dimensions m and n add up to D. There are four 9
cases that arise, namely (m; n) = (even,odd), (odd,even), (even,even) and (odd,odd). If we denote the Dirac matrices of the spacetime M m by, and those of the internal space K n by i, then the Dirac matrices ^A of M m K n can be written as: (even,odd) : ^ = l; ^i= i ; (odd,even) : ^ = ; ^i=l i ; (even,even) : ^ = l; ^i= i ; or ^ = ; ^i=l i ; (odd,odd) : ^ = l ; ^i = 2 l i ; (40) Note that in the nal case the extra Pauli matrices and 2 are needed in order to satisfy the Cliord algebra, in view of the fact that the Dirac matrices of D dimensions are twice the size of the simple tensor products of those in M m and K n. Note also in this case that the chirality operator in the total space is 3 l l. B Some low-dimensional examples In this appendix, we give explicit matrix expressions for the Killing spinors on the spheres S 2, S 3, S 4 and S 5. These examples arise in the near-horizon structures of Reiner- Nordstrm black holes, dyonic strings, M5-branes and D3-branes respectively. In each case, we may write the expression (7) for the Killing spinors on S n as = n 0. For S 2, taking i = i, where i are the usual Pauli matrices, we nd 2 = e i 2 cos 2 2 e i 2 sin 2 2 e i 2 sin 2 2 e i 2 cos 2 2! : (4) To avoid clumsy expressions later, we may dene t k = e i 2 k, tk = e i 2 k, c k = cos 2 k, s k = sin 2 k. The matrix 2 thus becomes 2 =! t c 2 t s 2 t s 2 t c 2 : (42) For S 3, S 4 and S 5 we obtain! t t 3 c 2 i t t 3 s 2 3 = ; (43) i t t3 s 2 t t3 c 2 4 = 0 B @ t t3 c 2 c 4 t t3 c 2 s 4 i t t3 s 2 s 4 i t t3 c 4 s 2 t t3 c 2 s 4 t t3 c 2 c 4 i t t3 c 4 s 2 i t t3 s 2 s 4 i t t 3 s 2 s 4 i t t 3 c 4 s 2 t t 3 c 2 c 4 t t 3 c 2 s 4 i t t 3 c 4 s 2 i t t 3 s 2 s 4 t t 3 c 2 s 4 t t 3 c 2 c 4 0 C A ; (44)
5 = 0 B @ t t3 t5 c 2 c 4 i t t3 t5 c 2 s 4 t t3 t5 s 2 s 4 i t t3 t5 c 4 s 2 i t t3 t5 c 2 s 4 t t3 t5 c 2 c 4 i t t3 t5 c 4 s 2 t t3 t5 s 2 s 4 t t 3 t 5 s 2 s 4 i t t 3 t 5 c 4 s 2 t t 3 t 5 c 2 c 4 i t t 3 t 5 c 2 s 4 C A : (45) i t t 3 t 5 c 4 s 2 t t 3 t 5 s 2 s 4 i t t 3 t 5 c 2 s 4 t t 3 t 5 c 2 c 4 C In these examples we have used the representations of Dirac matrices given in equations (38) and (39) of appendix A. References [] H. Lu, C.N. Pope and P.K. Townsend, Domain walls from anti-de Sitter spacetime, Phys. Lett. B39 (997) 39, hep-th/960764. [2] P. Breitenlohner and D.Z. Freedman, Stability in gauged extended supergravity, Annals Phys. 44 (982) 249. [3] Y. Fujii and K. Yamagishi, Killing spinors on spheres and hyperbolic manifolds, J. Math. Phys. 27 (986) 979. [4] M.J. Du, B.E.W. Nilsson and C.N. Pope, Kaluza-Klein supergravity, Phys. Rep. 30 (986).