CFT correspondence

25 November 1999

Physics Letters B 468 Ž1999. 65–75

Light cone gauge formulation of IIB supergravity in AdS5 = S 5 background and AdSrCFT correspondence R.R. Metsaev

1

Department of Theoretical Physics, P.N. LebedeÕ Physical Institute, Leninsky prospect 53, 117924 Moscow, Russia Received 19 August 1999; accepted 16 September 1999 Editor: P.V. Landshoff

Abstract Light cone gauge manifestly supersymmetric formulations of type IIB 10-dimensional supergravity in AdS5 = S 5 background and related boundary conformal field theory representations are developed. A precise correspondence between the bulk fields of IIB supergravity and the boundary operators is established. The formulations are given entirely in terms of light cone scalar superfields, allowing us to treat all component fields on an equal footing. q 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction A long-term motivation for our investigation comes from the following potentially important application. Inspired by the conjectured duality between the string theory and N s 4, 4 d SYM theory w1x the Green–Schwarz formulation of strings propagating in AdS5 = S 5 was suggested in w2x Žfor further developments see w3–5x.. Despite considerable efforts these strings have not yet been quantized Žsome related interesting discussions are in w6x.. Alternative approaches can be found in w7x. As is well known, quantization of GS superstrings propagating in flat space is straightforward only in the light cone gauge. It is the light cone gauge that removes unphysical degrees of freedom explicitly and reduces the action to quadratic form in string coordinates. The light cone gauge in string theory implies the correspond1

E-mail: [email protected]

ing light cone formulation for target space fields. The string theories are approximated at low energies by supergravity theories. This suggests that we should first study a light cone gauge formulation of supergravity theory in AdS spacetime. Understanding a light cone description of type IIB supergravity in AdS5 = S 5 background might help to solve problems of strings in AdS spacetime. Keeping in mind extremely important applications to string theory, in this paper we develop the light cone gauge formulation of IIB supergravity in AdS5 = S 5 and the associated boundary CFT and apply our results to the study of AdSrCFT correspondence at the level of stateroperator matching. Our method is conceptually very close to the one used in w8x Žsee also w9x. to find the light cone form of IIB supergravity in flat space and is based essentially on a light cone gauge description of field dynamics developed recently in w10x. A discussion of IIB supergravity at the level of gauge invariant equations of

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 0 6 3 - 1

R.R. MetsaeÕr Physics Letters B 468 (1999) 65–75

66

motion and actions can be found in w11x and w12x respectively. As is well known in the case of the massless fields, investigation of AdSrCFT correspondence requires the analysis of some subtleties related to the fact that transformations of massless bulk fields are defined up to local gauge transformations. These complications are absent in the light cone formulation because here we deal only with the physical fields, and this allows us to demonstrate the AdSrCFT correspondence in a rather straightforward way.

soŽ6. which is the isometry algebra of S 5. The odd part of the algebra consists of 32 supercharges which are responsible for 32 Killing spinors in AdS5 = S 5 space. We prefer to use the form of soŽ4,2. algebra provided by nomenclature of conformal algebra. In this notation we have, as usual, translations P a, conformal boosts K a, dilatation D and Lorentz rotations J a b which satisfy the commutation relations

w D, P a x s yP a , w D, K a x s K a , w P a , J b c x s h a b P c y h ac P b , w K a , J b c x s h a b K c y h ac K b ,

2. Light cone form of psu(2,2 <4) superalgebra and notation We use the following parametrization of AdS5 = S 5 space ds 2 s

1 z2

Ž ydt 2 q dx 12 q dx 22 q dz 2 q dx 42 .

q 14 dyi j dyi)j ,

z)0 .

Here and below we set the radii of both AdS5 and S 5 equal to unity. The boundary at spatial infinity corresponds to z s 0. The S 5 coordinates y i j are subject to the constraints y i k y k j s d ji ,

yi j s 12 e i jk n y k n ,

yi)j s yy i j ,

i , j,k ,n s 1,2,3,4 where e i jk n s "1 is the Levi-Civita tensor of suŽ4.. The coordinates yi j are related to the standard soŽ6. cartesian coordinates y M , M s 1, . . . ,6, which satisfy the constraint y M y M s 1 through the formula yi j s r iMj y M , where r iMj are the Clebsh-Gordan coefficients of suŽ4. algebra w13x. We use the coordinates y i j instead of y M as this allows us to avoid using various cumbersome gamma matrix identities. To develop light cone formulation we introduce light cone variables x "' Ž x 4 " x 0 .r '2 , x ' Ž x 1 q ix 2 .r '2 , x ' x ) , where x 0 ' t. In the following we treat xq as evolution parameter. Now let us discuss the form of the algebra of isometry transformations of AdS5 = S 5 superspace, that is psuŽ2,2 <4., which we are going to use. The even part of this algebra is the algebra soŽ4,2. which is the isometry algebra of AdS5 and the algebra

w P a , K b x s h abD y J ab , w J a b , J c d x s h b c J a d q 3 terms, a,b,c,d s 0,1,2,4

Ž 1.

where h a b s Žy,q, . . . ,q .. Throughout this paper instead of soŽ6. algebra notation we prefer to use the notation of suŽ4. algebra. Commutation relation of suŽ4. algebra are J i j , J k n s d jk J i n y dni J k j . In the light cone basis we have: J " x , J " x , Jqy, J x x , P ", P x , P x , K ", K x , K x . We simplify notation as follows P ' P x , P s P x , K ' K x , K s K x . The light cone form of soŽ4,2. algebra commutation relations can be obtained from Ž1. with the light cone metric having the following non vanishing elements hqys hyqs 1, h x x s h x x s 1. To describe the odd part of psuŽ2,2 <4. superalgebra we introduce 32 supercharges Q " i , Q i", S " i , Si" which possess D, Jqy and J x x charges. The commutation relations of supercharges with dilatation D D,Q " i s y 12 Q " i ,

w D,S " i x s 12 S " i ,

D,Q i" s y 12 Q i", D,Si" s 12 Si" ,

Ž 2.

tell us that Q are usual supercharges of Poincare´ subsuperalgebra, while S are conformal supercharges. The supercharges with superscript q Žy. have positive Žnegative. Jqy charge Jqy,Q " i s " 12 Q " i ,

w Jqy,S " i x s " 12 S " i ,

Jqy,Q i" s " 12 Q i", Jqy,Si" s " 12 Si" .

R.R. MetsaeÕr Physics Letters B 468 (1999) 65–75

The J x x charges are fixed by the commutation relations J x x ,Q " i s " 12 Q " i , J

xx

,Si"

s"

1 2

Si",

J x x ,Q i" s . 12 Q i", xx

w J ,S

"i

1 2

x s. S

"i

.

Ž 3.

Transformation properties of supercharges with respect to suŽ4. algebra are given by Q i", J j k s d i j Q k"y 14 d kj Q i", Q " i , J j k s yd ki Q " j q 14 d kj Q " i and the same for supercharges S. Anticommutation relations between supercharges are

 Q " i ,Q j" 4 s "P "d ji ,  Qqi ,Qyj 4 s Pd ji ,  S " i ,S j" 4 s "K "d ji ,  Syi ,Sqj 4 s K d ji ,  Qqi ,Sqj 4 s yJqxd ji ,  Qyi ,Syj 4 s yJyxd ji ,  Q " i ,S j. 4 s Ž Jqyq J x x . D . d ji . J i j . 1 2

Remaining commutation relations between supercharges and even part of superalgebra take the following form yi

qx

qi

yx

Q ,J

Q ,J

qi

yi

sQ ,

qi

w S , J x s yS ,

s yQ , yi

qx

qi

yx

yi

w S , J x sS ,

67

3. Light cone gauge formulation of IIB supergravity in AdS5 = S 5 background Our next step is to find a realization of psuŽ2,2 <4. superalgebra on the space of IIB supergravity fields propagating in AdS5 = S 5. To do that we use light cone superspace formalism. First, we introduce light cone superspace which by definition is based on position AdS5 = S 5 coordinates x ", x, x, z, y i j and Grassmann position coordinates u i and x i which transform in fundamental 4 irreps of suŽ4. algebra. Second, on this light cone superspace we introduce scalar superfield F Ž x ", x, x, z, y i j, u i , x i .. In the remainder of this paper we find it convenient to Fourier transform to momentum space for all coordinates except for the radial z and S 5 coordinates y i j. This implies using py, pq, p, p, l i , t i instead of xq, xy, x, x, u i , x i respectively. The l i and t i transform in 4 irreps of suŽ4.. Thus we consider the superfield F Ž p ", p, p, z, y i j, l i ,t i . with the following expansion in powers of Grassmann momenta l i and ti

F s pq2f q pq Ž l i c 1i q t i c 2i . qpq Ž l i l j f 1i j q l it j f 2i j q t it j f 3i j . i

q Ž el3 . c 3 i q l i l jt k c 1i jk q l it jt k c 2i jk i

Si., P " s Q i",

y Sy i , P s Qi ,

q Sq i , P s yQ i ,

Q. i, K " sS" i,

i

q Ž et 3 . c4 i q Ž el4 . f4 q Ž el3 . t j f j i j

Qyi , K s Syi ,

Qqi , K s ySqi .

The above generators are subject to the following hermitian conjugation conditions P " † s P ", K †sK ,

ŽJ

"x †

ŽQ .

1 q

s Q i",

ŽS

qy †

. s yJ " x , Ž J

. s yJqy,

† Ž J x x . s J x x , D† s yD, J i† j s J j i .

All the remaining nontrivial Žanti.commutation relations of psuŽ2,2 <4. superalgebra could be obtained by using these hermitian conjugation rules and Žanti.commutation relations given above.

4

ž yl Ž et i

i

1 q

"i †

. s Si",

pq

ij

k

. c 3)i q Ž el2 . Ž et 3 . c 1i jk )

jk

q Ž el3 . Ž et 2 . c 2i jk ) y Ž el4 . t i c4)i

†

Ž K " . s K ",

P†sP, "i †

ql i l jt ktn f i jk n q l i Ž et 3 . f j) i q Ž et 4 . f4)

pq

2

ž y Ž el . i

ij

/

Ž et 4 . f 1i j)

j

ij

q Ž el3 . Ž et 3 . f 2i j) y Ž el4 . Ž et 2 . f 3i j) 1 y

q2

p

3 i

ž Ž el . Ž et i

4

/

. c 1i )

4 q Ž elD . Ž et 3 . c 2i ) q

/

1 pq2

Ž el4 . Ž et 4 . f ) Ž 4.

R.R. MetsaeÕr Physics Letters B 468 (1999) 65–75

68

where G are the differential operators acting on F . Thus we should find representation of psuŽ2,2 <4. in terms of differential operators acting on light cone scalar superfield F . To simplify expressions let us write down the generators for xqs 0. The kinematical generators are then given by

where 1

Ž el4 . '

4! 1

i

Ž el3 . ' 2

e i jk nl i l j l k l n ,

3!

e i jk nl j l k l n ,

1

ij

Ž el . '

2!

e

P s p,

i jk n

lk ln ,

P s p,

Jqx s Ep pq,

Pqs pq,

Jqx s Ep pq,

and the same notation is adapted for t i . The field f i jk n satisfies the constraint

Kqs 12 z 2 y 2 Ep Ep pq,

f i jk n s 14 e i ji 1 j1e k n k 1 n1f i1 j1 k 1 n1 )

K s K 0 q 12 Ep q u i Sq i ,

In Ž4. the f is a complex scalar field, the c 1i , c 2i are ij two spin one half fields, the fields f 1,2,3 describe i jk i two Kalb-Ramon fields, the c 1,2 , c 3,4 are two gravitinos, while f4 , f i j , f i jk n describe graviton and self dual 4 form field. The reality constraint in terms of the superfield F takes the form

Qqi s pqu i ,

F Ž yp, z , y,y l ,y t . †

†

q

Ž F Ž p, z , y, l ,t . .

qx

P ,P,P,J

qx

,J

1

'2

Sqi s

Ž 8.

Qq i s li ,

zt i y l i Ep ,

1

'2

zpq x i q pq u Ei p ,

Ž 9.

†

In the light cone formalism the psuŽ2,2 <4. superalgebra has the generators q

Sq i s

/

1 1 q Jqys Ey p p y 2 ul y 2 xt q 2,

s pq4 d 4l†d 4t †e Ž l i l i q t it i .r p

H

ž

Ž 7.

q

, K , K , K ,Q ,

Ž 5. Ž 6.

Ž 10 .

where we use the notation K 0 ' 12 z 2 y 2 Ep Ep p

ž

which we refer to as kinematical generators, and yi y Py, Jyx , Jyx , Ky,Qyi ,Qy i ,S ,S i

1 1 1 q D s yEy p p y Ep p y Ep p q z Ez q 2 ul q 2 xt y 2 ,

J i jsli jqM i j ,

qi

qi q qy Qq ,J xx , i ,S ,S i , D, J

J x x s pEp y pEp q 12 ul y 12 xt ,

/

q y Ep yEy p p y Ep p y Ep p q z Ez ,

ž

/

1 2

which we refer to as dynamical generators. The kinematical generators have positive or zero Jqy charges, while dynamical generators have negative Jqy charges. For xqs 0 the kinematical generators are quadratic in the physical field F , while the dynamical generators receive corrections in interaction theory. In this paper we deal with free fields. At a quadratic level both kinematical and dynamical generators have the following representation in terms of the physical light cone superfield

l i j s y i k=k j ,

Gˆ s dpq d 2 pdzdS 5d 4l d 4t pq

l i m l m j s 14 l m n l n m d ji q 2 l i j ,

H

=F Ž yp, z , y,y l ,y t . GF Ž p, z , y, l ,t . ,

M i j ' u il j q x it j y 14 d ji Ž ul q xt . ,

=i j ' r iMj Ž d M N y y M y N . E y N , Ep"' ErE p ., ul ' u il i ,

Ep ' ErE p,

x t ' x it i .

Ep ' ErE p,

Ž 11 .

The orbital part of suŽ4. angular momentum l i j satisfies the following important relations

l i j ,l k n s d jk l i n y dni l k j ,

R.R. MetsaeÕr Physics Letters B 468 (1999) 65–75

frequently used in this paper. Dynamical generators are given by Pys y

Ez2

pp q

q

p

1

q

y

A,

2 z 2 pq

2p

2p

y i y Jyx s yEy p p q Ep P q u Q i y

y yi Jyx s yEy p p q Ep P y Q

q Ž ul q xt y 2 . i

yi

Q s pu q Qy i s

p q

p

1

'2

li y

ž

i

p pq

x Ez q

1 q

'2 p

ž

pq

Ž 12 . ,

li pq ,

Ž 13 .

l i2 j ' l i j l j i ,

ification of IIB supergravity do not satisfy conformally invariant equations of motion. Žii. Generators involve the terms linear and quadratic in Grassmann variables u i , l i and terms up to fourth power in x i and t i . The coordinate u i Žor l i . constitutes odd part of light cone gauge N s 4, 4 d SYM theory. The terms of the third and fourth powers in x i and t i are expressible in terms of the operator A and its commutators with x i and t i . Recall that the above representation was given for xqs 0. To study AdSrCFT correspondence we shall need the generators for arbitrary xq' i Eq p which are given by q y

1

i

x ,A

2z

t i Ez q

1 2z

/

q y

Gxqs eyE p P G xq s0 e E p P ,

,

Gxqs Gxq s0 y Eq Py,Gxq s0 p

/

wt i , A x ,

Ž 14 .

q 12 Eq2 Py, Py,Gxq s0 p

.

Using these relations we derive the complete expression for conformal supergenerators

where A ' X y 14 ,

69

2

X ' l i2 j q 4t l x q Ž xt y 2 . ,

t l x ' ti l i j x i .

q< q q y Sq i s S i x s0 q Ep Q i ,

Ž 15 .

yi Sqi s Sqi < xq s0 y Eq . p Q

For l i , t i and t i , x i we adopt the following anticommutation and hermitian conjugation rules

Below we shall need the complete expression for the following generators

 u i , l j 4 s d ji ,  x i ,t j 4 s d ji ,

Jqx s Ep pqy Eq p p,

u i† s

1 q

p

l i , t i† s pq x i ,

l†i s pqu i , x i† s

1 pq

ti .

y The remaining generators K, Syi , Sy are obi , K tainable from the above generators via commutation relations of psuŽ2,2 <4. superalgebra. Because these expressions are not illuminating we do not present them here. Following w10x we shall call the operator A the AdS mass operator. Few comments are in order. Ži. The operator A is equal to zero only for massless representations which can be realized as irreducible representations of conformal algebra w14,10x which for the case of AdS5 space is the soŽ5,2. algebra. Below we shall demonstrate that operator X Ž15. has eigenvalues equal to squared integers in the whole spectrum of compactification of IIB supergravity on AdS5 . This implies that operator A Ž15. is never equal to zero. From this we conclude that the scalar fields w15x as well as all remaining fields of compact-

Jqx s Ep pqy Eq p p,

y Jqys Jqy < xq s0 y Eq p P ,

Ž 16 .

Kqs 12 z 2 y 2 Ep Ep pq

ž

/

3 q y y Eq p yEp P y Ep p y Ep p q z Ez q 2 ,

ž

y D s D < xq s0 y Eq p P .

/

Ž 17 .

Making use of the expression for Py Ž12. we can immediately write down the light cone gauge action S l .c .s d 4 pdzdS 5d 4l d 4t pqF Ž yp, z , y,y l ,y t .

H

= Ž ypyq Py . F Ž p, z , y, l ,t . . Since the action is invariant with respect to the symmetries generated by psuŽ2,2 <4. superalgebra, the formalism we discuss is sometimes referred to as an off shell light cone formulation w9x. In what follows we shall exploit the following above mentioned important property of the operator

R.R. MetsaeÕr Physics Letters B 468 (1999) 65–75

70

X Ž15. – that its eigenvalues are squared integers. Let us demonstrate this important fact. First, we expand the scalar superfield Ž4. in S 5 coordinates y i j and Grassmann momentum t i `

Fs

i i jk , F l,3 do not depend on t i . The and superfields F l,1 i jk F l,3 is totally antisymmetric in i, j, k. Straightforward calculation gives the following eigenvalues of operator X

4

Ý Fl , Fls ls0

Ý F l,a ,

Ž 18 .

as0

where ‘spherical’ harmonic superfields F l, a satisfy the constraints l i2 jF l , a s l Ž l q 4 . F l , a ,

txF l , a s aF l , a .

Ž 19 .

The first constraint tells us that F l, a is an eigenvalue vector of square of suŽ4. orbital momentum l i j , while the second constraint tells us that F l, a is a monomial of degree a in t i . Note that superfield F 0 consists of fields of N s 8, 5d AdS supergravity, while the superfields F l ) 0 are responsible for Kaluza–Klein modes. Second, we evaluate eigenvalues of X for each superfield F l, a in turn. For the case of F l,0 , F l,4 the eigenvalues of X are easily found to be 2

XF l ,0 s Ž l q 2 . F l ,0 ,

2

XF l ,4 Ž l q 2 . F l ,4 .

Ž 20 .

In deriving the second relation one needs to use the relation like

where is totally antisymmetric in i, j, k. Next, we analyse the spectrum of X in F l,1 and F l,3 . In contrast to Ž20. it turns out that these superfields themselves do not diagonalize the operator X. Now we decompose the superfields F l,1 and F l,3 as follows Ž2. Ž2. F l ,1 s F lŽ1. F l ,3 s F lŽ1. ,1 q F l ,1 , ,3 q F l ,3 ,

where 2

Ž t l . i F li,1 ; lq4 2 F lŽ2. Ž t l . i F li,1 , l ) 0; ,1 ' t i q l 6 F lŽ1. t t Ž t l . k F li,3jk ; ,3 s t i t j t k y lq4 i j 6 F lŽ2. t t Ž t l . k F li,3jk , l ) 0; ,3 s t i t j t k q l i j

ž ž ž ž

F lŽ1. ,1 ' t i y

/

/

/

/

2

Ž2. XF lŽ2. ,3 s Ž l q 3 . F l ,3 .

Ž1. XF lŽ1. ,3 s Ž l q 1 . F l ,3 ,

2

2

Finally, we consider the most challenging case of the superfield F l,2 . It turns out that this superfield is decomposed into three superfields which are eigenvectors of operator X Ž2. Ž3. F l ,2 s F lŽ1. ,2 q F l ,2 q F l ,2 ,

4 Ž l q 3.

ž

F lŽ1. ,2 ' t i t j y

Ž l q 2. Ž L q 4. 4

q

Ž l q 2. Ž L q 4.

ž

F lŽ2. ,2 ' t i t j y

8 l Ž l q 4.

4 l Ž l q 4.

Ž 21 .

F li jk

Ž2. XF lŽ2. ,1 s Ž l q 3 . F l ,1 ,

y

t i Ž t l . j Ž t l . kF li jk s y 121 l Ž l q 4 . t it jt kF li jk ,

Žt l . i ' tj l j i ,

2

Ž1. XF lŽ1. ,1 s Ž l q 1 . F l ,1 ,

ž

F lŽ3. ,2 ' t i t j q

l Ž l q 2.

ti Žt l . j

/

l Ž l q 2.

4

/

Ž t l . i Ž t l . j F li,2j ;

Ž t l . i Ž t l . j F li,2j , l ) 0 ;

4 Ž l q 1.

q

ti Žt l . j

ti Žt l . j

/

Ž t l . i Ž t l . j F li,2j , l ) 0 ;

ij where the superfield F l,2 does not depend on t i and is totally antisymmetric in i, j. Relatively straightforward calculation gives the following eigenvalues of the operator X 2

2 Ž1. XF lŽ1. ,2 s l F l ,2 ,

Ž2. XF lŽ2. ,2 s Ž l q 2 . F l ,2 , 2

Ž3. XF lŽ3. ,2 s Ž l q 4 . F l ,2 .

Thus we have demonstrated that the operator X in whole space of superfield F take values which are squares of integers. This implies that the operator k ' 'X is well defined and possesses integer eigenvalues which are chosen to be positive in what follows.

R.R. MetsaeÕr Physics Letters B 468 (1999) 65–75

4. Light cone form of boundary CFT

Sqi s u i pq Ep q i

The next primary goal of this work is to find a light cone gauge realization of psuŽ2,2 <4. superalgebra at the boundary of AdS5 = S 5 which is M 3,1 = S 5, where M 3,1 is a Ž3 q 1. Minkowski space time Žfor a review of CFT see, for instance w16x.. At this boundary the superalgebra acts as the algebra of conformal transformations. Now we have to realize this superalgebra on the space of conformal operators. To this end we introduce boundary light cone superspace which is based on momentum variables p ", p, p, position S 5 coordinate y i j and the Grassmann momentum variables l i , t i . On this light cone superspace we define a superfield O loc which is collection of CFT operators with canonical dimensions Žcurrents. and superfield O˜loc which is a collection of shadow operators Žsources.. These superfields have a similar expansion in l i and t i as the superfield F Ž4. does. In general the representation of psuŽ2,2 <4. algebra in O loc differs from the one in O˜loc . It turns out however that if we introduce the new basis 1

1

O˜s q ky 2 O˜loc ,

O s qy ky 2 O loc ,

q 2 ' y2 Ž pq pyq pp .

Ž 22 . then the representations of psuŽ2,2 <4. in O and O˜ coincide. We found the following realization of psuŽ2,2 <4. superalgebra in terms of differential operators acting on CFT superfields O , O˜ P "s p ", qx

J

P s p, q

P s p,

q

s Ep p y E p ,

Jqx s Ep pqy Eq p , 3 q Kqs Kq 0 y 2 Ep q

Qqi s pqu i , Qyi s pu i q Qy i s

p q

p

pq 2 q2

Ž 23 . A,

'2

li y

'2 pq qt i ,

2'2 q i

2'2 q

yi x i , A y Eq p Q ,

y w t i , A x q Eq p Qi ,

Ž 26 .

1 1 y y q Jqys yEq p p q Ep p y 2 ul y 2 xt q 2,

J x x s pEp y pEp q 12 ul y 12 xt , 1 1 1 y y q D s yEq p p y Ep p y Ep p y Ep p q 2 ul q 2 xt y 2 ,

J i jsli jqM i j ,

Ž 27 .

where the AdS mass operator A is given in Ž15., the l i j and M i j are given in Ž11., while Kq 0 is given by q q q y Kq 0 s yEp Ep p y Ep yEp p y Ep p y Ep p .

ž

/

Ž 28 .

To be definite, here and below we assume that q 2 s p 02 y p 2 ) 0. Expressions for Jyx , Jyx are obtainable from Ž13. by inserting there the expressions for Py and Qyi , Qy given in Ž23., Ž25.. The i remarkable property of realization we constructed is that the dependence on AdS mass operator A in CFT is ‘dual’ to AdS representations. Namely, on AdS side the A appears in Py Ž12. having y1 D- and Jqy-charges, while on CFT side this operator appears in Kq Ž24. having opposite, i.e., q1 D and Jqy-charges. The same ‘duality’ is the case of AdS Qy generators Ž14. having y1r2 D- and Jqycharges and CFT Sq generators Ž26. having opposite, i.e., q1r2 D- and Jqy-charges. As before, the nonlinear dependence on Grassmann variables x i , t i is expressible through the operator A. As was said already, the above representation is applicable to both O and O˜. The price for this is that the generators are no longer local with respect to the transverse momenta p and p included in q. However, these nonlocal terms cancel when we transform from O and O˜ basis into the one of O loc and O˜loc .

5. AdS r CFT correspondence

q x i, i

pq

Ž 24 .

Qq i s li i

Sq i s yl i Ep q

71

Ž 25 .

After we have derived the light cone formulation for both the bulk superfield F and the boundary conformal theory operators collected in O and O˜ we are ready to demonstrate explicitly the AdSrCFT correspondence. We demonstrate that boundary values of normalizable solutions of bulk equations of

R.R. MetsaeÕr Physics Letters B 468 (1999) 65–75

72

motion are related to conformal operators O loc , while the ones of non-normalizable solutions are related to operators O˜loc Žsee w17,10x.. To this end let us consider solutions to light cone equations of motion which, as usual, take the form PyF s pyF . Taking into account the expression for Py Ž12. and rewriting these equations as

ž

yEz2 q

1 z2

Ž k 2 y 14 .

/

F s q 2F ,

k ' 'X ,

we immediately get the following normalizable and non- normalizable solutions

pq, p, p, y i j, l i and t i we use on AdS and CFT sides match. y Ž yŽ Now let us consider Pads 12. and Pcft 23.. Taking into account that solutions to equation of motion y Ž29., by definition, satisfy the relation Pads F s py F k y y y y we get PadsF s Zk Ž qz .i Pcft O . So Pads and Pcft also match. Taking this into account it is straightforward qy to see that the generators Jads , Dads Ž16., Ž17. and qy Jcft , Dcft Ž27. satisfy the relation Ž30.. Next we q Ž . consider kinematical generators Kq ads 17 and K cft Ž24.. Here we use the following relation Kq ads Zk Ž qz .

k

Fnorm s 'qz Jk Ž qz . i O , Fnon - norm s 'qz Yk Ž qz . i k O˜ ,

Ž 29 .

where O , O˜ are scalar superfields do not depending on z. The Jk and Yk are Bessel and Neumann functions respectively. The normalization factor i k is included for convenience. In Ž29. we use the notation O , O˜ since we are going to demonstrate that these are indeed the CFT superfields discussed above. Namely, we are going to prove that AdS transformations for F lead to conformal theory transformations for O Žand O˜. GadsF s Zk Ž qz . i k Gcft O ,

ž

s Zk Ž qz . Kq 0cftq

Zk Ž z . ' 'z Jk Ž z . .

Ž 30 . Here and below we use the notation Gads and Gcft to indicate the realization of psuŽ2,2 <4. algebra generators on the bulk field Ž7. – Ž10. and conformal operator Ž23. – Ž27. respectively. Now let us make a comparison of generators for bulk field F and boundary operator O . Important technical simplification is that it is sufficient to make comparison only for the part of the algebra spanned by generators P ", P , P , Jqy, J " x , J " x , Kq,Q " i ,Q i", J x x , D, J i j . It is straightforward to see that if these generators match then the remaining generators Ky and K, K, S " i , Si , " shall match due to commutation relations of the psuŽ2,2 <4. superalgebra. We start with a comparison of the kinematical generators Ž5.. As for the generators Pq, P, P, Jqx , Jqx , J x x , Qqi , Qq i they already coincide on both sides Žsee Ž7. – Ž10. and Ž23. – Ž27... In fact this implies that the coordinates

pq q q

pq 2 q2

3 A y Eq p Ž 2 q z Ez .

/

Ž Eq Zk Ž qz . . z Ez ,

q Ž . Ž . where Kq ads and K 0cft are given in 17 and 28 Ž . respectively. Using then 29 , the above relation and the fact that O in Ž29. does not depend z we get q immediately that Kq ads and K cft satisfy the relation Ž30.. y The last step is to match the generators Qyi ads , Q iads yi y and Qcft , Q icft. This is the most challenging part of the analysis. Let us consider Qyi . Generalization of our discussion to the case of Qy i is straightforward. As is seen from Ž14. and Ž25. requiring that these supercharges satisfy the basic relation Ž30. gives the following nontrivial relation

ž

x Ei z q

1 2z

x i, X

z Jk Ž z . i k s 'z Jk Ž z . i kx i ,

/'

Ž 31 . which is understood in weak sense, i.e., as an operator relation defined on the space of superfield O . This interesting relation is proved in Appendix. The same relation is valid in the case of the Neumann function. Taking into account matching Qyi , Qy i and Py we conclude that Jyx , Jyx Ž13. also match. Above analysis is obviously generalized to the case of non-normalizable solutions and shadow operators Ž29.. Thus we have demonstrated that boundary operators in O , O˜ in Ž29. are indeed the conformal operators. Because of multiplicative factors Jk and

R.R. MetsaeÕr Physics Letters B 468 (1999) 65–75

Yk involving powers of q the operators O and O˜, however, are not boundary values of solutions of the bulk equations of motion Ž29.. On other hand, by relations Ž22. it is easily seen that it is O loc Ž O˜loc . that is the boundary value of Fnorm ŽFnon - norm ., i.e. for small z one has the local interrelations 1

73

AdS5 background. In this perspective, we think that the results of this paper can also get interesting applications to massless higher spin field theory. Note that AdS5 = S 5 is a unique space where the consistent string and massless higher spin field theories may ‘‘meet’’.

1

Fnorm ; z kq 2 O loc , Fnorm - norm ; zy kq 2 O˜loc . Ž 32 . Note that for k s 0 the factor zyk in the second relation of Ž32. should be replaced by log z.

6. Conclusions We have developed the light cone gauge formulation of IIB supergravity in AdS5 = S 5 background and applied this formulation to the study of AdSrCFT correspondence Žfor review see w18x.. Because the formulation is given entirely in terms of the light cone scalar superfield it allows us to treat all fields of IIB supergravity on equal footing and in a manifestly supersymmetric way. Comparison of this formalism with other approaches available in the literature leads us to the conclusion that this is a very efficient formalism indeed. The results presented here should have a number of interesting generalizations, some of which are: Ži. extension of light cone formulation of IIB supergravity in AdS5 = S 5 background to the level of cubic interaction vertices Žsee w19x.. Žii. extension of light cone formulation of conformal field theory to the level of OPE’s and a study of light-cone form of AdSrCFT correspondence at the level of correlation functions. Žiii. generalization of light cone gauge formalism to the study of compactifications of 11-dimensional supergravity on AdS4 = S 7 and AdS7 = S 4 w20x. In these cases a formulation in terms of light-cone scalar superfields could also be developed. In view of previous experiences with massless higher spin fields in AdS4 space it is known that to construct self-consistent interactions for such fields it is necessary to introduce an infinite tower of massless fields w21x. For the case of AdS5 space it is expected that fields of IIB supergravity constitute lower spin multiplet of the infinite tower of N s 8 supersymmetric massless higher spin fields theory in

Acknowledgements I would like to thank A. Tseytlin for reading the manuscript and making comments improving it. This work was supported in part by INTAS grant No. 96-538 and the Russian Foundation for Basic Research Grant No. 99-02-16207.

Appendix A In order to prove Ž31. we start with

x ti ' e tt l xx i eyt t l x , t i t ' e tt l xt i eyt t l x , l i jt ' e tt l x l i j eyt t l x . Making use of the relations i

E t x ti s y Ž l t x t . ,

E tt i t s Ž t t l t . i , i

E t l i jt s Ž t t l t . j x ti y t jt Ž l t x t . , where Ž l x . i s l i j x j Žsee also Ž21., we get a closed differential equation for x ti which can be written in the following two equivalent forms i

E t2x ti s Ž 14 l n2 m q t l x . x ti q Ž 2 y xt . Ž E t x t . , i

E t2x ti s x ti Ž 14 l n2 m q t l x . q Ž E t x t . Ž 2 y xt . ,

Ž A.1 . i where the initial conditions are x ts0 s x i , E t x ti < ts0 i s yŽ l x . . Solutions to these equations are t 2 x ti s e

Ž2y xt .

ž

k cosh

2

tx i

k sinh q

k

2

t

Ž Ž xt y 2. x i y 2 Ž l x . i .

/

,

R.R. MetsaeÕr Physics Letters B 468 (1999) 65–75

74

and the formulas ŽA.2., ŽA.3. we get

k x ti s x i cosh t 2

ž

x Ei z Jk Ž z . i k 1

k sinh q Ž x i Ž xt y 2 . y 2 Ž l x . t

=e 2

i

.

2

t

k

s

/

sin w sin kw

k 1

.

Equating these two different forms of the solution we get the following basic formula

z

t

1

q

k

i

q Ž 2 y xt . x

i

.

/

q

.

Inserting t s i w and taking real and imaginary parts gives

x i cos kw s cos w cos kwx i sin w sin kw q

k

cos w sin kw

k

Ž Ž xt y 2. x i y 2 Ž l x . i . ,

sin kw

k

Ž 2 Ž l x . i q Ž 2 y xt . x i .

q sin w cos kwx i .

Ž A.3 .

Now we use the integral representation for Bessel function Jk Ž z . s

iyk

p

p

H0 d w e

i zcos w

cos kw ,

which is valid for integer k . Taking into account the relation

ž

x Ei z q s 'z

1 2z

ž

x i, X

x Ei z q

1 z

/

,

/

,

isin w Ž sin w cos kwx i

Ž Ž 2 y xt . x i q 2 Ž l x . i .

To derive the last formula we integrate by parts and since the operator k takes integer values the boundary terms cancel. Summing these contributions we get the desired relation Ž31..

References

Ž A.2 .

Ž 2 Ž l x . i q x i Ž 2 y xt . .

Ž Ž xt y 2. x i y 2 Ž l x . i .

i zcos w

cos w sin kw

k

k

Ž2Ž l x .

s

p

H dw e p 0

i

s e Ž cosh k t x sinh k t

sinh k t

icos w Ž cos w cos kwx i

Ž 2 Ž l x . i q x i Ž 2 y xt . . Jk Ž z . i k s

i

i zcos w

q

Ž 2y xt .

x i cosh k t q Ž 2 Ž l x . q x i Ž 2 y xt . .

p

H dw e p 0

/'

z Jk Ž z .

Ž 2 Ž l x . i q x i Ž 2 y xt . .

/

Jk Ž z .

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