CFT correspondence

10 September 1998

Physics Letters B 435 Ž1998. 291–298

Interacting spinors-scalars and AdSrCFT correspondence Amir M. Ghezelbash b

a,b,1

, Kamran Kaviani a,b,2 , Shahrokh Parvizi Amir H. Fatollahi b,c,4

b,3

,

a Department of Physics, Az-zahra UniÕersity, P.O. Box 19834, Tehran, Iran Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5531, Tehran, Iran c Department of Physics, Sharif UniÕersity of Technology, P.O. Box 11365-9161, Tehran, Iran

Received 25 May 1998; revised 20 June 1998 Editor: L. Alvarez-Gaume´

Abstract By taking the interacting spinor-scalar theory on the AdS dq1 space we calculate the boundary CFT correlation functions using AdSrCFT correspondence. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction The correspondence between field theories in Ž d q 1. y dimensional Anti-de Sitter space and d y dimensional conformal field theories has been studied in various aspects in the last few months. This correspondence has been conjectured in the large N limit of superconformal gauge theories and the supergravity on AdS dq 1 spaces w1x and also has been studied in w2x in connection with the non-extremal black-hole physics. This suggested correspondence has been made more precise in w3–5x. The partition function of any field theory on AdS dq 1 defined by, ZAdS w f 0 x s

yS w f x

Hf Df e

,

Ž 1.1 .

o

where f 0 is the finite field defined on the boundary of AdS dq1 and the integration is over the field configurations f that go to the f 0 when one goes from the bulk of AdS dq1 to its boundary. According to the

1

E-mail: E-mail: 3 E-mail: 4 E-mail: 2

[email protected]. [email protected]. [email protected]. [email protected].

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 8 1 5 - 6

A.M. Ghezelbash et al.r Physics Letters B 435 (1998) 291–298

292

above mentioned works, ZAdS is identified with the generating functional of the boundary conformal field theory given by,

HM d d x'g Of 0

:, ZCFT w f 0 x s ²e d Ž 1.2 . for a quasi-primary conformal operator O on the M d , boundary of AdS dq1. This correspondence were given in w4x for a free massive scalar field and a free UŽ1. gauge theory. Some other examples as interacting massive scalar, free massive spinor and massive interacting vector-spinor cases are studied in w6,7x and w8x respectively. Also for classical gravity and type-IIB string theory one can refer to w9–11x. In all these cases, the exact partition function Ž1.1. is given by the exponential of the action evaluated for a classical field configuration which solves the classical equations of motion. The result of calculations shows that the evaluated partition function is equal to the generating functional Ž1.2. of some conformal field theory with a quasi-primary operator with a certain conformal weight. The purpose of this letter is to investigate the above correspondence in the case of interacting spinors-scalars theory. The natural motivation for this study comes from the importance of non-linear SUSY theories living on the world-volumes of D-branes, namely supersymmetric DBI actions. The tension of D-branes is involved by eyf Ž f dilaton field. and by noting different powers of fermions in the action one can find vertices with different numbers of spinors and scalars. Even in the low energy limit which DBI action is approximated by SUSY gauge theories, the gauge coupling constant is involved by e f and so the terms with quadratic in spinors and different number of scalars will appear. In particular, we calculate the boundary CFT correlation functions using AdSrCFT correspondence to be in agreement with CFT expectations. In the case of correlation functions with quadratic in spinors we found that they are generated only from a surface term introduced in w7x to prevent the vanishing on-shell action. This observation is done also in w8x for the vector-spinor theory. 2. On-shell action Let us start with the following action for the interacting spinors-scalars theory which may be taken as a generalization of Fermi’s four fermion theory on the AdS dq 1: S s Sb q S f q Sint , Ž 2.1 . where Sb s

HAdSd

dq 1

x 'G

Sf s Sf 0 q Sf 1 s

1 2

Ž Ž =f . 2 q Mb2f 2 . ,

HAdSd

dq 1

x 'G c Ž Du y M f . c q k lim

ž

/

`

Sint s

HAdSd

dq1

x 'G

ž

Ž 2.2 .

Ý

n

H

e™0 M de

d d x Ge cc ,

(

Ž 2.3 .

/

l2 n , m Ž cc . f m ,

ns0, ms0

Ž 2.4 .

with l2,0 s 0, k is a constant and M de is a closed d y dimensional submanifold of AdS dq1 which approaches the boundary manifold M d of AdS dq1 as e ™ 0. The equations of motion in the bulk of the AdS dq1 are given by, `

Ž = 2 y Mb2 . f s

Ý

n

m l2 n , m Ž cc . f my1 ,

ns0, ms0 `

Ž Du y M f . c s y Ý ns0, ms0

n l 2 n , m Ž cc .

ny 1

`

cf m ,

c Ž yDu y M f . s y

Ý

n l2 n , m Ž cc .

ny1

cf m .

ns0, ms0

Ž 2.5 .

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293

The necessity of adding the S f 1 to S f in the interacting theory on AdS dq1 is as follows w7x: it can be shown easily that the quadratic spinor terms in action vanish with on-shell fields. So in calculating the partition function the correlation functions which are quadratic in spinors can not be generated. As we show in the following, only surface term in the action is responsible to generate the correlators with quadratic in spinors. In fact the surface term is sufficient to produce correlation functions quadratic in spinors and at tree level no modification of the action is required. We use the AdS dq 1 space which is represented in x s Ž x 0 , x . coordinates by the domain x 0 ) 0 and its metric is given by, ds 2 s

1 x 02

Ž Ž dx 0 . 2 q dx P dx . .

Ž 2.6 .

The boundary M d is the hypersurface x 0 s 0 plus a single point at x 0 s ` and the metric on the M d is y1 conformally flat dx P dx. Also we have 'G s xyd and Ge s eyd. By introducing the Green’s functions as 5 0

(

Ž = 2 y Mb2 . G Ž x , xX . s X

Ž Du y M f . S Ž x , x . s

d Ž x y xX .

'G

d Ž x y xX .

X

S Ž x , x . Ž yDu y M f . s

'G

,

Ž 2.7 .

,

d Ž x y xX .

'G

Ž 2.8 . ,

Ž 2.9 .

one can represent the solutions of equations of motion as w6–8x

f Ž x . s eyd

HM d

d X

x

d

EG E x0

< x 0s e fe Ž xX . `

HAdSd

q

c Ž x . s yeyd

dq1 X

x 'G G Ž x , xX .

Ý

n

m l 2 n , m Ž c Ž xX . c Ž xX . . f my1 Ž xX . ,

Ž 2.10 .

n , ms0

HM d

d X

x S Ž x , xX . cy Ž xX .

d

`

HAdSd

y

c Ž x . s yeyd

dq1 X

x 'G S Ž x , xX . c Ž xX .

Ý

n l2 n , m Ž c Ž xX . c Ž xX . .

ny 1

f m Ž xX . ,

Ž 2.11 .

n l2 n , m Ž c Ž xX . c Ž xX . .

ny 1

f m Ž xX . ,

Ž 2.12 .

n , ms0

HM d

d X

x cq Ž xX . S Ž xX , x .

d

`

HAdSd

y

dq1 X

x 'G c Ž xX . S Ž xX , x .

Ý n , ms0

where cq Ž x . and cy Ž x . are the boundary value of fields c Ž x . and c Ž x . with G 0 cy Ž x . s ycy Ž x . and cq Ž x . G 0 s cq Ž x ..

5

The explicit form of S , S and G can be found in w7,8,6x.

A.M. Ghezelbash et al.r Physics Letters B 435 (1998) 291–298

294

By defining

f Ž0. Ž x . ' eyd

HM d

d X

x

d

c Ž0. Ž x . ' yeyd

EG

< x 0s e fe Ž xX . ,

Ž 2.13 .

x S Ž x , xX . cy Ž xX . ,

E x0

HM d

d X

Ž 2.14 .

HM d

d X

Ž 2.15 .

d

c Ž0. Ž x . ' yeyd

x cq Ž xX . S Ž xX , x . ,

d

one finds that f Ž0., c Ž0. and c Ž0. satisfy the free equations of motion, so external legs in Feynman diagrams should be replaced by them. In a Born approximation one finds for the solutions of equations of motion as

f Ž x . s f Ž0. Ž x . q f Ž1. Ž x . q O Ž l2 . ,

Ž 2.16 .

c Ž x . s c Ž0. Ž x . q c Ž1. Ž x . q O Ž l2 .

Ž 2.17 .

c Ž x . s c Ž0. Ž x . q c Ž1. Ž x . q O Ž l2 .

Ž 2.18 .

with `

f Ž1. Ž x . '

HAdS

d dq1 xX'G G Ž x , xX .

m l2 n , m Ž c Ž0. Ž xX . c Ž0. Ž xX . .

Ý

n

Ž f Ž0. Ž xX . .

my1

,

Ž 2.19 .

n , ms0 `

c Ž1. Ž x . ' y

HAdSd

dq1 X

x 'G S Ž x , xX . c Ž0. Ž xX .

Ý

n l2 n , m Ž c Ž0. Ž xX . c Ž0. Ž xX . .

ny1

Ž f Ž0. Ž xX . .

m

,

n , ms0

Ž 2.20 . `

c Ž1. Ž x . ' y

HAdSd

dq1 X

x 'G c Ž0. Ž xX . S Ž xX , x .

Ý

n l2 n , m Ž c Ž0. Ž xX . c Ž0. Ž xX . .

ny1

Ž f Ž0. Ž xX . .

m

.

n , ms0

Ž 2.21 . By inserting the solutions Ž2.16., Ž2.17. and Ž2.18. in the action and using the definitions Ž2.13., Ž2.14. and Ž2.15. and equations of motion one finds for the action 6 SON SHELL s k lim

H

e™0 M de

d d x Ge cqŽ0.cyŽ0.

(

`

HAdSd

q

dq1

x'G

ž

Ý

n

Ž 1 y n q 2 n k . l2 n , m Ž c Ž 0 . c Ž 0 . . Ž f Ž 0 . .

ns1, ms0

m

/

q O Ž l2 . ,

Ž 2.22 .

which shows that the on-shell action finds quadratic terms in spinors Ž n s 1. only via the surface term in Ž2.1.. It also shows that the other non-quadratic terms are multiplied by a constant involving k .

6

From now we ignore the pure scalar terms which have been studied in detail in w6x.

A.M. Ghezelbash et al.r Physics Letters B 435 (1998) 291–298

295

3. Correlation functions

3.1. (2n,0)-point functions (pure spinor) The boundary term Ž2.22. gives the correlation function of a pseudo-conformal operator O a and its conjugate O on the M d with the following two-point function w7x: b

² O b Ž x . O a Ž xX . : s V a b Ž x , xX . ,

Ž 3.1 .

where V Ž x, xX . s 2p d r2G Ž M f q 1r2.rG Ž M f q d q2 1 . N x y xX Nydy1y2 M f Ž x y xX . P G. Now, the interaction term in Ž2.22. gives a contribution to the partition function as



`

Ý Ž 1 y n q 2 n k . lŽ 2 n ,0 . Hd dq 1 x x 0Ž ny1.Ž dq1.q2 n M

ZAdS cq , cy s exp y

f

ns2

2n

n

1

= Ł d d yi

dq1

is1

Ž x 02 q < x y yi < 2 .

2

Ł cq Ž y 2 jy1 . Ž y2 jy1 y y2 j . P Gcy Ž y 2 j .

qM f js1

0

,

Ž 3.2 .

m2 j .

Ž 3.3 .

from which one can find the Ž2 n,0.-point function, GŽ2 n ,0. Ž y 1 , PPP , y 2 n . m1 , PPP , m 2 n s Ž 1 y n q 2 n k . lŽ2 n ,0. d dq 1 x x 0Ž ny1.Ž dq1.q2 n M f

H

2n

n

1

=Ł

dq1

is1

2

Ž x 02 q < x y yi < 2 .

Ł Ž Ž y2 jy1 y y2 j . P G . m

qM f js1

2 jy 1

By using the Feynman parameter technics one can find w6x d N

Hd

dq1

x

2

p G

dq1.q Ý D i xyŽ 0 is 1

N

Ł Ž x 02 q N x y yi N 2 .

s Di

žÝ

Di

d y

Di

/ žÝ / G

2 2 2 Ł G Ž Di .

2

i

is1

Ł a iD y1 i

`

=

H0

d a 1 P PP d a n d Ž Ý a i y 1 .

Ý Di yi2j

ž Ýa a / i

j

.

Ž 3.4 .

2

i-j

Putting Di s DX ' d q2 1 q M f for all i one may change the Ž2 n,0.-point function Ž3.3. to a similar form which is suitable to investigate the conformal properties.

A.M. Ghezelbash et al.r Physics Letters B 435 (1998) 291–298

296

For n s 2, i.e. the Ž4,0.-point function we have d

G 2 DX y

ž

GŽ4 ,0. Ž y 1 , y 2 , y 3 , y4 . a , b ,g , ds Ž y1 q 4k . lŽ4 ,0.p 2

Ž a1 a 2 a 3 a4 .

=

DXy1

d

/

2

G Ž 2 DX .

4

4

ž

H is1 Ł da d Ý a y1

X

i

2G 4Ž D .

Ž y12 P G . ab Ž y34 P G . gd

Ž a 1 a 2 y122 q a 1 a 3 y132 q a 1 a 4 y142 q a 2 a 3 y 232 q a 2 a 4 y 242 q a 3 a 4 y 342 .

2D

X

,

i

is1

/ Ž 3.5 .

where it can easily be shown that GŽ4,0. has all appropriate conformal symmetries such as translation, scaling and inversion. To test inversion, we can write Ž3.5. as follows w6x:

G 2 DX y GŽ4 ,0. Ž y 1 , y 2 , y 3 , y4 . a , b ,g , d s Ž y1 q 4k . lŽ4 ,0.

ž

d 2

d

/

Ž 2p .

2

X

2

G Ž2 D .

ž hj Ł y / ij

3

DX

Ž y12 P G . a b Ž y34 P G . gd

i-j

= dzF DX , DX ;2 DX ;1 y

H

where h s

y12 y 34 y14 y 23

and j s

y12 y 34 y13 y 24

ž

Žhqj . Ž hj .

2

4

/

sinh2 z ,

y

hj

Ž 3.6 .

. Expanding the above relation in terms of yi j we have

G 2 DX y GŽ4 ,0. Ž y 1 , y 2 , y 3 , y4 . a , b ,g , d s Ž y1 q 4k . lŽ4 ,0.

ž

d 2

d

/

Ž 2p .

2

X

G Ž2 D .

2

ž hj Ł y / ij

3

Ž yˆ 12 P G . a b Ž yˆ 34 P G . gd Df

i-j

= dzF DX , DX ;2 DX ;1 y

H

ž

Žhqj . Ž hj .

2

4 y

hj

/

sinh2 z ,

Ž 3.7 .

where 2 D f ' DX y 1 s d q 2 M f and yˆi j is the unit vector along yi j , which shows the invariance of Ž4,0.-point function under inversion.

3.2. (2n,m)-point functions (spinor-scalar) Here, we calculate the spinor-scalar Ž2 n,m.-point functions of boundary CFT. By using the relations Ž2.4., Ž2.11., Ž2.12. and the following relation for the bosonic field configuration in the bulk of AdS dq 1 w6x: d d xXf 0 Ž xX .

f Ž x. sc

HM

d

ž

x0 x 02 q N x y xX N 2

Db

/

,

Ž 3.8 .

A.M. Ghezelbash et al.r Physics Letters B 435 (1998) 291–298

297

2

(

where f 0 is the bosonic field f Ž x . on the boundary and Db s dr2 q Mb2 q d4 and c s

G Ž Db.

, one can

p d r2G Ž D b y d r2 .

calculate the partition function to be `

ž

ZAdS cy , cq , f s exp y

dq1.qnq2 n D fqm D b Ž 1 y n q 2 n k . lŽ 2 n , m . Hd dq 1 x xyŽ 0

Ý ns1, ms1

m

=

H ks1 Łd

d

zk

2n

1

f0 Ž zk .

2 Db

Ž x 02 q < x y z k < .

H is1 Łd

n

d

yi

1

Ž x 02 q < x y yi < 2 .

D fq 12

/

= Ł cq Ž y 2 jy1 . Ž y2 jy1 y y2 j . P Gcy Ž y 2 j . , js1

Ž 3.9 .

from which one can find the Ž2 n,m.-point functions dq1.qnq2 n D fqm D b GŽ2 n , m. Ž y 1 , PPP , y 2 n ; z 1 , PPP , z m . m1 , PPP , m 2 ns Ž 1 y n q 2 n k . lŽ2 n , m. d dq 1 x xyŽ 0

H

m

=Ł

ks1

2n

1

Ł

D < 2 b is1

Ž x 02 q < x y z k .

n

1

Ž x 02 q < x y yi < 2 .

D fq

1 2

Ł Ž Ž y2 jy1 y y2 j . P G . m js1

2 jy 1 , m 2 j

,

Ž 3.10 .

By using the relation Ž3.4. and taking

D1 s . . . s Dn s D f q 12 ,

Dnq1 s . . . s DNsnqm s Db ,

Ž 3.11 .

one can obtain the explicit form of the Ž2 n,m.-point functions. For n s 1 and m s 1 one finds d

p2 G G 2,1 Ž y 1 , y 2 ; z . g , d s 2 kl 2,1

ž

1 q 2 D f q Db y d

/ ž G

2 2G

2

1 q 2 D f q Db 2

1 2

Ž D f q . G Ž Db . 1 2

/H

`

0

d a1 d a 2 d a 3 d Ž Ý ai y 1.

1 2

a 1D fy a 2D fy a 3D by1

=

Ž a 1 a 2 Ž y1 y y 2 . 2 q a 1 a 3 Ž y1 y z . 2 q a 2 a 3 Ž y 2 y z . 2 .

Ž1q2 D fq D b .r2

Ž y12 P G . gd . Ž 3.12 .

The integrals over a ’s can be done d

p2 G G 2,1 Ž y 1 , y 2 ; z . g , d s 2 kl 2,1

ž

=

Ž y1 y y 2 .

2 2G

B D f q 12 ,

ž

1 q 2 D f q Db y d

Db 2

/ž

2 D fy D b

B

2

1 q 2 D f q Db 2

/

1 2

Ž D f q . G Ž Db .

Db 2

/ ž G

, Df q Db

1 y Db 2

Ž y1 y z . Ž y 2 y z .

/ Žˆ

Db

y 12 P G . gd ,

where B is the beta function and yˆ 12 is the unit vector in the direction Ž y1 y y2 .. The calculation of Ž2,2. correlation function can be done as above.

Ž 3.13 .

298

A.M. Ghezelbash et al.r Physics Letters B 435 (1998) 291–298

4. Conclusion In this paper we apply the conjectured AdSrCFT correspondence to spinor-scalar interacting field theory. In this way we calculated the pure spinor and mixed spinor-scalar boundary CFT correlation functions. In all cases were done explicitly we found agreement with CFT expectations. Our main result is related to the correlators with two spinor and any number of scalars which were introduced in the text as Ž2,m.-point functions. It is observed that Ž2,m.-point functions come only from a surface term introduced in w7x Žwhich is added to prevent the vanishing of the on-shell free action., and no more modification is required. So we found that no modification is required to get Ž2,m. CFT correlators up to the first order of perturbation. Also the effect of the surface term on the other correlators Ž2 n,m. Ž n G 2. found to be a multiplicative constant.

Acknowledgements We are grateful to K. Sfetsos for his helpful comment about the terms coming from the surface term and a missed factor two.

References w1x w2x w3x w4x w5x w6x w7x w8x

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