There has been great interest recently in type IIB theory on ads 5 S 5 ë1, 2, 3ë, due to its possible relation to N=4,d=4 Yang-Mills theory. This vacu

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1 SLAC-PUB-7840 SU-ITP-98è36 May 1998 Near Horizon Superspace Renata Kallosh 1a, J. Rahmfeld 1b 12 cd and Arvind Rajaraman 1 Department of Physics, Stanford University, Stanford, CA Stanford Linear Accelerator Center, Stanford University, Stanford, CA ABSTRACT The ads p+2 S d,p,2 geometry of the near horizon branes is promoted to a supergeometry: the solution of the supergravity constraints for the vielbein, connection and form superelds are found. This supergeometry can be used for the construction of new superconformal theories. We also discuss the Green-Schwarz action for a type IIB string on ads 5 S 5. Submitted to Physics Letters B a b c d kallosh@physics.stanford.edu. rahmfeld@leland.stanford.edu. arvindra@leland.stanford.edu. Supported in part by the Department of Energy under contract No. DE-AC03-76SF00515.

2 There has been great interest recently in type IIB theory on ads 5 S 5 ë1, 2, 3ë, due to its possible relation to N=4,d=4 Yang-Mills theory. This vacuum is the near horizon limit of a D3-brane geometry. Related to this, we also have the ads 7 S 4 and ads 4 S 7 vacua of M-theory, which are the near horizon geometries of a M5-brane and an M2-brane respectively. We would like to formulate a description of these vacua in superspace. This is of great use, since the knowledge of the supervielbeins enables us to write down worldvolume actions of branes in these backgrounds, in particular, the Green-Schwarz action for a type IIB string on ads 5 S 5. A rst step was taken in ë4ë, where the torsion and curvature superelds were found in the framework of the standard supergravity superspace ë5, 6, 7ë. This knowledge enables one to prove that these vacua are exact solutions. In this paper, we will nd in addition the vielbeins, connections and form superelds. In the ads 5 S 5 case, this problem was attacked by Metsaev and Tseytlin ë8ë using the group manifold approach. We shall extend their construction to the M-theory vacua, and nd closed form expressions for the supervielbeins and superconnections. This in particular enables one to directly write down a closed form expression for the Green-Schwarz action for a type IIB on ds 5 S 5, using ë9ë. We dene the forms, following ë10ë where supergravity was dened in a group manifold approach: where L a L a èx; çè = E a çèx; çèdx ç + E a èx; çèdç L èx; çè = E çèx; çèdx ç + E èx; çèdç! ab èx; çè =! ab çèx; çèdx ç +! ab èx; çèdç è1è èl è corresponds to the bosonicèfermionicè components of the vielbein èe a ;E in ë5ëè and! ab is the connection supereld. The lowest èç =0ècomponents of E a ç and! ab ç are the space-time vielbein and connection. The space-time form-elds of supergravity A ç1 ;:::ç p+1 èxè are also introduced in superspace with the help of the superspace forms. In terms of the vielbein L ^A = 1

3 èl a ;L è the form elds are A = L ^A 1 p+1 :::L^A A ^A 1 ::: ^A p+1 èx; çè; F = da; df =0 è2è In the group-manifold approach, the curvature and torsions are dened somewhat dierently. In eect, non-zero value of the curvature and torsion are absorbed by redening them in such a way that the new curvature and torsion are zero. For example, the standard denition of ads geometry has a non-zero curvature R ab cd =, ëa c bë d è3è which translates in form language to R cd ç èd! +! ^!è cd =,L c ^ L d : è4è Instead, in the group manifold approach one uses the anti-de-sitter algebra ëp a ;P b ë=j ab ; ëp a ;J bc ë=ç ab P c, ç ac P b ; ëj ab ;J cd ë=ç bc J ad + 3 terms è5è and introduces the 1-form dierential operator D = d +! ab J ab + L a P a : è6è The ads curvatures are now dened by D 2 çr ab J ab + R a P a è7è and it is easily veried that R ab = R ab + L a ^ L b è8è R a = T a è9è which vanish for this geometry. Hence, the ads geometry can be dened by requiring vanishing generalized ads curvatures. For the analogous description in superspace we must generalize the dierential operator D to include all the generators of the appropriate superconformal algebra: OSpè8j4è for the ads 4 S 7 vacuum of M theory, OSpè6; 2j4è for the ads 7 S 4 vacuum 2

4 of M theory and SUè2; 2j4è for the ads 5 S 5 vacuum of IIB string theory. In general, the superconformal algebra is of the form ëb A ;B B ë = f C AB B C ëf ;B B ë = f B F ff ;F g = f C B C è10è where B A and F are the bosonic and fermionic generators, and the f are the structure constants. The dierential operator is then D = d + L A B A + L F : è11è L A and L are the left-invariant Cartan forms. The equation D 2 = 0 leads then to the Maurer-Cartan equations dl A + L B ^ L C f A BC, L ^ L f A =0 è12è dl + L A ^ L f A =0: è13è We will now solve these equations using the standard method which was also employed in ë8ë. Let G be an element of the superconformal group, then G,1 dg = L A B A + L F : è14è Writing G = gèxèe çf, we nd that where with L A 0 G,1 dg = e,çf De çf ; D = d + L A 0 B A = L A èx; ç = 0è. It is now useful to decompose L as L = L 0 èxè+ Lèx; ~ çè: è15è è16è è17è To nd explicitly ~ LA and ~ L one then replaces ç! tç and introduces a t-dependence into the vielbein components èwhich will be eventually be removed by setting t = 1è, giving rise to the modied relation e,tçf de tçf = ~ L A t B A + ~ L t F : è18è 3

5 Dierentiating both sides and utilizing the superalgebra è10è one nds the dierential t ~ L A t = ç f A ~ L t t ~ L t = Dç, ç f A ~ L A t : è20è which have the structure of coupled harmonic oscillators. Together with the initial conditions ~L A t=0 = ~ L t=0 =0 è21è and è17è, we can easily write down the explicit solution for the supervielbein èsetting t = 1è in closed form. One nds that and where L =è sinh M! èdçè M! L A = L A 0 +2ç fè A sinh2 M=2 èdçè M ; 2 ç M 2ç =,ç f A ç f A : Note that M 2 is quadratic in ç and that all higher order terms in ç in eqs. è22è, è23è are given by even powers of M, i.e. by powers of M 2 up to èm 2 è 16. è22è è23è è24è If we choose ç to be the standard fermionic coordinates of superspace, the above solution gives the superspace geometry in the Wess-Zumino gauge. This can be readily veried by noting that hence L èx; ç =0è=. èdçè = dç +èl A 0 B Açè ; è25è However, it turns out that there is an often more convenient gauge which simplies the geometry considerably. Consider space-time dependent ç of the form ç èxè =e èxèç è26è so that èdçè = e èxèdç + ç èd + L A 0 B Aè e èxè ç ç 4 è27è

6 The second term drops if we choose ç to be the Killing spinors of the background! As is well-known, those are precisely dened by the equation èd + L A 0 B Aè èxè Kill =0 è28è and the solution is of the form ë11ë èxè Kill = e èxè const: è29è where const is a constant spinor. Hence, choosing ç = Kill èxè in è15è leads to èdçè = e èxèdç : è30è With the explicit Killing spinors, constructed for all relevant ads S spaces in ë11ë, the superspace structure simplies as follows: L = e èx; çèdç L A = e A Mèx; ç =0èdx M + e A èx; çèdç ; è31è where and e èx; çè =è sinh M! e M èxè ;! e A èx; çè =ç èxèfè A sinh2 M=2 e M èxè 2 where M now contains the space-time dependent spinors ç èxè èm 2 è =,ç èxèf A ç èxèf A : è32è è33è è34è Instead of the standard Wess-Zumino gauge with e èx; ç =0è= wehave a Killing spinor gauge since the fermion-fermion part of the vielbein at ç = 0 is `curved' with E èx; ç =0è=e èxè: è35è Like in the at superspace, there are no fermions, the gravitino vielbein component è çèx; çèdx ç is not generated at all levels of ç. This also explains why the boson-boson component ofthevielbein e A Mèx; ç =0èisç-independent. 5

7 After developing the general formalism let us now turn to a specic example, the type IIB theory on ads 5 S 5. This vacuum is a special case of the IIB superspace ë7ë. It was presented in ë4ë in eqs. è46-51è. The expressions for the constant Lorentz curvature and torsion of the string vacuum can be reinterpreted via the superconformal group manifold approach. Following ë8ë we consider a coset superspace SUè2;2j4è with the even part ads SOè4;1èSOè5è 5 S 5 = SOè4;2è SOè6è. The even generators are SOè4;1è SOè5è then two pairs of translations and rotations, èp a ;J ab è with a = 0; 1; 2; 3; 4 for ads 5 and èp a 0;J a0 b 0è;a0 = 5;6;7;8;9 for S 5 and the odd generators are the two D = 10 Majorana-Weyl spinors Q 0 I. The dierential operator related to SUè2; 2j4è superalgebra is given by D = d + G,1 dg = d + L a P a + L ab J ab + L a P a + L ab J ab + L 0I Q 0 I è36è satisfying D 2 =0:Taking care of the change of notations with respect to the dierent representation for the fermions and splitting the ten-dimensional ^a into two vedimensional ones a; a 0 one can verify that the supergravity constraint for the vector part of the torsion can be brought tothe form R a = dl a + L b ^ L ba, iç L I a ^ L I =0 R a0 = dl a0 + L b0 ^ L b0 a 0, iç L I a0 ^ L I =0 è37è where we follow the conventions of ë8ë. For the spinorial part of the torsion one nds the following form èupon changing the spinorial representationè R 0 I = dl 0 I +è i 2 a IJ L J ^ L a, 1 4 ab L I ^ L ab è 0 + :::=0 è38è The Lorentz curvature form of supergravity has a non-vanishing fermion-fermion component as well as the boson-boson component. It can easily be shown that these equations are equivalent to the constraints of the ads 5 S 5 discussed in ë4ë. Thus the results for the Lorentz-valued curvatures and torsions can be rewritten as D 2 =0 where D is based on superconformal algebra. In addition, the supergravity ads 5 S 5 vacuum is dened by the presence of the 3-form and 5-form. F è3è =,il^a ^ L èç^a è ^ L + c:c: è39è 6

8 G è5è = L a ^ L b ^ L c ^ L d ^ L e abcde + L ^ a0 L ^ b0 L ^ c0 L ^ d0 L e0 a0 b 0 c 0 d 0 e 0 + L^a ^ L^b ^ L^c ^ L èç^a^b^c è ^ L è40è Here we use the notation of ë7ë but for convenience we split the ten-dimensional vector index ^a = a; a 0. We now turn to the supergeometry of this space following our general procedure and using the conventions of ë8ë. The complete superspace forms in the Killing gauge specied for the ads 5 S 5 are L 0I = e 0 I 0 Ièx; çèdç 0 I è41è L^a = e^a mèx; ç =0èdx m + e^a 0 Ièx; çèdç 0 I è42è L^a^b =!^a^b m èx; ç =0èdx m + e^a^b 0 Ièx; çèdç 0 I è43è Here e 0 I Ièx; çè =è sinh M! 0 I 0 M 0 J e 0 J 0 Ièx; ç =0è; è44è where e^a 0 Ièx; çè =,2iè ç ç I èxè^a è 0è sinh2 M=2 M 2 e^ab Ièx; çè =2 IJ èç I èxè^a^bè 0 0è sinh2 M=2 M 2! 0 I 0 J! 0 J 0 K e 0 J 0 Ièx; ç =0è e 0 K 0 I èx; ç =0è è46è è45è èm 2 è I L = ë IJ è, a ç J èxè ç ç L èxè a + a0 ç J èxè ç ç L èxè a0 è KL è ab ç I èxè ç ç K èxè ab + a0 b 0 ç I èxè ç ç K èxè a0 b 0 èë è47è We use the simplifying notation ç 0 I èxè = ç 0I e 0 I 0 Ièxè. We expect that this form of the superspace will be most suitable for the construction of the superconformal D3 brane action. 7

9 It may also be useful to work within the Wess-Zumino gauge in the superspace. The solution is than given by and L I = "è sinh M! èi Dç M ^aè!i L^a = e^a mèxèdx m, 4iç I sinh2 M=2 Dç M 2 where M 2 is given by eq. è47è with constant ç. Here, èdçè I = çd è!ab ab +! a0 b 0 a0b 0è ç ç I, 1 2 iij èe a a + ie a0 a 0èç J è48è è49è è50è The complete form of the GS type IIB ç-symmetric string action in the generic supergravity background was presented in ë9ë: S =, d 2 ç p gg ij L^a i L^a j +i Z M 3 s IJ L^a ^ çl I ^a ^ L J ; where S IJ has non-vanishing elements S 11 =,S 22 = 1. This action in ads 5 S 5 background was given in ë8ë up to terms of the order ç 4. The complete action in this background is the one in eq. è51è where the pull-back of the space-time forms to the world-sheet L^a i ;LI i is obtained from eqs. è48è, è49è with dç ç dç i ç and dx ç dç i x. In conclusion, we have solved the supergravity superspace constraints for ads p+2 S d,p,2 backgrounds. We have found here the complete expressions for the supereld values of the vielbeins, connections and forms in two dierent gauges. One gauge that utilizes the space-time Killing spinors of the background leads to a vanishing gravitino supereld E ç èx; çè = 0, and a curved fermion-fermion vielbein. In this gauge we observe a dramatic simplication of the superspace structure. The constraints can also be solved in the standard Wess-Zumino gauge. Here, the gravitino supereld picks up terms of linear and higher order in ç, and the fermion-fermion vielbein is at at ç =0. For various brane actions both of these gauge choices may beuseful. è51è Thus we have made here a necessary step for constructing the brane actions in ads p+2 S d,p,2 backgrounds. As explained in ë12ë the classical actions of M2, M5, D3 and some other branes in their near horizon superspace geometry will have two 8

10 types of symmetries: global superconformal symmetry due to the isometry of the background, and local reparametrization and ç-symmetry. Upon gauge-xing, the local symmetries of the gauge-xed theory will have some form of non-linearly realized superconformal symmetry. It will be a challenge to construct such superconformal actions as well as to nd the gauges in which the actions take the simplest form. We have also made progress towards understanding of the GS type IIB string theory in ads 5 S 5 background. We extended the results of ë8ë, where the action was explicitly given up to ç 4 terms, by solving the constraints exactly which enabled us to give a complete and closed form of the string action in this background. We had stimulating discussion with P. Howe, J. Kumar, J. Schwarz, E. Sezgin, D. Sorokin, M. Tonin, P. Townsend, A. Tseytlin, A. Van Proeyen and P. West. The work of R.K and J.R is supported by the NSF grant PHY The work of A. R. is also supported in part by the Department of Energy under contract No. DE-AC03-76SF References ë1ë J. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, hep-thè è1997è. ë2ë S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Gauge Theory Correlators from Noncritical String Theory, hep-thè è1998è. ë3ë E. Witten, Anti-de Sitter Space and Holography, hep-thè è1998è. ë4ë R. Kallosh and A. Rajaraman, Vacua of M Theory and String Theory, hep-thè è1998è. ë5ë E. Cremmer and S. Ferrara, Formulation of eleven-dimensional Supergravity in Superspace, Phys. Lett. 91B è1980è 61. 9

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