JHEP10(2009)051 3 3 2), | S 1 Ads , (1 × 3 Heory P SU August 10, 2009 October 20, 2009 : : September 20, 2009 : Ads Received , Published Unting in D1-D5 Systems

JHEP10(2009)051 3 3 2), | S 1 AdS , (1 × 3 heory P SU August 10, 2009 October 20, 2009 : : September 20, 2009 : AdS Received , Published unting in D1-D5 systems. boundary superconformal Accepted s in the worldsheet theory ´ Ecole Normale Supérieure clude an infinite dimensional 10.1088/1126-6708/2009/10/051 sigma-models on trum of string theory on de l’ doi: c , Published by IOP Publishing for SISSA and Jan Troost c with Ramond-Ramond fluxes. We also indicate how 3 space-times with boundary conditions that allow for S 3 [email protected] × , 3 AdS AdS Raphael Benichou 0907.1242 a,b AdS-CFT Correspondence, Conformal Field Models in String T String theory on [email protected] SISSA 2009 Institute of Mathematical Sciences,Taramani, C.I.T Chennai, Campus, 600113 India Perimeter Institute for Theoretical Physics, Waterloo, Ontario, ON N2L2Y5, Canada Laboratoire de Physique UnitéMixte Théorique, du CRNS et soie a’nvri´Per et àl’UniversitéPierre associée Marie Curie 6, UMR24 8549 Rue Lhomond, Paris 75005,E-mail: France [email protected] b c a c ArXiv ePrint: with Ramond-Ramond fluxes, and in the microscopic entropy co Keywords: we explicitly construct thedescribing string R-symmetry theory on andto Virasoro construct charge thealgebra full plays boundary an superconformal important role algebra. in classifying The the full spec

Abstract: Sujay K. Ashok, black hole states has global asymptotic symmetries which in conformal algebra. Using the conformal current algebra for Asymptotic symmetries of string theory on with Ramond-Ramond fluxes JHEP10(2009)051 3 3 4 5 6 7 8 8 9 1 3 5 9 12 18 19 21 22 25 16 20 21 26 27 s gravitational theories has ries related by holography trong coupling phenomena with RR flux 3 – 1 – AdS 2) | 1 , with Ramond-Ramond fluxes (1 3 S ) global symmetry P SU R × , 3 = 4 super Yang-Mills theory, and type IIB string theory on SL(2 N AdS × ) R , A.1 Matrix generatorsA.2 for Explicit formA.3 of the worldsheet The currents action 5.1 Superconformal5.2 generators Further remarks 4.1 R-symmetry generators4.2 from non-trivial diffeomorphism Computation of the spacetime R-current algebra 3.1 Conformal current3.2 algebra The SL(2 3.3 Left current3.4 algebra primaries Representations of the global bosonic symmetry group 2.1 The hybrid2.2 formalism The supergroup2.3 sigma model The brane2.4 configuration The physical Hilbert space shed light both onin non-perturbative gauge quantum gravity theory. andis The on s the quintessential example four-dimensional of a pair of theo 1 Introduction The holographic correspondence between gauge theories and C The Virasoro algebra B Primary operators 6 Summary and conclusions A Worldsheet action for strings on 4 The spacetime R-current 5 The spacetime Virasoro algebra 3 The worldsheet conformal current algebra Contents 1 Introduction 2 String theory on JHEP10(2009)051 ins algebra ] for the ra in the 13 x operators , 10 ] as well as the , bulk superstring 4 egrability of the t of string theory background with 9 3 3 S 2) supergroup model | 1 × , AdS 3 oup which includes the (1 ty in solving string theory ounds arise as near-brane t algebra of the model is x (except in a plane wave ensional symmetry algebra upergroup sigma-model on , one can make headway via rator that plays the role of xtend the finite dimensional y generators were explicitly AdS classifying the spectrum. It lude fundamental strings and al symmetry is central in the f the worldsheet model is the t therefore has the exceptional tor product expansion (OPE) y directly and explicitly in the ing area formula for the black P SU o algebra in [ ]. The further development of ] which renders eight spacetime 1 ]. Technically, our analysis is a 11 12 ]. 10 we review key features of the worldsheet ]. The integrable structure is coded in – 6 2 8 , 5 charges. Indeed = 4 super Yang-Mills theory [ – 2 – we indicate how to construct the full space-time N local 5 -model [ ]. For string theory on an backgrounds of string theory with non-zero Ramond- 7 σ 3 S × space-times supplemented with boundary conditions that 3 3 2) supergroup (or rather, its universal cover), which conta | 1 AdS , AdS . In section (1 4 . The construction of the vertex operators for the R-current P SU subspace. The main building block in constructing the verte 3 3 ]. We believe it is important to construct the symmetry algeb ]. S 7 space-time with Ramond-Ramond flux [ 17 × – 5 3 S 14 ]). A tool that has stimulated significant progress is the int × 2) at the Wess-Zumino-Witten points and beyond in the contex AdS 3 | 5 1 , , 2 (1 This article is organized as follows. In section Here we concentrate on In a particular instance of the holographic correspondence Quantum gravity on To that end we work in the hybrid formalism of [ AdS of the spacetime symmetry generators are the curents of the which satisfy a conformal current algebra found in [ a bosonic Neveu-Schwarz-Neveu-Schwarz flux, the space-timeconstructed symmetr in terms of worldsheet operators in [ the generators in space-time andis the performed calculation in of section super their Virasoro opera algebra and discuss the properties of the ope supercharges manifest. In thatsigma-model formalism on the the central part o sigma-model in the hybrid formalism. The worldsheet curren the correspondence has been somewhaton hampered non-trivially by curved the backgrounds difficul limit with [ Ramond-Ramond flu spectrum of the dilatationintegrability operator of in the bulk worldsheet the existence of an infinite number of non-chiral version of thecase construction of of F1-NS5 the super backgrounds.P SU Virasor For interesting studies of the s recalled in section the existence of an infinite number of non-local charges. bulk string theory explicitlywill and moreover to be exploit interestingintertwines it to with maximally the see integrability in how of the the model. local infinite dim two-dimensional conformal algebra [ allow for black hole solutions has an asymptotic symmetry gr theory is dual to a conformalproperty field theory of in having two an dimensions. infiniteisometry I number group of [ local charges which e we refer to [ geometries of D1-D5 brane configurations (whichNS5-branes). may also After inc introducing amicroscopic third counting charge, that the reproduceshole conform the entropy. Bekenstein-Hawk We believe itD1-D5 is near-brane useful geometry. to exhibit this symmetr Ramond flux with eight or sixteen supercharges. These backgr JHEP10(2009)051 e 2) | are and 1 (2.1) , ¯ σ ]. i aα (1 2) has a θ 11 ]. | − d certain 1 ¯ psu 11 , ρ − (1 = so that the action ¯ φ P SU . In the presence of ˜ φ and e ac K uadratic in the fermionic iσ bd ) related to the bosonized and e and make our concluding δ ll as the ghosts) [ σ the worldsheet theory in the − It renders eight supercharges upergroup sigma-model that s order, the action pertaining ollected in the appendices. l string Hilbert space. φ + ρ e tions are given by s well as two chiral interacting 2) superalgebra. The latter is | − ad ms). The ghosts appear in the and ¯ is embedded in a supergroup [ (2 K = ρ 3 bc ]. The supergroup S psl δ φ brid string which renders sixteen supercharges × 11 is embedded in a super-coset. We concentrate − 3 3 bc S 2) [ . aα | AdS SU(2). There is a corresponding K 1 × S cd , with positive exponent. In other words, the 3 ad × bc δ K δ ¯ (1 φ – 3 – 1) − AdS , − is a non-linear sigma-model with target space the ] allows one to covariantly quantize string theory abcd with Ramond-Ramond fluxes ǫ bd 3 bα and P SU 18 3 K S S αβ φ ǫ S ac ac × 2 1 δ δ 3 × = 3 ]= ]= } in terms of the spin-fields in the RNS formalism. There are AdS cd cα bβ aα ] in which the manifold ,S p , K ,S AdS The formalism is based on defining space-time fermions background with Ramond-Ramond fluxes, and its relation to th 18 ab ab background the ghost fields aα 3 1 K S K [ S 3 [ { . Some technical details regarding the worldsheet action an S × 6 3 × (and their right-moving counterparts ¯ 3 σ AdS . It is then possible to integrate out the fermionic momenta, AdS aα ]. In this formalism, the space-time p and 19 ρ In the There exists another formulation of the six-dimensional hy 1 maximal bosonic subgroup which is SU(1 on the formalism defined in [ manifest [ defined by its generators and their (anti-)commutation rela remarks in section the central extension in the spacetime algebra. We summariz 2.2 The supergroup sigmaWe will model work to lowest orderto in the the six-dimensional ghost exponentials. space At thi with Ramond-Ramond fluxes in a six-dimensional space-time. aspects regarding worldsheet operators and their OPEs are c 2 String theory on in space-time manifest. superalgebra which is a particular real form of the near-brane geometries. We also discusshybrid the formalism, BRST whose operator cohomology of determines the physica 2.1 The hybrid formalism The hybrid formalism introduced in [ (covering of the) supergroup manifold In this section, wedescribes briefly the review the hybrid formalism, the s their conjugate momenta six bosons corresponding tobosons the six space-time directions a ghost systems of the RNS formalism. respectively chiral and anti-chiralaction (up as exponentials to of interaction the ter fields worldsheet action is a perturbation series in the variables non-zero Ramond-Ramond flux, themomenta worldsheet Lagrangian is q only depends on the bosonic and fermionic coordinates (as we JHEP10(2009)051 is 2) | 3 (2 (2.4) (2.3) S are the × psl 3 ba adius. The K through the s the form AdS − 5 o the number D 2) for a total of = 2) superalgebra. , | Q 2 1 ab run over the states , , (which has the real ) (2.2) K n in much more de- (1 g γ out α, β su licit calculations can be ∂ s: een-Schwarz superstring an element of the group 1 2) algebra (which is the . . . | (2) − -D1 branes, the couplings the six bosonic generators + sl (2 phere. When . From the kinetic term in gg indicates the non-degenerate , and results for the n the β sl he second factor the fermions, ′ bβ out ∂ A.2 1 2) algebra, but in the appendix ¯ and D5-branes ∂θ T r | − (2) , 5 ] aα (2 sl 3 gg g α NS µ ∂θ ⊕ psl ∂ ∂ AdS Q 1 1 αβ 2). g 2) and | − ǫ − (2). The generators | 3 (2) g g 2) as 1 S 2)) and the indices (2 ab | ( sl µ | , ′ g sl 1 1 ∂ , × (1 , aα psl z δ ⊕ T r − S 2 [ (1 ′ – 4 – d aα (2) θ αβγ (2) is anti-symmetric. Under the bosonic subalgebra SU(1 P SU e sl Z psu sl zT r yǫ α ∈ 2 2 3 αβ e ∼ ǫ d d g 1 πf = WZ B 4 Z (4) S Z g 2 = so + π ik 1 πf 24 1) and the eight fermionic generators as (2 kin , 16 − S 3 , 4 matrix parameterization of the = = = fermionic . The tensor × S S kin WZ out S S multiplying the kinetic term is the square of the spacetime r 1) + (1 (2) , 2 1 1 sl f , ), we find that the quadratic fermionic term in the action take in the case of the algebra multiplying the Wess-Zumino term is quantized. It is equal t k are vector indices of out A.35 are related to the number of NS5-branes takes values in the supergroup (2) k ). We spell out the non-linear supergroup sigma-model actio a, b g R 2) algebra augmented with a central bosonic generator). Exp su | , There is no fundamental representation of the The action of the non-linear sigma-model on the supergroup i and (2 2 1 f we give an explicit 4 psl form fourteen generators that span the adjoint of where The main advantage of theis hybrid that formalism it compared is to covariantly quantizable. the Gr 2.3 The brane configuration The coefficient equation ( bosonic generators. There is an outer automorphism algebra We parameterize the group element coefficient transform as (3 where bi-invariant metric. It can be thought of as the supertrace i in a doublet of performed with the matrix parameterization of this algebra and the outer automorphism algebra tail in terms of a global coordinate system in appendix of units of Neveu-Schwarz-Neveu-Schwarz flux on the three-s where the first factor representsthe the U(1) third to one be an dividedSL(2 out, element t of the group SU(2) and the last factor realized as the near-brane geometry of a system of NS5-F1-D5 model are obtained by dividing out by the central generator. JHEP10(2009)051 2) | 1 , (2.5) (2.6) (1 belong rongly. 2) is the | PSU 1 T , . At zeroth + (1 ¯ G = 4 topological s, and ) (2.7) P SU N . and φ reads: ) since the D5-branes e correlation functions ( + + p,q O G ts, the principal chiral constraints that can be G ( 1 + puting Q x operators for the genera- nts . The correlation functions arge associated to the cur- anes. We thus have: + C old. mpared to F1-strings to be xteen or eight supercharges, m and it therefore falls into G a-model on igma-model, which is given in )= + 1 string theory. We will show that D 3 ) 5 S φ ,Q 2 D 2 1 × Q ∂ F ) fixes the volume of the compactifica- 2 s 3 5 g Q + ]. Q + AdS 12 5 , = 4 superconformal model at central charge 5 ∂φ∂φ 2 NS ( – 5 – Q 2 1 NS N ) and ( Q q + = = (divided by p,q 2) | string theory with Ramond-Ramond flux in the hybrid ( 1 1 5 , 2 k 3 1 f (1 Q , the holomorphic BRST current Q S φ e × 3 PSU )= 5 T D AdS = 2 superconformal algebra. In particular, physical states iσ ,Q e 5 N ]. lighter than the NS5-branes, they curve the geometry less st = NS s 18 + Q ( G /g ]: depends only on the compactification variables and the ghost 11 is the ten-dimensional string coupling constant. Note that ). As such, we will be able to insert these vertex operators in + C s g G 2.7 Our goal is to construct, in the worldsheet theory, the verte The physical string Hilbert space is given in terms of a set of = 6 associated to the compactification manifold. Via the ghos to the cohomology of the charges associated to the BRST curre formulas [ are a factor of 1 tors of the asymptotic symmetry group for the defined in terms of an to generate space-time Ward-identities. 3 The worldsheet conformalIn current the algebra previous section we reviewed that the supergroup sigm Supersymmetry requires the relativeequal number to of the relative D1-branes number co of D5-branes compared to NS5-br where order in the ghost exponential where these vertex operators arerent BRST ( closed with respect to the ch central building block for formalism. The supergroup sigma-model has zero Killing for is the holomorphic stress-energyterms tensor of of a the generalized supergroup Sugawara s construction [ model with Wess-Zumino term is coupledof to the the model compact can theory string be amplitudes defined [ in terms of the prescription for com and via the attractor mechanism tion manifold. For a type IIB superstring background with si the internal manifold can be either a four-torus2.4 or a K3 The manif physical HilbertThe space worldsheet theory also contains an c JHEP10(2009)051 ¯ z ], ]. R, 12 12 j (3.5) (3.4) (3.2) (3.3) (3.1) and . . . and right nvariance L + R,z j G ] (0) 12 ¯ z c L, j ) ) g 2 − ), the coefficients of the ) c − z ...... − 2 , + c 2.2 c − + + ( ( c orphic coordinates replaced r product expansions to de- ts have been derived in [ − + i i nt components + erivatives of the current. We 2 − c c − currents are given by + oduce current algebra primary hnical tool in constructing the c nt algebra was analyzed in [ + c , are given by [ (0) (0) (2 + c + L ¯ + z ± ( nerate right multiplication: c − c c (0) + + G c c c L, L,z c i − j j c . L,z by a group element in 2 2 ¯ z z j , ¯ = = = z z ) g ) 1 1 ) ) 2 ¯ ˜ c 4 3 ∂g . ∂g g g g − − c c 1 1 2 kf − − ¯ − z − − f g g 2 4 ¯ 2 4 ∂gg ∂gg ± + − c c c we give explicit expressions for left-invariant c c ( + − c c (1 − − – 6 – c − = = . The classical A.2 = = ab ¯ z R ) = ¯ 2) supergroup model in a particular coordinate sys- z f (0) + ( (0) + ( | L, L,z 2 − ¯ z G 1 ) ± j j R, R,z c 2 , c c j j c L, L,z ) − )+ j j c − − (1 − 4 2 + 2 w c ¯ c z z c c c + 2 + h h + + − + + are given in terms of the couplings by: c c c + c c ( z c 2 + P SU + − ( ( ab ab c + c c c ( f f (2) i − i + + πδ = = = and 2 2 1 3 2 2 1 ¯ c c z z g ) we can calculate the classical currents associated to the i c c + ab c ab ab ˜ κ cκ κ 2.2 ∼ ∼ ∼ denotes a super Lie algebra valued index: (0) (0) (0) a ¯ ¯ z z b b b L,z L, L, j j j ) ) ) z z z ( ( ( ¯ z a a a L,z L,z L, j j j tem. For the supergroup non-linear sigma-model in equation the class of models for which the worldsheet conformal curre The ellipses refer towill subleading only terms need proportional the torive leading the the singular d spacetime behaviour superconformal of algebra. the The operato right curre where now, conformal current algebra, expressed purely in terms of right-moving currents in the Since the worldsheet current algebraspace-time Virasoro will algebra, be we the reviewfields central it and tec here. some We useful also notations. intr 3.1 Conformal currentFrom algebra the action ( of the theory under left multiplication of the field Similarly, we also have the left-invariant currents that ge The operator product expansions satisfied by the left curren satisfy similar operatorby product anti-holomorphic expansions ones. with In the appendix holom multiplication by a group element in where the constant JHEP10(2009)051 ) elf, R , (3.6) (3.8) (3.9) (2 (3.10) sl 0) with , 1 − ) generators are these coefficients R g , submanifold of the . . 3 ) 0 ) i. ¯ z ¯ − z L, SL(2 (which leads to the first two J = z, c AdS z, , which is a subgroup of x × ( rticular, we can apply this ( 3 g ed field e ompose observables in the ) directions, we obtain the ) ) R up. We introduce auxiliary ¯ − z R,z − − L,z and ˜ R , j ion of the group: , j , ergroup element by an element 4 3 2 2 AdS z, , the c ( x , c 0) x 1 , + c + L,z (0 2 j ) (3.7) ) + ¯ )+ uct expansions are then derivatives of this c is of conformal weight ( 0 ¯ z ¯ ¯ z z O − L, 0 a L, ween + L z, J z, j − R, j ( ( x ¯ J z − − ¯ x e 3 3 R, ). The Ward identity for the left translation of the L,z − 0 and a L,z xj xj − L, to just the SL(2 j 2¯ 2 )= ( J – 7 – i ¯ z a x − − − − ) ) z, 3 e ) that normalize the currents. For future purposes, ¯ ¯ ; z z c = ) isometry group of − as having a logarithmic operator product expansion with its x, z, z, a c − 3.2 L R ( ( ( g )= ˜ , ¯ c z J ¯ x component as well as the right-currents. We have for + + L,z L,z + R, j j j x, global symmetry = 0 by acting with the lowering generators of the z ( ) in an irreducible representation of the group, can be gen- SL(2 ) = ≡ x ¯ O x R ) )= =1= × , z z = ¯ x, 1 ) ; ; ( c x x x R O + ( (¯ ) in terms of which the global SL(2 , c − ¯ z SL(2 ¯ x ˜ c L,z R, 2 × j j x, ) gives rise to the third identity. R , g SL(2 − are the factors defined in ( L,R J ± c A given observable Roughly speaking, we can think of log 2 group valued field identities, in a particular normalization for the currents 3.2 The satisfy the equations where as in an abelianmore theory. basic logarithmic The OPE. current That component gives operator rise prod to relations bet we note that due to the existence of the elementary group valu generates the symmetry which is theof left the translation of supergroup. a sup Restricting the index The zero-mode of the global current erated from its value at algebra where the 0 index indicatesoperation the to zero a mode current of component the in current. the adjoint In representat pa instance: the left/right space-time Virasoro algebra. In holography generators of the SL(2 respect to the zero-modes of the Virasoro algebras. expressed simply in terms of differential operators. In space-time the + component of the currents Similar relations hold for the ¯ supergroup plays a specialspacetime role, theory which in makescomplex terms it variables of ( useful representations to of dec this subgro JHEP10(2009)051 3 0) , n of + h AdS (3.12) (3.13) (3.14) D transform esentation ... . ) defined in h 1). ¯ z . + − h h z, ¯ z :). h ation properties ; ( L,z L, c z x h j )]Φ )]Φ j j ( b ¯ z x x L ) j ) , j ) is x : x − − x R ∂ , y y − bc − ( ( − When we continue y h h ) is a field satisfying the y 1) rrent = in which case the operator 2( . We think of the bosonic − 2( 3 3.4 h uction of the space-time R- + 2 + 2 o-boundary propagator for a of SL(2 − − ) a subspace of a representa- ect to the left current algebra. − t y y ¯ y y + y : +( h ∂ ∂ + less singular (3.11) + less singular ∂ ∂ 2 2 D b ¯ z y, 2 ) ) 2 ; j ) ) ) ) x x c z ¯ ¯ ¯ x w w w w x j hx tion − − 2 − − − − (: w, w, y y ( ( y ¯ bc z z y − [( [( ( ( φ φ a x ¯ a a f transforming in the representation ( w w t t ∂ ) c i 2 2 c 1 1 h in the representation in which Φ − − − − w ab x have trivial operator product expansion with c c ab + ¯ a z z f − − t f + − ) a φ z – 8 – + + c c − − 2 ¯ 2 z w c c = ) c ( ¯ ∂j + + i c c + − w − − + + + c c t c + − − z SU(2) of the supergroup. − + + c c 2 − z ) (¯ 2 ) ; × ( a ¯ g z ) y i )= )= ) h w ¯ ¯ − w w ∂j )= )= R − − , ¯ ¯ 2 + − w w x c c x w, w, ( ( z ( ( 1 ( w, w, φ φ c x∂ + ; ; ) ) ]. It is also useful to rewrite the operator product expansio ¯ ¯ ¯ ¯ y y − z z with respect to the current algebra ( 10 y, y, z, z, = ) = ( ( are the generators of the Lie super-algebra taken in the repr ( ( φ ¯ 3 z h h w a t t ; a a L,z L, )Φ )Φ y j j ) as parameterizing (via the variables ( ¯ ¯ z z ¯ w z, z, L,z ; ; j w, ) read: ; x x · transforms. The coefficients are fixed by the global transform ( ( ) components of the left-currents as follows: ¯ y ) ¯ z 3.9 R z φ y, , L,z L, ; ( j j x h ( L,z In our conventions, the quadratic casimir of the representa j 3 We have used the fact that the generators 3.3 Left currentIt algebra will primaries be useful to define fields which are primaries with resp A left primary field can be written as the following differential operators: the SL(2 operator product expansions: to Euclidean signature itscalar will field function coupling as to the a unique space-time bulk-t operator of dimension in the conventions of [ of the bosonic subgroup SL(2 field Φ 3.4 Representations ofFor the later global purposes, bosonic we define symmetry a group bosonic field Φ equation ( where the matrices in which After these worldsheetcurrent. preliminaries, we turn to the constr tion of the supergroupproduct corresponding expansions to of this a field left with primary the field, components of the cu of the field and the demand that the field the Maurer-Cartan operator JHEP10(2009)051 4 is in 3 µν , the (4.3) (4.1) (4.2) g µ ξ ] in the and one AdS 22 3 directions. , 3 R-symmetry S 21 AdS R mensions. These and ) index, and s 3 R SU(2) , undary. From the bulk × . Since the coordinates ). If we further reduce AdS L γ r mode expansion in the = 2 supergravity. In the 2.7 n at infinity act non-trivial- ms for which the generating rrents on the boundary. The tions. Diffeomorphisms that N with the Gaussian coordinate ed by the vector field which act non-trivially on the and ¯ re analyzed in [ 3. In the following discussion , roup. For the supersymmetric ere. is an SL(2 3 or for an SU(2) gauge boson is which is a solution of type IIB γ K ] by computing the cohomology - or of the right-current is a matter of am an SU(2) m g , 20 = 6, ¯ γ. X or AdS ) , φ 4 metric takes the form: × D 11 − m T L,z dγd 3 3 e j nd right-currents generate a suitable basis for φ ( ¯ z S 2 e O a L, × AdS j + 3 + 2 )+ ¯ z ¯ ) in powers of γ – 9 – dφ ¯ γ m L, AdS γ, j ( = γ, ( f 3 a L,z can be either f j = X 2 AdS z µ factor. The massless excitations of the string give the to be the SU(2) index, 2 ξ ds d 3 a S Z × ) which admits an analytic continuation to euclidean 3 , the fluctuations of the metric with one index in R 3 , AdS AdS , runs over the six indices corresponding to the µ ) on SL(2 behaves near the boundary as give rise to an SU(2)-valued massless vector field in three di ξ ¯ γ, φ 3 S γ, ) parameterize radial slices, this is equivalent to a Fourie Diffeomorphisms that induce a non-vanishing transformatio The R-symmetry algebra extends to a current algebra on the bo Whether we write down the vertex operator in terms of the left ¯ γ 4 γ, hybrid formalism, this has been shown in detail in [ supergravity multiplet and one tensor multiplet of where, from now on, we specify we will focus on the system ( the cotangent bundle in spacetime. convention. Indeed both one-forms associated to the left- a 4 The spacetime R-current 4.1 R-symmetry generators from non-trivial diffeomorphism We consider the space-time background ( boundary coordinates. The integrated string vertex operat of the BRST operatorthe associated theory down to to the current in equation ( supergravity approximation. ly on the Hilbert space of the theory. We parameterize We can further expand the function Poincarécoordinates. In these coordinates the The asymptotic symmetry group isvector generated field by diffeomorphis string theory. The compact space where the index the six-dimensional metric. Under a diffeomorphism generat group corresponding to the bulk isometries of the three-sph point of view, this isspace due to of the bulk existence configurations offall diffeomorphisms with off particular slowly generate boundary abackgrounds condi non-trivial under asymptotic consideration, symmetry these g diffeomorphisms we index in gauge bosons in the bulkspace-time are conformal associated field with theory dimension exhibits, one cu in particular, JHEP10(2009)051 (4.5) (4.7) (4.6) (4.8) (4.9) (4.4) ). (4.10) 4.8 . the fermionic ) can also be in- ) ¯ x coordinates. Such . φ equation ( is not BRST exact − 3 x, i )) e ¯ z φ ( . We can then define itten out in detail in | φ R O z, In order to write down 2 ) AdS ; e φ R + ¯ ) 2 x , e ¯ BRST x ent in the fully interacting does not vanish fast enough ) x x, , Q ¯ x − . ξ . We define the left-moving ) 1 − me two-dimensional conformal ) Λ( a γ − n ξ γ ∂ ( . ¯ )(¯ z γ m ) as: x a ∂ξ ∂ L, a )(¯ → ∞ ∂ a ¯ j z n x ¯ z − J ∂ξ a ). The variables ( L, a +1 φ n ¯ a z L, j γ − φ m n L,z J j )+ a j L, − x γ ¯ z + ¯ j z e ( + ( ) related to SL(2 a L, z, n + 1 + ( j ; ( O −∞ x ∞ ) a ¯ x X . Asymptotically, the left-moving R-current = ) in a convenient way we define the function φ + ¯ + ∂ξ n x L ¯ x ∂ξ − ¯ z x, ) n ( – 10 – e 1 a a m L,z − L, γ ( R that parameterizes the eigenvalue of the parabolic a L,z j Λ( j , J )= j ( γ ¯ O ∂ = ¯ x ( x z a ( L,z z − ) ]: 2 + j a 2 a n L,z d ( d j J 10 x ξ ( z )= Z z 2 1 Z − 2 . So the integrated vertex operator that generates a gauge d ) d = γ ν ¯ γ, φ ξ a Z n Z µ J ( γ, ¯ ∂ ) near the boundary at 1 ; π φ ∇ ) as: ¯ x a − L,z x j e ( = x, ( a )= x J z O µν Λ( ( 2 has an SU(2) index, and depends only on the a d δg + a J ξ n -th mode of the boundary R-current Z does not belong to the Hilbert space. Working at first order in γ n i ), first introduced in [ φ = )= , one can derive the explicit expression for the R-current in | x A ) ( ¯ γ, φ n a ( γ, J ξ ; ¯ x ) using diffeomorphisms with Applying the diffeomorphism transformation to the action wr x (¯ x, a ¯ We define the currents, we can rewrite the vertex operator as: generator of the symmetry group SL(2 where we have introduced the variable if the state where therefore has the form metric changes as Non-trivial diffeomorphisms forthe the exact interacting expression theory. for the R-current Similarly, we can also introduce the variable ¯ We propose the followingtheory: expression for the boundary R-curr The vector field transformation for the gauge boson is a vertex operator can be BRST non-trivial if the vector field Λ( boundary R-current at infinity. This is related to the fact that a state of the form J terpreted to parameterize the manifoldfield on theory which is the spaceti defined. appendix JHEP10(2009)051 ], 10 . We (4.11) (4.12) R current ] for the R ) ) (4.13) 10 , with zero . Since this ¯ R SU(2) 3 w j , 2 | × − x elta-function in L | ¯ z AdS SL(2 )] (4.14) ) by a trivial gauge w, ¯ x w × . ( − a L 2 w, alar from the boundary z ) ry group of the theory. J ; ( e very useful in our later R ¯ y φ , O all that in the case of the ) reads perators in the continuous ree-point function or from euclidean − y, e ( group SU(2) when shifting the contour of emi-classical grounds in [ ehaviour is that the function 1 ytically continued to discrete gebra. Notice that a different eet and space-time conformal )+ he expressions of [ + ike a delta-function identifying 3.12 l supergroup. Φ ¯ w 2 φ ) e to assume that the delta-function y w, ) ; ¯ x ¯ − y 1 − x ). The expression we put forward for y, ( γ [( , 1 y , it coincides with the expression derived 1 4.8 )(¯ ∂ x x Φ )Φ 1 − w y π γ − 1 = 1, equation ( − − γ ( z h – 11 – x Λ= ≈− ( ¯ x ∂ 1 (2) π )= representation of the SL(2 δ ). ¯ ] the three-point function for fields in the continuous w + 1 ) group theoretic properties, namely from the analytic 23 )= w, )= ¯ R is a primary field with respect to the current algebra. It γ, φ ; , ¯ w ¯ y 1 × D γ, ¯ γ, φ y, w, would produce an operator related to + 1 ( ]. In [ γ, ; ) 1 ; of the Clebsch-Gordan coefficient of the form ¯ D y n ¯ 23 x ( j )Φ ξ y, ( z x, is defined as: 1 ( ; 1 x 1 ( )Φ Φ ¯ z L,z z, j ; ¯ x ) arose from SL(2 x, ( ) to the bulk point ( 1 4.13 ¯ x Φ . It is thus the bulk-to-boundary propagator of a massless sc ) representation theory. What is important to us is that the d w x, x R → lim , z We collect some important formulae below that will prove to b We wish to argue that the following equation holds true: In the worldsheet theory Φ and where the function Φ equation ( Since Λ behaves near the boundary as Λ in the weak coupling, near-boundary region in ( transformation. The advantage ofΛ our has choice of a subleading simpleIndeed, b behaviour the function under Λ the satisfies action the of equation the global symmet and was made precise in [ representations in the Wess-Zumino-Witten model were anal the R-current is the natural non-chiral generalization of t values of the spin. Inintegration doing over so, the one spins picksrepresentation. that up appear residues These of in poles poles theSL(2 product can of arise two from o the dynamical th vertex operators which generate thechoice asymptotic for symmetry the al parameter continuation in the spin point ( calculations. We begin by noting that for part of the three-point functionappearing is in universal, it the is product natural above is universal as well. The first justification for thisdimensions is of the the matching left ofWess-Zumino-Witten model, and the this worldsh right equation was hand argued sides. for on s Moreover, we rec This function is an eigenvector of the laplacian operator on can extend the representation to a representation of the ful algebra and with spin zero under the action of the bosonic sub transforms in the discrete eigenvalue. Near the boundary, thisγ wave function behaves l JHEP10(2009)051 . B (4.15) (4.18) (4.20) (4.21) (4.19) (4.22) ation is true , ) generators in i R ) , ¯ w ) is the R-current of volving an anti-holo- w, ; . . ¯ y )] (4.16) )] (4.17) )] )] rents. Similarly we have . ¯ ¯ ¯ z 4.10 z z ¯ z product singularity inside ) ) y, ¯ ¯ z z some detail in appendix z, z, z, z, ; ; ; Λ( ; z, z, ht-current, we obtain similar ¯ ¯ ¯ ¯ x x x ¯ w x ; ; n of the algebra of currents in ∂ ¯ ¯ x x ) for the SL(2 x, x, x, x, b L,z x, x, : ( : ( : ( : ( j , 3.13 1 1 1 : ( : ( 1 . ) Φ Φ Φ 1 1 Φ ) y ¯ ¯ )+ z z ( y Φ Φ ¯ b ( w L, ¯ z R,z R, L,z b j J j j j R,z R, · J w, [: j j [: [: [: · ) ; , we deduce the following equations satisfied ¯ x ¯ x : : x x ¯ ) y x x ∂ ∂ ∂ ∂ ( . We will therefore compute the OPE between x − + a − π π ( − + y, + 1 1 1 x c c 1 c – 12 – c c a c ) defined in equation ( J ¯ x − J − Λ( x − − − − ( ∂ y w a y ∂ → J ¯ z lim )= )= x → )= )= ) = ) = ¯ ¯ lim z z x b ¯ L, z ¯ z ¯ ¯ z z j z, z, h z, z, z, z, ; ; ; ; ; ; ¯ ¯ w x x ¯ x ¯ x ¯ ¯ 2 x x d x, x, x, x, x, x, ( ( ( ( 1 Z Λ( Λ( 1 1 1 ¯ z z Φ Φ 1 π Φ Φ ) reads, using the expression ( ∂ ∂ ¯ z z ¯ z z ∂ ∂ − ∂ ∂ B.7 )= y to be embedded into a primary of the current algebra, this equ ( b 1 J If we assume Φ up to first order in the fermionic currents as demonstrated in From these relations and by integrating over ¯ However it turns outmorphic to derivative be of one more of convenient the to currents, compute namely an OPE in and to integrate the result with respect to ¯ terms of differential operators: Moreover equation ( by the operator Λ: Using the expression ofrelations: the stress-tensor in terms of the rig All these relations willthe be boundary repeatedly theory. used in the derivatio 4.2 Computation of theIn spacetime order R-current to algebra show that the operator Thus, from now on wethe work equation at leading order in the fermionic cur the spacetime CFT, we want tocorrelation compute functions the in spacetime the operator following limit: the current JHEP10(2009)051 ]. ) ¯ w 24 ): − ntour (4.26) (4.23) (4.24) (4.25) ¯ z )] We can ¯ w 4.19 w, . 5 ) ) − w, ¯ ¯ ) ; w z w ( ¯ ¯ y w , − w, O y, ; ¯ z ) ( w, ¯ y ) ¯ 1 ; z ¯ z ¯ )+ y w, y, z, ¯ )Φ w z, 1 ; y, y − ; Λ( w ¯ x Λ since the later trans- Φ ¯ x ¯ z : ( w, w ¯ − w ( ; − x, ∂ 1 ∂ a L,z ¯ x, y ( x O l disc cut out around the z ), can be written as ( ( 1 Φ b 1 L,z y, Φ j dzj (2) Φ o boundary terms. ¯ s are close one to another on z ween these two operators, we ities with the other operators R,z )+ t there are no singular terms δ : ( 4.11 w j ∂ ¯ 2 This implies that we can freely we can integrate by parts freely, but Λ and w a 1 L,z I ) )+ o the presence of other operators w ¯ . Moreover there is no singularity d to the OPE between the oper- y a L,z s procedure can be found in [ ) : Φ ¯ w + ∂ j w, ¯ z dzj ; , − 1 R,z ¯ 1 y w, z, er operators will contribute. w j Φ x ; ; )+ ¯ I z ¯ y, [(¯ y ¯ ¯ x z ¯ a x ) : L, y, ∂ y Λ( z, x, )+ ¯ ], we regularize by cutting small holes in zj ( ; w ¯ z − 1 d ¯ − Λ( x ∂ c 10 ) Φ w w z, x − y x, ∂ ( ; I + w Λ. To prove this we use the formula ( c ( ¯ z ¯ x will pick up the singular terms in the OPE be- ¯ 1 ). The OPE we wish to compute is w − → + – 13 – (2) b lim L, · y z ∂ c Φ δ j x, ( x w ) as being inserted within a worldsheet correlation z ( b ( − − − ∂ x 1 π π π Λ, J c c c ¯ ( z (2) Φ a w a ¯ L, − δ − − z w ∂ , which, using equation ( J j 2 a L, ¯ x x d = = ∂ z ¯ zj ) = 2 Z and d ¯ d w 1 w 1 πi I Z w, ) and − ; y − ¯ i y 1 ( b derivative of the R-current as y, J − ) = )= )= x y x Λ( ( ( x b w ( a ∂ a J J Λ. We conclude that the only singular terms picked up by the co ) · J ¯ x ¯ ¯ z are SU(2) indices. Following [ w ¯ ) x ∂ ∂ ∂ x b z, ( ; a ¯ x J and ¯ x and x, ( ∂ 1 1 a Φ w We can also work with a worldsheet without holes. In that case 5 → lim z The previous computation isators straightforwardly generalize Φ where the contact terms between the equations of motion and the oth arising between the SU(2) currents and the operators Φ tween the integrated composite operators. First notice tha where the contour integralposition runs of over the the integrated operator boundary in of the smal and its derivative with respect to ¯ The contour integrals around the point think of the operators So we will write the ¯ the worldsheet, at the points where operators are inserted. use the equation of motionon (that the have contact worldsheet), terms but singular the integration by parts gives rise t form in the trivial representation under theeither action in of SU(2) the OPE between Φ function. Since we are interestedwill in only the keep (spacetime) track OPE ofthe bet worldsheet, the and terms discard arising the when possiblein contribution these the due operator correlation t function. A general justification for thi JHEP10(2009)051 ) 12 ¯ x A ) ) x, w w ; (4.29) (4.28) (4.27) ( ( ) (and ¯ z c c ¯ ¯ z z and j j w w z, ) ) 8 g g ) 4.26 − − : ( A ¯ ) w z z − − 2 1 , w using the for- ) . We can now 2 6 2 2 − Φ ) c c ¯ w z 1 i − A 12 ( ( ¯ z ) w . c c , L, z ¯ − )(¯ j w 3 ) can be simplified i ab ab − )( z ) w A ( f f 11 ( z ¯ w, w w c ¯ (¯ z ( . ; A + + j ,...,A c z ¯ ) y ) ) ) j w, 1 y + i g ) ; ( w w y, ) A 5 ¯ g c ( ( y ¯ c c − w z z ivative of Φ A e J ¯ ¯ Λ( j w w j − y, c . 2 ) ) w c w, 4 g g ab − − ∂ ( ; c Λ( ) c ( t the terms ¯ ¯ ¯ z z y w ¯ if − − z c ivative of Λ). Using the relation ) ) ) ) ab , which involve double poles. We ∂ ) ¯ ¯ 4 4 ¯ ¯ y y y y y, ) and ( ) ab f z, y c c 10 9 ¯ z ; f ( ( y, y, y, y, + can be simply performed. The regu- Λ( ¯ c c x A − ; ; ; ; + z, ) , A w w ¯ ¯ ¯ ¯ ; ab ab x 7 w w ) w w 11 + ¯ x, ∂ w w ( ¯ x f f ( ) ( 2 w − A 1 c ¯ z z ( , A w, w, w, w, − (2) j x, A z c ¯ z 4 Φ )+ )+ ¯ j z, : z 2 Λ( Λ( Λ( Λ( : ( ; c w πδ w 4 w ¯ ¯ and , A c w w w w ¯ c x ) can then be read from equation ( 1 1 → c ∂ ∂ ∂ ∂ ¯ − − 5 y lim ab z ) ) ) ) Φ A – 14 – : = 2 x, ab z z ¯ f z A ¯ ¯ ¯ ¯ x x x x ( h y, ( ( f 1 , ; L, + 11 2 9 x, x, x, x, j ¯ ) Φ + (2) (2) 2 w ; ; ; ; A 3 ¯ : z ) A ¯ 2 ¯ ¯ ¯ ¯ w c z z z z 1 : ∂ ) , πδ πδ w c + w, : 3 w 2 ab ¯ 2 2 ), the terms ( z, z, z, z, − w c w 5 ab ( ( ( ( κ − → A Λ( ab ab lim z 1 1 1 1 ab κ z → A − : (¯ lim z 3.6 w z κ Φ Φ Φ Φ : h ( ˜ ¯ ˜ cκ cκ z z ∂ : : : : + h (¯ ) x d w w w w w 9 ¯ x 2 w w ¯ z → → → → A 2 d lim lim lim lim w∂ I ¯ d dz dz z z z z z x, : : : : d ) come from the OPE between two SU(2) currents. We obtain: 2 d ; + w w w d w Z ¯ z 2 I w I I 2 Z × × × × d I Z 3 4.25 z, A πi 1 + + + ( c 3 2 1 Z c ab − × Φ − κ ab 1 : c πi 2 κ w from equations ( 2 − − → explicitly first: i ) = z : y = = = 1 ( = ) We have to compute the regular term in the OPE between 1 b A R g J A ) − 4.16 x 4 ( c a R J + ¯ x 2 ∂ mula ( where we performed the contour integral and replaced the der We get twelve terms (three on each double-line) that we denot integral in equation ( This leaves us with computing the terms separately and these four terms combine to give similarly for the limitc involving the anti-holomorphic der deal with explicitly perform the contour integrals. One observes tha lar limit lim vanish. The contour integrals in JHEP10(2009)051 (4.32) (4.36) (4.34) (4.35) (4.33) (4.31) . ) ) ), we get . . ¯ ¯ . , one can check ) ) w w k , we find that ¯ ¯ 3.6 ) 1 . w w i )] /k ) . ) ¯ w, w, ¯ w A ( ( w ¯ ¯ ) w w w, w, f f , ¯ ; ; w w, = 1 ¯ ) w, w w ¯ ¯ y y ; w, w, ¯ ; 2 y ¯ ; ; y w, i∂ i∂ ¯ y y, y, f ¯ ¯ y y ; y, ) − − y, ¯ y ( y, ¯ y, y, w I en algebra involving the other : ( y, oefficients ( ) Λ : ( Λ : ( Λ( luation of Λ( 1 )= )= y ¯ w, w w w ¯ ¯ z z ; w of the R-current algebra: Λ : ( Φ ∂ ∂ ∂ ¯ − y ∂ ¯ ¯ z ¯ Λ : ( w w I z, z, ) ( ( x ∂ ¯ w L, L,z R,z z y, L, ( ) (4.30) j j j f f ∂ j ) ¯ ¯ z : : z, L,z w ¯ :: ( (2) w w We can also check that the operator ; ¯ j ) : w L, δ ¯ w w x 1 : ¯ 1 1 j z )+ x w, − − 6 2 2 leads to : ; Φ ¯ ∂ w, d d ]. w ¯ x, z z w z, ¯ ; y ( 2 ; 10 ab ¯ w ¯ y w w 1 10 R,z d ¯ Z Z 2 x w, y, A j Φ ; d ) ) y, πκ : ¯ z ¯ y x, y y Z 2 dz∂ dz∂ ∂ ¯ z ) Z Λ( : − : ( y, w w Λ : ( − − y – 15 – ) ] by an overall normalization factor of L, w w I I 1 w j ), we have: y x x ∂ = Λ( 10 → − ∂ ( ( ) lim 1 1 Φ ¯ z π π ¯ z w ¯ − : ¯ z z ) : x 2 2 10 − − ∂ h (2) (2) ( L, y 4.19 x L, z, δ δ defined in [ j A ( j w ; x x (2) − : 2 L,w I ¯ x ∂ ∂ + δ (2) )= )= j contribute due to the following integrals: ) : : d x [ x ¯ ¯ δ z z 7 − + y w ˜ ˜ c c ( 7 x, x c c w Z A → z, z, ∂ π∂ A − lim 2 (2) : ( z ( ˜ ( c ab ab : 3 d 1 δ − + + x 1 f f κ κ c c ab c c ( ) ) 4 − − 2 2 π π Φ κ Z c c and ¯ w w ab z ab A − − (2) 1 κ κ π − 4 δ − L, − − + j = = = 2 A = z : = = z 1 7 4 ( ( = 2 = 2 : differs from the one of [ )= A A A w 1 ). Using formula ( ¯ I y (2) (2) 10 ¯ y → A lim y, z A ¯ zδ : ( y, dzδ d ; I and ¯ w w w I I w, Λ( w ∂ Our definition of 6 that this reduces to the operator The contact terms in We shall turn tobosonic the currents study on of the this boundary. operator For now, after we deriving observe that the wh We deduce that Combining these four terms and using the relations between c and where we have introduced the central extension operator Using these results and following the steps we did for the eva A similar analysis for the double pole term in JHEP10(2009)051 4) , (5.1) (4.39) (4.38) (4.37) = (4 . N urrents and the operator ) )] )) ¯ ¯ w ¯ )) w , z ¯ z ) z, w, y w, ; )] ; z, ( ; ; ¯ ¯ a ¯ y x ¯ y w ¯ x J . y, c x, y, complex conjugate of ) w, x, ( ed the current conser- ab ; y ¯ 1 Λ( Λ( . Similarly we can prove ( ¯ y ¯ z 1 Λ( c w Φ ghosts and therefore it is if ¯ z ∂ ¯ ∂ J ) w y, ¯ z ∂ c e that the operator is closed ( y L, 1 rs that correspond, in space- L, ¯ be constructed similarly. We ab j s of the boundary operators. w j L,z [ − Φ d 2 j x ¯ r of the boundary stress tensor: ) dual conformal field theory. In Λ=Φ if ¯ w 2 x x w ssion for the stress tensor for the ∂ : x of left generators. The generators ( ∂ ¯ ∂ d y I y ∂ ¯ w (2) 1 − y, − I L, πi c j ) + 2 x i 1 πδ ¯ ) + 2 z − = 4 superconformal algebra is at level ¯ z + − z, )+ + 2 ) ; z, I ¯ N )] w ; ¯ ¯ x I w 2 ¯ ¯ x w ) ) x, w, y y w, ; x, w, ; 1 ¯ ¯ y Λ defined as: Λ( − − ; ¯ – 16 – y z Λ( ¯ y x ¯ x y, z ∂ y, ( ( ( ∂ x y, 1 x ∂ ( ab ¯ Λ( (2) ¯ z 1 Φ ∂ κ δ w and find: ¯ w Φ L, x ∂ j ∂ L,z ∂ ). They are part of the generators of the small ∼ x j x x ) L,w x ab ∂ ( = 4 superconformal algebra, this implies a central charge j L,w y ∂ a [ dw j ( symmetry of the ( [ b πκ +( J N 2 J R I z w dw · 2 2 ]. First we will compute the OPE between this operator and the + d d ) πi c I . ∼− x 10 ( i 1 Z Z − ) a y π J ( = = = 0 1 b 2 )= J ¯ y − · . We will confirm the value of the central charge by evaluating ) ). We shall show that, given our basic OPEs of the worldsheet c y, ) that involves the operator ( I x )= ( I ) = 0. ¯ a x y ¯ y 4.10 ( ∂ J = 6 4.19 ¯ T x y, c ( ∂ I y ) does not depend on the spacetime coordinates ∂ ¯ y Before we delve into this calculation, it is important to not Putting together what we have so far, we find the OPE y, ( . By the structure of the Once again, this is acase non-chiral with generalization pure of the NS expre flux [ I We observe that the SU(2) in the hybrid cohomology. The operator is independent of the primary operators, this reproduces the superconformal OPE R-current ( that We can integrate with respect to ¯ From the first linevation. to the From the second,equation ( second we integrated to by the parts third, and we us used the (spacetime) I equal to product expansion of the stress-tensor with itself. 5 The spacetime VirasoroIn algebra the previous sectiontime, we to have the identified (left) the R-currents vertex operato superconformal symmetry of the holographically (boundary this section we will address theof construction of the the space-time full set right-movingput superconformal forward algebra the following can expression for the vertex operato JHEP10(2009)051 - x Λ. eet ¯ z ∂ (5.2) (5.3) ) ¯ w )) ¯ w, w . ; ¯ y :. The right ) come from w, ; 1 y, )) symmetry. So ¯ y ) 5.2 Φ ¯ w ¯ z Λ( L y, w R,z z, w, and the integrated j ∂ ; Λ( ; y ¯ ¯ x ¯ ming from the short 1 w y ∂ ∂ i of the bracketed terms x, y, e OPE between the ¯ g terms. We suppress ) ( r Φ )] L,z 1 ¯ )] w ¯ Λ( j w ssible to show that this is ¯ Φ z 2 y w n the integrated operators. rid superstring amplitudes, ¯ z ormal dimension one — this ∂ w, ∂ w, z, ) current: ; a ¯ z L, ; ; ish to evaluate the OPE: e to the OPE ( ¯ y R t-distance singularity between ¯ es we have already performed. y ¯ L, x ¯ zj , j ). One can alternatively fix the y, d ) + 2 2 y y, under the SU(2) x, ¯ ∂ z ( w w 5.1 Λ( 1 Λ( I w − Φ w . w, ∂ 2 ∂ ; y w ) ) + 2 y )+ ¯ y ) symmetry group (up to a BRST exact ∂ y ¯ ¯ ∂ z w ¯ ) w y, L − ¯ z L,z z, ) w, j ; x Λ( w, ; and the SL(2 y z, R ¯ x operators integrated in the derived R-current, ; ¯ ¯ w [( y ; , ∂ ¯ 1 y 1 ) x ∂ ¯ x x, ¯ y y, z ∂ ( – 17 – y, ∂ 1 ) we can rewrite the later as : x, z, ( Λ( Φ ; Λ( 1 y L,z w ¯ x ∂ w j ∂ , but the coefficient cancels against the delta-function 4.19 : a L,z ∂ y )Φ 1 y ¯ w x, z ∂ x ∂ ) as a sum of eight terms. Consider, for instance, the ( will not contribute to these poles since all of the inte- ( ¯ z a L, → 1 dzj lim − z j : L, Φ 5.2 w w j y ¯ ¯ a z z L,z 2 I y j w a a L, L, d ∂ 2 [2( operators come close to each other. The same is true for d x ) by demanding that the energy-momentum tensor transform 1 Z +( πi 1 ∂ dzj dzj and : ). Using the operator product expansion between the worldsh 2 Z 5.1 w w w ) ¯ z × − I I y 2.7 → a L, lim z j : − w w ¯ z ) = 2 2 x a L, y ( d d j ( )-left currents integrated in the stress-tensor. h (2) T Z Z R δ · w , x 2 ) πi πi 1 1 ∂ d x 2 2 2 ( Λ. Indeed using equation ( a z − Z − − ∂ J ¯ ). As argued before, the contour integral picks up the pole co x = = = = ∂ 1 We can expand the OPE ( As for the previous computation, it is simpler to consider th 5.2 and B 1 We conclude that the only singular terms that will contribut and the SL(2 operator). As a BRST-closedand operator, compute we Ward can identities insert for it the resulting in correlator. hyb BRST closed when it is a worldsheet conformal primary of conf follows from formula ( appearing when the two Φ operators in the stress-tensor.Φ Notice that there is no shor the short-distance singularity between the Φ in ( term obtained by taking the OPE between the first terms in each coefficients in formula ( distance singularity between the operator Φ as a weight 2 tensor under the global SL(2 energy-momentum tensor, the currentstrue and for the the field particular Λ, coefficients it chosen is in po equation ( derivative of the current and the stress-energy tensor. We w current has a pole in its OPE with Φ The contour integral will pickThe up SU(2) the poles currents in the OPEs betwee grated operators in the stress-tensor transform trivially the singular terms only come from the OPE between the operato Using the samethe techniques, details we since can the calculations compute are all very of similar to the the remainin on JHEP10(2009)051 (5.5) (5.4) (5.7) (5.8) (5.6) (5.9) (5.10) (5.11) . It is 0, and 1 2 , 1 ± ± 0) in space-time, , . 2 1 . 1 1 − Φ Φ have charges R , ) algebra gives rise to R j ) j R l for the vertex operators ) y , , ⊗ y ⊗ ( ) ) -dependent currents: (2 PE − ) y ome of the properties of the x . symmetry transformations of x e how our discussion may be ( x sl ( x ) r satisfies the OPE that codes ∂T ( a r eneral form ( x x ˜ aα J F − ( 0 ˜ aα F ) , j a + J j ) generators of charges − x y . x J x ˜ 2 aα ∂ , 2 y , r x ) + ∂ ) 2 − 1 2 ) ) e ∂ y is the central extension. The details b y y (2 y ( x − F ( ( ˜ aα I − + 2 + + sl , T a − xj , we get that the current is a conformal 2 1 1 2 1 x 2 (2) J x x ( . ) + ) F δ Φ Φ ( − 1 j y x x x x b + ( − ∂ ] in the zero ghost picture. ∂ 0 ∼ ˜ aα − a , π∂ J R 4 R ) 2 1 J 10 j ) x 2 to give: j x , where = 2 – 18 – y ( y + − I F ( 2 x I ⊗ ⊗ j e b 3 T − ) ∼ ) ∼− ) x = x x ) x = 6 ) ( ( ( ( x ) = y ( c a ( ˜ aα x ˜ aα F F a ∼ ( J j j T 0) in space-time. Imposing that the operator transforms J 1 ) ) , Σ ) ˜ aα b F y x 2 3 y Z j ( ( Σ ( and confirms the value of the central charge obtained using b a T doublet. The current must have weight ( Z T J · C ¯ x R ) ∂ ) = x ) = ( x (2) x ( T ( su ˜ aα ˜ aα G G indicates an a (2) charges which are inert. The eight fermionic generators -current algebra. This is a consistency check on our proposa su R are given in the appendix By evaluating the right-hand side at the point of the boundary currents.central extension In in the more following,extended detail. we to will also But discuss describe first, s the we superconformal briefly generators. 5.1 outlin Superconformal generators With respect to the scaling operator, we have the which can be integrated with respect to ¯ primary of dimension one in space-time: which gives for the central charge Combining these terms, we eventually obtain the spacetime O the the natural then to gather the eight fermionic currents into fou therefore, the superconformal generators should be of the g appropriately with respect to the raising operator in the where ˜ One can also check thatthe with Virasoro our algebra: definition, the stress-tenso the relation These operators have weight ( We thus propose the followingthe operator boundary to generate theory: the super This is the analogue of the expressions in [ JHEP10(2009)051 . . . (5.12) (5.13) ] have )+ ¯ γ 11 ¯ ∂ erators in ∂γ ). The proof in correlation − ¯ = 4 supercon- x takes the form c I . . I x, + N diagonal mode of /k )] x ¯ ∂γ ; ¯ γ ] for a detailed analy- w erator ∂ + 13 Λ( stress tensor and the R- c es of the central extension . mentum tensor, we know ( w 1 as superfields in spacetime l extension operator. φ ∂ attempt to compute it, such 2 nes of the calculation of the bra in space-time we need to eory is given by the operator ) ur derivation of the R-current appears as the coefficient of l fields described in [ erfields, and we need to work x ) within a correlation function. ze ; I 2 y ( w d ( bβ ¯ ˜ w Z limit, the operator G L, · that we found agrees with the central 1 π j ) I x − ( ) )+ →∞ ˜ aα γ x ). Thus it behaves like a constant in corre- φ ; G − w ¯ x ∂ x 4.37 ( Λ( – 19 – We refer the reader to [ ¯ w (2) ∂ δ 7 ) ) ] by an overall normalization factor of 1 In order to prove that we have an ]. x ¯ γ ; 10 ¯ ∂ 10 w ( ∂γ − L,w c j [ + w ¯ γ ), we have chosen to truncate to a bosonic component of the 2 ) is independent of the spacetime coordinates ( d ¯ x 5.1 ¯ Z ∂γ∂ x, ( + 1 π I c ( z = 2 ]). However, as is clear from our expressions of the vertex op I ) and ( d 16 Z is the operator , 4.10 I − 15 = I functions. lation functions. sis regarding the behaviour and the interpretation of the op From the form ofthe the metric vertex (mixed operator with we an can anti-symmetric see tensor). that it is an off- for this has been given in equation ( extension operator found in [ We leave this for future work and proceed to discuss properti This calculation can guide us in lifting the limitations on o Our definition differs from the one in [ 2. Near the boundary of AdS space, in the 1. The operator 3. For the case of pure NSNS flux, the operator , where 7 I equations ( Note in particular that by the fact that the operator 6 the boundary two-point function for the boundary energy-mo Higher superfield components. a full supermultiplet worth(see of e.g. fields and [ should be thought of take into account theat first least fermionic to correction first to order these in sup the fermionic currents. superfield in spacetime. To compute the super conformal alge operator. 5.2 Further remarks From the operator productcurrents, we expansion found that involving the the6 central charge boundary of the boundary th We would like to make a few observations regarding the centra formal algebra in space-time,R-current we algebra and should compute proceed the OPE along of the li and Virasoro algebras. The vertex operators for the physica There are some qualitatively newas features the that OPE arise between when the we fermionic currents and the operator Φ JHEP10(2009)051 ] 2 3 at ]). S 13 n of with , 13 × AdS , 2 3 10 S 10 , ] or from 9 xploit the AdS 7 × space-time, 3 3 AdS AdS further our analysis. tten points (where the e extended to rst step in the construc- background the vacuum is to extend our analysis round backgrounds that can be s. We have verified that rmal anomaly coefficient 3 ndary conditions. Indeed, 3 l gauge theory in the Higgs cide with those obtained in or the boundary supercon- n more detail in [ ole excitations) in the gly curved regime, and codes o algebra directly [ ll be the natural tool to clas- AdS d by the fluxes. ing theory on AdS ting primary states. The latter orldsheet techniques. A crucial e coefficient of the Wess-Zumino xplicit expression for the central ] satisfied by the currents in the nce of a supergroup valued field s which are near-brane limits of l to the Brown-Henneaux central fficients appearing in the current r in excited states (as in [ 12 ]. The target space in that case is the 28 ]. Since the gauging involves only the right 29 with Ramond-Ramond fluxes can be described – 20 – 2 S U(1)) [ × 2 × ]. These imply that in an 27 ]. Technically, our results are a non-chiral generalizatio AdS (U(1) 13 2). It is important to note that most of the simplifications th / , | 1 2) 10 , | 1 , (1 , 9 (1 P SU P SU = 4 superconformal algebra. A related problem is to further e N 2) superisometries of the hybrid formalism to covariantize | 1 , (1 There are many directions for future work. A clear challenge We would like to argue that our construction is a significant fi It is a good check on our construction that at Wess-Zumino-Wi Our construction of the spacetime symmetry generators may b that it governs the value of the conformal anomaly. In a backg both Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarzthey fluxe satisfy the conformal operatortechnical algebra ingredient using was stringy the w conformal current algebra [ there are various otherin ways the to supergravity approximation, compute namely the from value the of Virasor the confo geometric coset for the NS-NS case.corrections Our to calculation the is central extension also operator valid that in can the occu stron 6 Summary and conclusions Our main result isformal the algebra, construction in the of worldsheet the theory vertex that operators describes f str to the full the holographic Weyl anomaly [ supergroup model those references. Our generalizationD1-D5 D-brane includes systems background for whichphase the can be infrared studied limit directly. ofextension An the operator. interesting dua result is the e expectation value of the central extensioncharge operator as is determined equa in supergravity. This has been argued i tion of the full spectrum of string theory (including black h described in a Berkovits type formalism. P SU arise in the calculations followalgebra. from constraints These on constraints, the in coe throughout turn, the follow two-dimensional from moduli the space existe parameterize coefficient of the kinetic term isterm), equal in which absolute is value to the th casethe with NSR only formalism NS-NS in [ flux, our results coin One can also attempt to generalize our arguments to other background with Ramond-Ramond fluxes, withthe appropriate full bou local asymptoticsify symmetry the algebra spectrum, in thus space-time reducingoften wi the reduces spectral to problem a to mini-superspace lis problem. backgrounds. String theory in using the four-dimensional hybrid formalism [ JHEP10(2009)051 (A.2) (A.1) (A.4) (A.3) . ! 1 ! σ 3 i 0 2 σ 0 i 2 − 1 3 . σ σ i i 0 0 2 2 ! − − 1 mis, Robert Myers and !

− 2 ! . gma-model is tied in with σ ! 1 0 0 2 = = 0 2 1 eralgebra which we will use I σ ! 0 gma-matrices are given by i 2

l, and the inﬁnite set of con- 1 2 0 34 − 23 I two-dimensional CFT dual to 0 i nd better the relations between 2 − Notice that we obtain a set of = 2 2 K K algebra will be left intact. Then σ I σ I 3 i 0 0 i 1 2 2 2 2 1 σ 0 0 . c − − − − ! with RR ﬂux σ 2 !

3 σ 0 2 abc i 2 σ = = = = 0 ! iǫ i 2 2 i in terms of the Pauli matrices: 21 41 22 42 σ 0 + 2 − AdS 0 i 2 S S S S σ 0 ab I i i 0 2 − ! ! K ab 2) – 21 – | δ

3 1

! ! 1 σ σ , 1 3 = = 0 0 1 2 2 1 = = σ σ 0 0 b (1 2 1 1 2 2 − − σ 13 24 σ a 1 3 1 3 K K σ σ σ σ σ 0 0 0 0 1 1 2 2 2 2 1 1 P SU − − − − spacetime. ! ! 3

2 ! σ 1 0 i 2 = = = = 0 1 1 0 σ 0 i 2 − AdS

11 31 12 32 3 1 S S S S = σ σ i i 0 0 2 2 1 σ − −

= = 12 14 K K Finally, the conformal current algebra of the supergroup si A.1 Matrix generators for both the worldsheet integrableserved structure charges in of space-time. the It sigma-modethese would be two important useful to features understa of the theory. Acknowledgments We would like to thankAmit Costas Sever for Bachas, helpful Denis discussions. Bernard, Jaume Go A Worldsheet action for strings on We will now give anto explicit write down matrix the representation worldsheet of action. the Recall sup that the Pauli si Then we can write the bosonic generators They satisfy the algebra our previous construction willholomorphic generators extend in straightforwardly. spacetime,quantum as gravity suited on for an a chiral symmetry, a reasonable hypothesis is that the left-current Similarly, we represent the fermionic generators as JHEP10(2009)051 S 2): | 1 (A.8) (A.6) (A.7) , (A.11) (A.12) (A.13) (A.10) (A.14) (1 inations . ) ) (A.5) psu ) 24 bβ 23 . S we choose the K K    aα 3 + I ! ! 0 S + ( 2 0 13 ) φ σ 14 i AdS 1 e K i 2 2). The generators icitly. ) ( 2) as Str σ K | | i − ( 0 0 1 2 − 2 1 1 i (2 = γ , 1 − 2 − 0 0 0 − sl )(¯ γ . ab i

3 rization of the sigma-model = se generators from the ones = δ 4 )(¯ metric: of K i 5 − 2    SU(1 ) = = αβ iS 2 osonic generators of ǫ γ ) (A.9) K K θ 2 5 + ϕ ∈ θ ( j ± − − φ γ K K g K 1 3 e i cos sin ϕ = S ( 1 2 + + ( K i ) 3 − ( ) ! = iϕ iϕ φ φ e 24 bβ 1 4 − 14 − − ! α AdS ± σ K e e 3 aα Str ,S 0 K g i 0 0 θK 2 S S θ e ) 3 2 φ − 2 i θ ie aα = S − e − aα − σ 0 0 ) θ g S e 13 i + 2 1 sin ij h e 0 0 0 3 23 F η cos 2 K γ g K + α

K – 22 – = 2 1 1 ) ( α iϕ 2 ( 2 )(¯ i γ e i iϕ 1 − ϕ F 2 1 2 = = = e iS ) )(¯ g + ie 1 4 = i + i 1 cd = = j ± ϕ g K K γ I ( 0 − 4 1 ( K α − 1 φ , K γ K K ab    e i e S ! ! K K = = + ( + ( = h 3 0 by a simple linear change of basis: φ φ σ 3 α 3 ) ) i 2 S ± − − 1 Str σ ab g e e S 34 34 i 0 0 2 − K = K K − 0 0 0    1 2 − +

abcd ǫ 12 12 = = = + + + ++). For the fermionic generators, we use the linear comb − K K 3 0 3 ( ( − = 2 2 1 1 K K AdS i g = = cd 0 3 , K K K ab is an overall phase that we will gauge away eventually. For ). These will prove to be useful in writing out the action expl K α h 2.1 with and Poincar´eparameterization: They give the generators in the fundamental representation with signature ( A.2 Explicit formOur of goal the in worldsheet this currents on section the will supergroup. be to We parameterize obtain the a group concrete element paramete One can easily obtainin the ( commutation relations between the These generators satisfy square to a multiple of the identity. We choose the invariant It will turn out to be useful to choose the following basis of b These are related to the JHEP10(2009)051 (A.17) (A.18) (A.15) (A.23) (A.20) (A.16) (A.21) (A.19) (A.22) I 4 5 1): dα , K K . The other dγ 3 )] )] + S . 2 2 S 3 ) 2 dϕ dϕ abcd ... . AdS ǫ ¯ ¯ ∂φ . + + ∂ϕ 2 2 + dg 1 1 αβ ) to SU(1 3 ∂φ ǫ K R ] dϕ dϕ 1 cδ , cγ θ∂ϕ θ θ − AdS )( )( dγ + 2 0 2 rget space ) g 2 2 ba bβ 2 K ϕ ϕ ¯ θ γ ] nsider . + γ K 3 − − aα    ): dγ + sin + S ) 1 1 ¯ I ∂γ∂ 0 R 1 1 + ¯ 2 dθ ϕ ϕ , aδ dg γ 2 1 − + 1 θ 3 γ ¯ ( ∂ϕ φ − S φ cb 1 φ . + sin( e g cos( 1 2 K e θ θ (1 + ¯ ! ¯ cd γ∂γ φ K + on our basis of generators. We have: ] φ ¯ i θ∂ϕ ∂ + + 2 K 3 i , ¯ γe sin 2 − sin e 0 † bδ γ dg φ ¯ γ e θ θ ¯ γdγ 2 θ 1 1 1 − AdS φ abcd 3 e φ . γdφ + − cgc g ǫ e ac 2 i

2¯ 3 g K cos φ e + cos ] = S K αβ 2 K − → 1 – 23 – − g ǫ γdφ i i 1 − − + cos e ) j 3 √ ¯ g ∂θ ¯ γ γδ bβ 2 ǫ , d aα θ dθ dθ θdϕ + 2¯    dφ 1 ∂θ ) = F , ) AdS − 2 2 αβ ¯ aα 2 γ are real while they become complex conjugate in Eu- c X aα = ǫ d ϕ =0 ¯ ϕ ∂g θ +2[ +[ i dθ γ ) cos 3 cγ e = 2( − ( R 2 1 − θ , = [ = 1 i d − 1 1 . 3 bβ − AdS 3 + 2 i ϕ ϕ S aα and ¯ θ SL(2 F , g K g ¯ aα 3 γ i aα ∂g AdS aα j θdϕ 1 S cos( sin( 3 θ 5 dθ 2 − S dg , − AdS aα − − 4 1 6 3 e , , g 1 both 1 ∂g 3 + dθ X =3 − 3 S − AdS +2[ +2[ i 3 ) is given by: g 3 = 1 c ∂g = 2[sin = − AdS 1 3 F g 3 AdS − AdS S h S g g dg h 3 1 . We have used the following isomorphism from SL(2 S dg 3 − F 1 g 3 g − S g part is given by = ( AdS 3 dg S 1 − g We now expand the left-invariant one-form The This follows from the following parameterization of SL(2 clidean For Lorentzian purely bosonic part is given by So we recover the worldsheet action for a sigma-model with ta where the matrix The action for the AdS part thus takes the form One can check that: The remaining term can be tackled as follows. First, let us co JHEP10(2009)051 : 3 uffi- . AdS (A.24) (A.25) (A.30) (A.26) (A.29) (A.27) (A.28) (A.32) (A.31)      θ ρ ρ ρ ρs sinh sinh θ cosh θ cosh s c θ c d . ) ) ) c 2 2 1 ) ϕ ϕ 6= 1 6= ϕ − ϕ + + b b t ϕ − ( ϕ ( t i ( i ( i i . − ie − e − and and AdS θ ) parameterization of ρ . ρ ie ρ e g onomical way to express the 0 rmation of the fermionic gen- c c ρs 3 K ) S = = sinh sary to obtain the result of con- ϕ sinh cosh g t, ϕ, ρ . θ i a a θ − θ cosh c t s . c ) ( . K ) ± ) aα )      1 i i 2 − 2 1 . b S ϕ α α α α ϕ e ϕ ϕ − + − + βα 3 3 1 1 1 aα + + 2 ǫ − ab − + t S S S S − ϕ t ( δ ϕ ( ) ( i ab ( i i ρK 3      i ) for aα e bβ − 2 − ) for S dg,S ie S f M − g bd − 1 ( bα e ± − aα αβ ρ ie 0 ρ e θ g ǫ 1 xK f aβ xS K h . The fermionic currents can then be written in – 24 – ρ − ) f i ρs −→ M θ ) = = ϕ sinh + = sinh = t F      θ aα ( θ + sin AdS + sin α α α α cosh s c − cosh − + − + ± aα ) g 3 3 1 1 Π dg ) aα θ e 2 ) 1 = cos 1 ad S S S S c Π aα such that 2 ϕ Π ) ϕ − F θ ϕ 1 = ( = +      g c − ϕ xS xK − ϕ 3 ϕ t ( B M ( + ( i i t i g ( − − i and AdS ie ± e aα , we can obtain all other possibilities easily. This proves s ie g − S θ − ba − 1 = (cos = (cos Let us first compute the fermionic currents K − B ρ e θ ab ab ρ g ρ − is given by ρs = sin xK xK = θ sinh − − M cosh s e e sinh θ ab θ s cosh θ cd cα ) c c ) K 2 ) ) S 2 K 1 ϕ 1 ϕ ϕ ab ϕ ab + − + ϕ t − t ( ( xK ( ϕ i xK i i ( e e i − − ie e e ie −      Let us write this compactly as follows: In order to obtainjugating the the left-invariant one-forms, generators by it the isconjugation bosonic relations neces group is elements. as follows: An ec cient to compute the bosonic and fermionicFermionic currents. currents. Recalling that where the matrix For this we have to compute Under conjugation by the bosonicerators group is element, encoded the in transfo a matrix Then, In this calculation, we find it convenient to use the global ( the form We have denoted JHEP10(2009)051 } 2 4 he 4 , M b 1 b , . 0 )) )) π 2 2 (A.35) (A.36) (A.37) (A.33) (A.34) 2 K 1 1 ϕ ϕ { b b k/ he purely − + 1 1 )) )) . . . e matrix ϕ ϕ φ φ + − − . t sin( t cos( i . θ θ K 5 5 ¯ i . ∂φ b b ℓ generators than with ) sin( ) sin( bβ . ∂φ )) )) i φ i nts from the fermionic φ f 2 2 2 ) cos 2 X ) cos 2 2 K 2 ϕ ϕ f + 2 + aα ϕ t = ϕ t f one can write out the full − − + WZW ... . 1 3 + ab 1 1 2 S 5 2 ϕ S agrangian (up to the overall 1 itly in terms of the Poincaré δ b + b sin( ϕ ϕ g ¯ γ cos( ϕ , the action coincides with the ¯ ∂ θ b αβ   ρ )) − )) 2 )+ ǫ i /k φ ) φ sin( ) and using the fact that ¯ ∂γ sin( ∂φ ¯ K ∂g ) sin( φ θ ρ b i )+sin( − 1 − 2 b 2 − t = 1 ) sin 2 ∂φ − t − ϕ X 5 k ) ϕ , 2 2 a,b,α,β t 2 A.24 2 4 1 , + ϕ f − − ϕ f + ) cos 2 1 X 1 ∂g g ) + cosh 2 2 =3 ) sin( + 2 − ) sinh 2 i 1 1 + n ) cos( ϕ ϕ φ f 1 φ ) 1 ϕ φ = 2 and rewriting the partial derivatives of − ) cos( φ   j ϕ g ϕ ¯ γ φ − 1 ¯ z 3 + + + + ¯ z sin( ∂ t − + S t t + 1 + ) sin( t η g ℓ – 25 – 2 − t ϕ ( ∂γ STr( ) cos( ϕ ¯ + ) ∂γ − sin( m sin( 2 z 2 sin( + cos( 3 ¯ c ) γ ) cos( + 2 ρ ϕ ϕ j ρ cos( 4 1 ∂ φ d − − + ρ ϕ 1 − AdS + + 1 θ b b Z g k 1 ℓ ) + cos( t ¯ ϕ ∂γ ) ( 2 2 ϕ   cosh 2 ¯ γ i + φ ϕ 1 ∂ mn πf 2 − K + 1 − η + i ) sin 2 +(cos( 4 f ) cos( )+sinh2 c b 2 t 3 1 φ + (cosh 2 b θ φ 2 + (sin( +(cos( ϕ =0 , − = ϕ ) 5 3 0 1 , we get 0 , − 2 − b with NS-NS flux with an overall multiplicative factor of b X + g b , we obtain this bosonic contribution to be cos( φ φ t m,n ) t X ϕ } 2 2 ) 1 WS 3 =0 ) ρ It is much easier to directly work with the 2 i 5 e e ) and combining this with an antisymmetric part coming from t , φ S ϕ − φ ) cos 2 = ) cos( ϕ 4   1 2 , 2 In order to check the normalization, it is useful to focus on t ). Taking into account 3 3 − ’s are calculated, then, using the explicit components of th − − ϕ L z z ϕ AdS ϕ ) cos( ) sin( b t 1 2 2 t 1 K A.22 φ φ sinh 2 )) can be written in the form d d + ϕ { + − AdS ¯ γ, φ 0 + sin( 1 g cos( + + 1 cos( sin( 2.2 3 Z Z t ϕ t γ, θ ϕ sin( ρ b ρ ρ π π ’s and 1 1 θ b θ 4 2 f sin 2 generators to derive the contributions to the bosonic curre (cos( (sin( = = − +(cos( − − +(cos( ab in terms of that on WS K = cosh 2 = sinh 2 = sin 2 = cos 2 = sinh 2 = S 1 0 1 4 3 2 5 -dependent) terms in the action. Starting from ( ℓ ℓ ℓ ℓ ℓ ℓ − θ Bosonic currents. WZW term, we find that ( the commutes with All the currents haveaction been to all explicitly orders written in out the fermionicNormalization. above coordinates. and, Using equation ( bosonic part. Letcoordinates us ( first write the worldsheet action explic One can check that, at the WZW point, putting bosonic action on Using the commutation relations, we find that g A.3 The action Once the to simplify the fermionicconstants currents, in ( the kinetic part of the L JHEP10(2009)051 ) (B.5) (B.7) (B.6) (B.4) (B.3) w z ( − ∂φ transforms a w t a φ t − ) c ) w z + w + ) c : ( : ( : − ) as a field satisfying w + φ φ φ φ c a w : ( t 3.4 . is annihilated by all the n in which ) a L,z − ) eld j x φ z a that the pole of order two ) L,z ( : e exact values of the coef- soro primary. We can read . w : ( t conservation and with the φj ) z a − L,z n implies that a primary field : φ j : ( ) w z ld with respect to the Virasoro − z . w ( − w φ − ) + less singular+ less singular (B.1) (B.2) a L,z ( c − ) w z − j − ) ) φ ∂φ x + . and the holomorphic stress-tensor: a istics. They can be consistently restored. L,z : ) ) + w a w a ¯ ¯ ) c x : ( j w w w ( t t z − a φ w w : + a φ t t − c a w, w, : ( a w b L,z c − − t . ( ( + 2 t 1 j 2 a c − + φ φ φ z z f 2 t − + c ( ( a a ) a c a t t L,z + t − a + − L,z j w + 2 c j + : − − ( 2 ) : ) c ) 2 2 : c c ) f φ ) z + w w z ( a + − a ab L,z ( c w + + z t φ c c j = – 26 – ) κ φ : ( − a ) z a a + + + 1 − 8 φ t 1 z − w t t x c c c c φ 1 a ( z w ( z a − t 2 ∆ ( ( w − t a − − φ a = − L,z t 2 a L,z a w x 2 j + − t j f x c ) : )= − ∂φ x + ) = ) = z a + c ( . This implies that the operator : c ( t ¯ ¯ w w )= φ w + + T + + z c 1 c → c ( aL,z w, w, lim + x j ( ( φ OPE gives the holomorphic derivative of : with respect to the left-current algebra ( c ) ) + φ φ ) ) z ) ) φ − w ( ) 2 − − ¯ ¯ z z ( ) c c φ T.φ w : : T ( z, z, w w w + + ( ( φ + + ¯ z a − → → c c + + t lim lim a a x x L,z L, c c z : : a j j ( ( ( t 1 1 1 1 2 c c c c 1 1 2 f 2 2 2 2 = = = = )= is a generator of the Lie super-algebra in the representatio w ( a t T ) z We will not be careful about minus signs due to fermionic stat ( 8 φ We observe that there isis no proportional pole to of the order operator greater than two, and positive modes of theoff stress the tensor, conformal and dimension thus of the that operator it is a Vira B Primary operators We define a primary field The simple pole in the the OPEs: We deduce the OPE between the stress-tensor and the primary fi where Let us compute the OPE between a left-primary field on the left. Theficients structure is of fixed these byMaurer-Cartan demanding OPEs equation. compatibility is We both postulated, will now with and showwith curren that th respect this to definitio the left-currentalgebra. algebra The is worldsheet also stress a tensor primary is: fie JHEP10(2009)051 ¯ z L, j ) (C.1) (C.2) 2 y 1 do not ∂ Φ ) ¯ z w y L, ) Λ) − j − ¯ w w 2 x z ¯ z x ) replaces the ∂ ∂ ( ¯ L, z ). As we did − ¯ w j ab L, z c 2 y + 2 j ( − 5.7 f ∂ i 2 y ) 1 z ) ∂ (¯ g y Φ / x − -tensor with its anti- − ∂ 2 2 ¯ z ) x are identical except the c ensor ( Λ + 2 L, w )( 1 ¯ j licit. In order to compute 2( w ¯ x w ote is that the mixedrphic terms pieces in ∂ − ∂ y + ( z − ∂ ( ) will pick up the poles in the ¯ z ¯ z s quite tedious. The key point ¯ z i z lculation for the case with pure d L, L,z L, j )(¯ j j C.1 ¯ z 2 y 2 g y I y ¯ z ∂ ∂ L, ∂ ) ) − j L, ab c 2 y j y w )+ 2 1 ¯ f z 2 ) y ) ∂ 1 c ) − ∂ − ¯ L, w g w Φ ab ]Φ 1 ) c 2( j x z y Λ) + ( f ¯ 2 y 2 w ( ( − − − ¯ ) ∂ w ) L,z i ∂ + 4 g 2]Φ j 2 z ab ) ∂ z c − -dependent combinations of the currents (and w 2 ( 2 c (¯ x ¯ f ) i w i with the operator Φ x − − − z ∂ ) L,z L,z − 2( w g 2 2 j ¯ y y – 27 – j z )(¯ − c 2 z 2 y − ) ∂ ∂ y g L, ( − + 2 − ) ) z ∂ ∂ i j w 2( ) y y 1 4 z + − )(¯ y ( c g − Φ 2 i + − − c x ) and the antiholomorphic factor 1 2( − z ) satisfied by the coefficients of the current algebra, the L,z − x x ∂ ( Λ + 2 j − 2( x i 2 L,z ¯ 2 c y . ( w c j 3.6 − [2( [2( L,z ∂ 2 )+ ∂ 1 2 1 y + ) j c y w 2( w w ) w w ∂ ab w x ¯ c Φ 2 Φ + w ∂ + 2 ¯ ∂ 2 f y y w 1 1 c ab − d − c − − ( + ) − ( ∂ ∂ + − f L,z ) g z z / 2 2 z z − L,z z ) ) j dz Z z 2 j ( w − g y (¯ z ) L,z − − − − − 2 y w c i (¯ 1 ∂ j c c c c i ∂ I w 4 (2) c − πi ( 1 − 2 y 2 c − + + + + 2 4 ∂ + + + + z 4 c cδ c c c c − ( z 2 c 2( 2 − 2˜ × × − i ( c + + + + z i − c c c c ( − 2( 2 ∼ ∼ ∼ ∼ ∼ ∼− ∼ ∼ ∼ ∼ ) = y ¯ ¯ ¯ 1 1 1 1 z z z ( Φ Φ Φ Φ L,z L, L,z L, L, L,z ]. T j j j j j j y y · 2 2 y y y y y y ∂ ∂ L,z L,z 10 ) ∂ ∂ ∂ ∂ ∂ ∂ j j · · · · · · x 2 x x L,z L,z ( j j ∂ ∂ 2 x x T L,z L,z L,z L,z L,z L,z ¯ x ∂ ∂ j j j j j j ∂ 2 2 x x x x x x The OPEs of the current component ∂ ∂ ∂ ∂ ∂ ∂ their derivatives), which we list below for convenience: holomorphic factor. The contour integrals in equation ( where we kept thethis coordinate OPE, dependence we require of the the OPEs between operators the imp overall factor outside is C The Virasoro algebra In this appendix we will consider the self-OPE of the stress t OPEs between the integrated operators.in the One OPEs, useful which point contain to both n holomorphic and anti-holomo computation is a simple non-chiralNS generalization flux of [ the ca in the bulkholomorphic of derivative the : paper, we will evaluate the OPE of the stress contribute to the contour integrals.is that The because full of computation the i relations ( JHEP10(2009)051 ) y ]. ] − (C.3) (C.5) (C.4) (C.6) x , ( Adv. . , (2) δ SPIRES Λ Λ ] [ ¯ w w , ∂ ∂ y y superstring ∂ ∂ ]. 5 · · ) S backgrounds which hep-th/9711200 1 1 w × ) Φ Φ 3 ; x x 5 y S w ), we get : OPE. It is obtained by ∂ ∂ ; ) ) SPIRES Λ( her terms leading to the y × the second and the third w w AdS w C.1 ] [ 3 hep-th/0202109 Λ( ∂ [ − − ¯ y w ... , i.e. the doubles poles and the um tensor. super Yang-Mills ]. ∂ z z ∂ ( ( · AdS ion ( y + (1999) 1113] [ s to ) ∂ I (2) (2) =4 · . . . ) w ) y 38 ; cδ cδ ] to N + x 2˜ 2˜ w SPIRES Strings in flat space and pp waves from − ; 10 4 )( ) occurs when both partial derivatives ) ) that we already encountered in the − − x ] [ x 1 ). The regular limit leads to a y ( ]. )( Φ I Λ Λ (2002) 126004 ¯ 1 hep-th/0202021 z C.1 ¯ 3 − w 3.6 w [ (2) 4.36 Φ δ L, ∂ ∂ x j 3 y y x ( ( ∂ ∂ L,z 2 x · · j D 65 π∂ ∂ SPIRES ( – 28 – 1 1 2 − x )= : ] [ Φ Φ ∂ y Hidden symmetries of the w ( x x : ∂ ∂ → (2002) 013 T )= lim w z 2 2 · : Int. J. Theor. Phys. y Exactly solvable model of superstring in plane wave hep-th/0305116 ) ) ( [ → ) 1 3 ) lim ¯ 04 The Bethe-ansatz for z w w c c ˜ c : x T , we find limit of superconformal field theories and supergravity 2 2 ( Phys. Rev. · ) x − − − − , ˜ c T ) − − c N z z 3 x ( (¯ + c − ( ( JHEP c 1 T , c ¯ z ¯ x ( + (1998) 231 [ hep-th/0212208 dz d ∂ [ 2 I I w (2004) 046002 2 The large- w w d . This generalizes the result of [ 2 2 I d d Z D 69 Z Z (2003) 013 = 6 ]. c πi πi 1 1 4 4 super Yang-Mills 03 − − is the operator defined in equation ( =4 I Phys. Rev. SPIRES N [ Ramond-Ramond background JHEP Theor. Math. Phys. Integrating with respect to ¯ Let us illustrate this for the most singular term in the above act on it. Finally we obtain : [2] D.E. Berenstein, J.M. Maldacena and H.S. Nastase, [1] J.M. Maldacena, [3] R.R. Metsaev and A.A. Tseytlin, [5] I. Bena, J. Polchinski and R. Roiban, [4] J.A. Minahan and K. Zarembo, 2 x Let us first combineterms. the After first performing and the the contour fourth integration, terms, this followed lead by where The coefficients simplify thanks to the relations ( self-OPE of the R-current. taking the most singular termscontact in terms). the current-current Collecting OPEs the ( four relevant terms in equat ∂ include RR fluxes.standard A operator similar product analysis expansion can for be the done energy-moment for allReferences the ot factor and the most singular term in the OPE ( which gives JHEP10(2009)051 , , , , , , (1999) 008 , Adv. Theor. 04 , 3 -models (1998) 009 , σ 3 02 ]. (1995) 123 ensional background ]. 2), JHEP | AdS 1 , , 3 -WZNW model (2000) 31 (1 representations + 3 JHEP ]. B 433 AdS SPIRES , 2) (2002) 303 SPIRES | 282 ] [ PSU ] [ (2 ]. , ]. ]. psl SPIRES B 621 String theory on AdS ] [ and 2d conformal field theory sigma-model as a conformal field ]. 3 ) Nucl. Phys. background with Ramond-Ramond flux n | , SPIRES ]. SPIRES SPIRES ]. 3 n AdS ( ]. ]. ]. ] [ Annals Phys. ] [ ] [ ]. , × AdS 3 SPIRES hep-th/9902098 arXiv:0711.1174 [ Nucl. Phys. PSL [ S Asymptotic dynamics and asymptotic symmetries ] [ , Boundary Spectra in Superspace SPIRES SPIRES 3 SPIRES SPIRES SPIRES – 29 – Conformal Current Algebra in Two Dimensions ] [ ] [ SPIRES Comments on string theory on Integrability for the Full Spectrum of Planar [ The WZNW model on Tensor products of hep-th/9902180 ] [ ] [ ] [ ]. [ Conformal field theory of AdS background with AdS The space-time operator product expansion in string theory topological strings (1999) 018 (2008) 064 hep-th/9906215 [ hep-th/9908041 hep-th/9806104 Central Charges in the Canonical Realization of Asymptotic [ [ SPIRES [ 03 01 =4 More comments on string theory on (1986) 207 Notes on ]. ]. ]. ]. ]. (1999) 205 hep-th/9806194 N ]. Vertex operators for 104 JHEP JHEP arXiv:0712.3549 hep-th/9910205 arXiv:0903.4277 hep-th/0610070 hep-th/9812046 , [ [ [ [ [ , (2000) 555 B 559 (2000) 333 (1999) 139 SPIRES SPIRES SPIRES SPIRES SPIRES SPIRES [ ] [ ] [ ] [ ] [ ] [ Black hole entropy from near-horizon microstates (1998) 733 [ Quantization of the type-II superstring in a curved six-dim B 571 2 Operator product expansion and factorization in the H B 565 B 548 Six-dimensional supergravity on arXiv:0901.3753 (2007) 003 (2008) 024 (1999) 003 (2009) 017 (1998) 026 , Nucl. Phys. 03 10 11 06 12 , Nucl. Phys. hep-th/9712251 hep-th/9910013 hep-th/9407190 hep-th/0106004 hep-th/9903219 [ of three- dimensional extended AdS supergravity duals of field theories Nucl. Phys. [ [ Nucl. Phys. JHEP JHEP JHEP [ hep-th/0506072 JHEP Symmetries: An Example from Three-Dimensional Gravity JHEP [ Ramond-Ramond flux Commun. Math. Phys. theory Math. Phys. AdS/CFT [6] N. Gromov, V. Kazakov and P. Vieira, [9] J. de Boer, H. Ooguri, H. Robins and J. Tannenhauser, [7] J.D. Brown and M. Henneaux, [8] A. Giveon, D. Kutasov and N. Seiberg, [18] N. Berkovits and C. Vafa, [20] L. Dolan and E. Witten, [22] M. Henneaux, L. Maoz and A. Schwimmer, [23] J. Teschner, [24] O. Aharony and Z. Komargodski, [25] A. Strominger, [17] T. Quella, V. Schomerus and T. Creutzig, [19] N. Berkovits, [21] J. de Boer, [13] A. Giveon and D. Kutasov, [14] M. Bershadsky, S. Zhukov and A. Vaintrob, [16] G. T. Götz, Quella and V. Schomerus, [10] D. Kutasov and N. Seiberg, [11] N. Berkovits, C. Vafa and E. Witten, [12] S.K. Ashok, R. Benichou and J. Troost, [15] G. T. Götz, Quella and V. Schomerus, JHEP10(2009)051 ] (1998) 023 07 n a Calabi-Yau ]. , Springer, Berlin Superstring theory on hep-th/9907200 [ JHEP , SPIRES ] [ wiebach, (2000) 61 B 567 Conformal Field Theory hep-th/9404162 [ – 30 – Nucl. Phys. , The holographic Weyl anomaly (1994) 258 ]. B 431 SPIRES ] [ Covariant quantization of the Green-Schwarz superstring i Nucl. Phys. as a coset supermanifold , 2 ]. S × 2 SPIRES hep-th/9806087 Germany (1997). [ background AdS [ [27] M. Henningson and K. Skenderis, [28] N. Berkovits, [29] N. Berkovits, M. Bershadsky, T. Hauer, S. Zhukov and B. Z [26] P. Di Francesco, P. Mathieu and D. Senechal,