- Email: [email protected]
Antisymmetric tensor field on AdS5
26 November 1998
Physics Letters B 441 Ž1998. 173–177
Antisymmetric tensor field on AdS5 G.E. Arutyunov 1, S.A. Frolov
2
StekloÕ Mathematical Institute, Gubkin str.8, GSP-1, 117966 Moscow, Russia Received 30 July 1998; revised 28 August 1998 Editor: P.V. Landshoff
Abstract By using the Hamiltonian version of the AdSrCFT correspondence, we compute the two-point Green function of a local operator in D s 4 N s 4 super Yang-Mills theory, which corresponds to a massive antisymmetric tensor field of the second rank on the AdS5 background. We discuss the conformal transformations induced on the boundary by isometries of AdS5. q 1998 Elsevier Science B.V. All rights reserved.
The recent Maldacena’s conjecture w1x relates the large N limit of certain conformal theories in d-dimensions with classical supergravity on the product of anti de Sitter space AdS dq 1 with a compact manifold. According to w2,3x the precise relation consists in existing the correspondence between supergravity fields and the set of local CFT operators. Then the generating functional of the connected Green functions of the CFT operators is identified with the on-shell value of the supergravity action. With this identification at hand, the AdSrCFT correspondence was recently tested by explicit computation of some two- and three-point correlation functions of local operators in D s 4 N s 4 super Yang-Mills theory, which correspond to scalar, vec-
1
E-mail: [email protected] Address after September 1, 1998: The University of Alabama, Department of Physics and Astronomy, Box 870324, Tuscaloosa, Alabama 35487-0324. 2
tor, symmetric tensor and spinor fields on the AdS5 background w4–12x. D s 4 N s 4 super Yang-Mills is related to the S 5 compactification of D s 10 IIB supergravity. Except the fields mentioned above, the spectrum of the compactified theory also contains the massive antisymmetric tensor fields of the second rank w13,14x. These fields obey first-order differential equations and their bulk action vanishes on shell. Thus, the bulk action is not enough to compute the CFT Green functions and one has to add some boundary terms. This is quite similar to the case of fermions on the AdS background w5x. The origin of boundary terms in the AdSrCFT correspondence was recently clarified in w15x, where it was shown that they appear in passing from the Hamiltonian description of the bulk action to the Lagrangian one. The idea was to treat the coordinate in the bulk direction as the time and to present the bulk action in the form HŽ pq˙ y H Ž p,q . q total deriÕatiÕe.. Here a choice of coordinates and momenta is dictated by the
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 1 1 3 6 - 8
G.E. ArutyunoÕ, S.A. FroloÕr Physics Letters B 441 (1998) 173–177
174
transformation properties of gravity fields under isometries of AdS. In the Hamiltonian formulation the total derivative term should be omitted while from the Lagrangian point of view it can be compensated by adding to the bulk action a proper boundary term. In this note we demonstrate how this general approach works in the case of antisymmetric tensor fields of the second rank and compute the two-point function of the corresponding local CFT operators. We start with the following action for a massive complex antisymmetric tensor field of the second rank 3 S s y d5 x
H
ž
i 2
) ) ´ mnrls amn Er als q 'y g mamn a mn .
/
Ž 1. 0 g s yxy1 0 2 1 Ž
Here metric: ds s
x 02
is the determinant of the AdS dx 02 q hi j dx i dx j .. Because of in-
frared divergencies one should regularize the action by cutting AdS5 space off at x 0 s ´ and leaving the part x 0 G ´ . We use the convention ´ 01234 s y´ 01234 s 1. Action Ž1. vanishes on shell and, therefore, can not produce the two-point functions in the boundary CFT. According to the general scheme discussed above, we need to rewrite action Ž1. in the form suitable for passing to the Hamiltonian formulation. To this end one has to establish a proper set of variables that can be treated as coordinates and their conjugate momenta. It can be done by studying solutions of equations of motion coming from Ž1. i 2
´ mnrlsEr als q 'y g ma mn s 0.
Ž 2.
Acting on Ž2. with an operator y 2i ´mnrls= r q y g mgml gns we arrive at the second-order equation
'
= r Ž =r amn y =m arn q =n arm . y m2 amn s 0.
The last equation implies the constraint = mamn s 0 and, therefore, can be written in the form
= r =r amn q Ž 6 y m2 . amn s 0.
Ž 4.
Specifying Ž4. for a i j , we obtain x 02 E 02 a i j q x 0 E 0 a i j q x 02 I a i j y m2 a i j y 2 x 0 Ž E i a 0 j y E j a0 i . s 0,
Ž 5.
ij
where I s h E i E j . The derivatives E i a 0 j can be expressed from Ž2.: i
xy1 m ´ i jk l a k l , Ž 6. 2 0 where in the last formula and below the indices are raised with respect to the Minkowski metric. Therefore, Eq. Ž5. reduces to
E i a0 j y E j a0 i s E 0 a i j q
x 02 E 02 a i j y x 0 E 0 a i j q x 02 I a i j y m2 a i j y im ´ i jk l a k l s 0,
Ž 7.
To solve Ž7. we introduce the projections Žanti.self-dual parts of a i j : a" ij s
1 2
ž
ai j "
i
a" ij
on the
i " kl ´ i jk l a k l , a " . i j s " ´ i jk l a 2 2
/
Ž 8. Then Eq. Ž7. splits into equations for x 02 E 02 a i"j y x 0 E 0 a i"j y m
Ž m " 2.
aq ij
and
ay ij :
a i"j q x 02 I a " i j s 0.
Ž 9. Momentum space solutions of Ž9. for k 2 ) 0 obey"Ž . Ž . ing the boundary conditions a " i j ´ , k s a i j k and 4 vanishing at x 0 s ` read as a i"j Ž x 0 , k . s
x 0 K m " 1Ž x 0 k .
´ K m " 1Ž ´ k .
a" ij Ž k. ,
Ž 10 .
where K m " 1 is the Mackdonald function and k s < k <. However, we can not assign arbitrary boundary yŽ . Ž . values for both aq i j k and a i j k since these com-
Ž 3. 4
3 It follows from w13,14x that antisymmetric tensor fields arising in S 5 compactification of IIB supergravity are classified by complex representations of SO Ž6..
As was noted in w16x for k 2 - 0 there are two independent solutions regular in the interior. However, both of them are nonvanishing at infinity. A proper account of these solutions may be achieved by introducing an additional boundary at x 0 s1r ´ and requiring the vanishing of the solution on this boundary. Then the solution is unique and in the limit ´ ™ 0 delivers the same contribution to the two-point function as the solution for k 2 ) 0 does. In the sequal, we restrict ourselves to the case k 2 ) 0.
G.E. ArutyunoÕ, S.A. FroloÕr Physics Letters B 441 (1998) 173–177
ponents are related to each other. To find this relation, note that the components a 0 i can be directly found from Ž2.: a0 i s y
i 2m
x 0 ´ i jk l E j a k l .
Ž 11 .
Then substituting into Ž6. one obtains the constraint x0 Ž E E k Ž aqi k y ayi k . y E i E k Ž aqjk y ayjk . . m j m q s E 0 ai j q Ž 12 . Ž a y ayi j . , x0 i j which after projecting on its Žanti.self-dual part results into the following equations x0 k . " I a i"j q I a i.j q 2 Ž E i E k a . jk y E j E a i k . 2m m s E 0 a i"j " a i"j . Ž 13 . x0
ž
/
With the solution for a i j at hand one can compute the derivative E 0 a i j . By using the following properties of the Macdonald function Knq1 Ž z . y Kny1 Ž z . s
2n z
q
E 0 ay ij
Ž x0 , k . s
ž
ž
q x0
2m
/
2 m ´ K mq 1 Ž ´ k . q
x0
k 2 x0 2m
/
aq ij Ž k. ,
2 m ´ K my 1 Ž ´ k .
K my 1 Ž ´ k . K mq 1 Ž ´ k .
q2
ž
j sy
x 02 y ´ 2 2 1 2
1 ay ij
q
Ž k . . Ž 14 .
aq ij Ž k.
Ž k i aqjl Ž k . y k j aqi l Ž k . . k l k2
i
Ai
Ž Ai x 2 y 2 x iA k x k . Ž 17 .
k q 0 q d aq i js j E k a i jq j E 0 a i j
Finally, substituting Eqs. Ž10. and Ž14. into Ž13. qŽ . Ž . we find the relation between ay i j k and a i j k : ay ij Ž k. sy
j 0 s x0 Ž Ak x k q D . ,
where Ai , D, L ij , P i generate on the boundary special conformal transformations, dilatations, Lorentz transformations and shifts respectively. Since E i j 0 ; x 0 and a 0 i ,ay i j tend to zero when ´ ™ 0, in this limit one finds the following transformation law for the boundary value of aq i j:
ay i j Ž x0 , k .
k 2 x 02 K mq1 Ž x 0 k .
Ž 16 .
Note that the Killing vectors of the AdS background can be written as
/
aq i j Ž x0 , k .
k 2 x 02 K my1 Ž x 0 k .
m
y
k 2 x0
d a i j s j rEr a i j q a i r E j j r y h j r E i j r
qDx i q L ij x j q P i ,
one finds m
K mq1Ž ´ k .
for m G 1 and as Ž ´ k . 2 m for 0 - m - 1. Thus, y if we keep aq i j finite in the limit ´ ™ 0, then a i j y tends to zero. Otherwise, keeping of a i j finite leads q to divergency of aq i j . Therefore, only the a i j component can couple on the boundary with the CFT operator Oi j . This conclusion can be also verified by considering the conformal transformations of aq i j on the boundary induced by isometries of AdS. Denote by j a a Killing vector of the background metric. Under diffeomorphisms generated by j the antisymmetric tensor a i j transforms as follows
q y
Knq1 Ž z . q Kny1 Ž z . s y2 KnX Ž z .
E 0 aq i j Ž x0 , k . s y
In the sequal, we restrict ourselves to the case m ) 0. Ž . When ´ ™ 0 the ratio K my1 ´ k behaves as Ž ´ k . 2
ž
Kn ,
175
2
q ik
ž a ŽE j j
k
k k y E kj j . y aq jk Ž E i j y E j i .
qE k j k aq ij .
/
m q Ž . Recalling that E 0 aq i j s y x 0 a i j q O 1 and taking into account the explicit form of the Killing vectors we finally arrive at
k q k q q k d aq i js j E k a i jq Ž 2 y m . Ž A k x q D . a i jq a i k L j
/
.
Ž 15 .
k y aq jk L i ,
Ž 18 .
G.E. ArutyunoÕ, S.A. FroloÕr Physics Letters B 441 (1998) 173–177
176
where L i j s L i j q x iA j y x j Ai. Eq. Ž18. is nothing but the standard transformation law for an antisymmetric tensor with the conformal weight 2 y m under the conformal mappings. Thus, on the boundary aq ij couples to the operator of conformal dimension D s 2 q m. In particular, for m s 1 the antisymmetric Ž . tensor field aq i j transforms in 6 c irrep of SU 4 and couples on the boundary to the following YM operator w17x: Oi Aj B s c Asi j c B q 2 i f A B Fiqj that obviously has the conformal weight 3. It is clear from the discussion above that aq ij plays the role of the coordinate. Now rewriting action Ž1. in the form HŽ pq˙ y H Ž p,q .. we get
When ´ ™ 0 and for m integer one finds m
K my 1 Ž ´ k .
Ž y1. s 2 my1 K mq 1 Ž ´ k . 2 Ž m y 1 . !m! =Ž ´ k .
2m
log ´ k q ...,
while for non-integer m: K my 1 Ž ´ k .
G Ž 2 y m.
sy
K mq 1 Ž ´ k .
2
2m
Ž m y 1. G Ž m q 1.
=Ž ´ k .
2m
q ...,
where in both cases we indicated only the first non-analytical term. Hence, from Ž21. we deduce the two-point function of O in the boundary CFT: m
S s y d5 x
H
ž
Ž ayi j .
)
)
q y E 0 aq i j q E 0 Ž ai j . ai j
² O i j Ž k . O k l Ž q . : s yd Ž k q q .
qi ´ i jk l Ž a0)i E j a k l y a )i j E k a0 l . m q x0
Ž
a )i j a i j q 2 a0)i a 0 i
.
=Ž ´ k .
/ q2
)
yi j q d 5 x E 0 Ž aq . ij . a
H
ž
Ž 19 .
/
The last term in Ž19. is a total derivative, which is omitted in passing to the Hamiltonian formulation. Thus, the action one should use in computing the Green functions is given by S s y d5 x
H
žŽ
)
)
q q y ay i j . E 0 ai j q E 0 Ž ai j . ai j
qi ´ i jk l Ž a 0)i E j a k l y a )i j E k a0 l . m q x0
Ž a)i j ai j q 2 a0)i a0 i .
/
In the Lagrangian picture the total derivative term can be compensated by adding to action Ž19. the following boundary term I s d4 x Ž
H
) yi j aq a . ij
Ž 20 .
.
ž
log ´ k h i w kh l x j
Ž k ih j w k y k jh i w k . k l x k2
/
and a similar result for m non-integer. The last expression exhibits the structure of the correlation function for an antisymmetric tensor field of the conformal weight 2 q m in the D s 4, N s 4 SYM theory. Note that on shell instead of Ž20. one can use the following boundary term Is
.
2m
Ž y1. 2 my2 2 Ž m y 1 . !m!
1 2
Hd
4
xa )i j a i j .
Finally, we remark that in the case m - 0 the component ay i j should be regarded as the coordinate that leads to the change of the sign in the last formula. The authors thank L.O. Chekhov for valuable discussions. This work has been supported in part by the RFBI grants N96-01-00608 and N96-01-00551.
Thus, the on-shell value of Sis given by S s y d4 k
H
ž
K my1 Ž ´ k . K mq 1 Ž ´ k .
= aq ij q2
Ž aqi j .
Ž k i aqjl y k j aqi l . k l k2
References
)
/
.
Ž 21 .
w1x J. Maldacena, The large N limit of superconformal field theories and supergravity, hep-thr9711200. w2x G.G. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory correlators from noncritical string theory, hep-thr9802109.
G.E. ArutyunoÕ, S.A. FroloÕr Physics Letters B 441 (1998) 173–177 w3x E. Witten, Anti de Sitter space and holography, hepthr9802150. w4x I.Ya. Aref’eva, I.V. Volovich, On large N conformal field theories, field theories in anti-de Sitter space and singletons, hep-thr9803028. w5x M. Henningson, K. Sfetsos, Spinors and the AdSrCFT correspondence, hep-thr9803251. w6x W. Muck, ¨ K.S. Viswanathan, Conformal field theory correlators from classical scalar field theory on anti-de Sitter space, hep-thr9804035. w7x D. Freedman, Samir D. Mathur, A. Matusis, L. Rastelli, Correlation functions in the CFTd r AdS dq1 correspondence, hep-thr9804058. w8x H. Liu, A.A. Tseytlin, Ds 4 Super Yang-Mills, Ds 5 gauged supergravity and Ds 4 conformal supergravity, hepthr9804083. w9x G. Chalmers, H. Nastase, K. Schalm, R. Siebelink, R-Current Correlators in N s 4 super Yang-Mills theory from anti-de Sitter supergravity, hep-thr9805105. w10x W. Muck, ¨ K.S. Viswanathan, Conformal field theory correlators from classical field theory on anti-de Sitter space II. Vector and Spinor fields, hep-thr9805145.
177
w11x A. Ghezelbash, K. Kaviani, S. Parvizi, A. Fatollahi, Interacting spinors - scalars and the ADSrCFT correspondence, hep-thr9805162. w12x S. Lee, S. Minwalla, M. Rangamani, N. Seiberg, Three-point functions of chiral operators in Ds 4, N s 4 SYM at large N, hep-thr9806074. w13x H.J. Kim, L.J. Romans, P. van Nieuwenhuizen, Phys. Rev. D 32 Ž1985. 389. w14x M. Gunaydin, N. Marcus, The spectrum of the S 5 compactification of the chiral N s 2 Ds10 supergravity and the unitary supermultiplets of UŽ2,2 <4., Class. Quan. Grav. 2 Ž1985. L11. w15x G.E. Arutyunov, S.A. Frolov, On the origin of supergravity boundary terms in the AdSrCFT correspondence, hepthr9806216. w16x V. Balasubramanian, P. Kraus, A. Lawrence, Bulk vs. boundary dynamics in anti-de Sitter spacetime, hepthr9805171. w17x S. Ferrara, C. Fronsdal, A. Zaffaroni, On N s8 supergravity on AdS5 and N s 4 superconformal Yang-Mills theory, hepthr9802203.
Physics Letters B 441 Ž1998. 173–177
Antisymmetric tensor field on AdS5 G.E. Arutyunov 1, S.A. Frolov
2
StekloÕ Mathematical Institute, Gubkin str.8, GSP-1, 117966 Moscow, Russia Received 30 July 1998; revised 28 August 1998 Editor: P.V. Landshoff
Abstract By using the Hamiltonian version of the AdSrCFT correspondence, we compute the two-point Green function of a local operator in D s 4 N s 4 super Yang-Mills theory, which corresponds to a massive antisymmetric tensor field of the second rank on the AdS5 background. We discuss the conformal transformations induced on the boundary by isometries of AdS5. q 1998 Elsevier Science B.V. All rights reserved.
The recent Maldacena’s conjecture w1x relates the large N limit of certain conformal theories in d-dimensions with classical supergravity on the product of anti de Sitter space AdS dq 1 with a compact manifold. According to w2,3x the precise relation consists in existing the correspondence between supergravity fields and the set of local CFT operators. Then the generating functional of the connected Green functions of the CFT operators is identified with the on-shell value of the supergravity action. With this identification at hand, the AdSrCFT correspondence was recently tested by explicit computation of some two- and three-point correlation functions of local operators in D s 4 N s 4 super Yang-Mills theory, which correspond to scalar, vec-
1
E-mail: [email protected] Address after September 1, 1998: The University of Alabama, Department of Physics and Astronomy, Box 870324, Tuscaloosa, Alabama 35487-0324. 2
tor, symmetric tensor and spinor fields on the AdS5 background w4–12x. D s 4 N s 4 super Yang-Mills is related to the S 5 compactification of D s 10 IIB supergravity. Except the fields mentioned above, the spectrum of the compactified theory also contains the massive antisymmetric tensor fields of the second rank w13,14x. These fields obey first-order differential equations and their bulk action vanishes on shell. Thus, the bulk action is not enough to compute the CFT Green functions and one has to add some boundary terms. This is quite similar to the case of fermions on the AdS background w5x. The origin of boundary terms in the AdSrCFT correspondence was recently clarified in w15x, where it was shown that they appear in passing from the Hamiltonian description of the bulk action to the Lagrangian one. The idea was to treat the coordinate in the bulk direction as the time and to present the bulk action in the form HŽ pq˙ y H Ž p,q . q total deriÕatiÕe.. Here a choice of coordinates and momenta is dictated by the
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 1 1 3 6 - 8
G.E. ArutyunoÕ, S.A. FroloÕr Physics Letters B 441 (1998) 173–177
174
transformation properties of gravity fields under isometries of AdS. In the Hamiltonian formulation the total derivative term should be omitted while from the Lagrangian point of view it can be compensated by adding to the bulk action a proper boundary term. In this note we demonstrate how this general approach works in the case of antisymmetric tensor fields of the second rank and compute the two-point function of the corresponding local CFT operators. We start with the following action for a massive complex antisymmetric tensor field of the second rank 3 S s y d5 x
H
ž
i 2
) ) ´ mnrls amn Er als q 'y g mamn a mn .
/
Ž 1. 0 g s yxy1 0 2 1 Ž
Here metric: ds s
x 02
is the determinant of the AdS dx 02 q hi j dx i dx j .. Because of in-
frared divergencies one should regularize the action by cutting AdS5 space off at x 0 s ´ and leaving the part x 0 G ´ . We use the convention ´ 01234 s y´ 01234 s 1. Action Ž1. vanishes on shell and, therefore, can not produce the two-point functions in the boundary CFT. According to the general scheme discussed above, we need to rewrite action Ž1. in the form suitable for passing to the Hamiltonian formulation. To this end one has to establish a proper set of variables that can be treated as coordinates and their conjugate momenta. It can be done by studying solutions of equations of motion coming from Ž1. i 2
´ mnrlsEr als q 'y g ma mn s 0.
Ž 2.
Acting on Ž2. with an operator y 2i ´mnrls= r q y g mgml gns we arrive at the second-order equation
'
= r Ž =r amn y =m arn q =n arm . y m2 amn s 0.
The last equation implies the constraint = mamn s 0 and, therefore, can be written in the form
= r =r amn q Ž 6 y m2 . amn s 0.
Ž 4.
Specifying Ž4. for a i j , we obtain x 02 E 02 a i j q x 0 E 0 a i j q x 02 I a i j y m2 a i j y 2 x 0 Ž E i a 0 j y E j a0 i . s 0,
Ž 5.
ij
where I s h E i E j . The derivatives E i a 0 j can be expressed from Ž2.: i
xy1 m ´ i jk l a k l , Ž 6. 2 0 where in the last formula and below the indices are raised with respect to the Minkowski metric. Therefore, Eq. Ž5. reduces to
E i a0 j y E j a0 i s E 0 a i j q
x 02 E 02 a i j y x 0 E 0 a i j q x 02 I a i j y m2 a i j y im ´ i jk l a k l s 0,
Ž 7.
To solve Ž7. we introduce the projections Žanti.self-dual parts of a i j : a" ij s
1 2
ž
ai j "
i
a" ij
on the
i " kl ´ i jk l a k l , a " . i j s " ´ i jk l a 2 2
/
Ž 8. Then Eq. Ž7. splits into equations for x 02 E 02 a i"j y x 0 E 0 a i"j y m
Ž m " 2.
aq ij
and
ay ij :
a i"j q x 02 I a " i j s 0.
Ž 9. Momentum space solutions of Ž9. for k 2 ) 0 obey"Ž . Ž . ing the boundary conditions a " i j ´ , k s a i j k and 4 vanishing at x 0 s ` read as a i"j Ž x 0 , k . s
x 0 K m " 1Ž x 0 k .
´ K m " 1Ž ´ k .
a" ij Ž k. ,
Ž 10 .
where K m " 1 is the Mackdonald function and k s < k <. However, we can not assign arbitrary boundary yŽ . Ž . values for both aq i j k and a i j k since these com-
Ž 3. 4
3 It follows from w13,14x that antisymmetric tensor fields arising in S 5 compactification of IIB supergravity are classified by complex representations of SO Ž6..
As was noted in w16x for k 2 - 0 there are two independent solutions regular in the interior. However, both of them are nonvanishing at infinity. A proper account of these solutions may be achieved by introducing an additional boundary at x 0 s1r ´ and requiring the vanishing of the solution on this boundary. Then the solution is unique and in the limit ´ ™ 0 delivers the same contribution to the two-point function as the solution for k 2 ) 0 does. In the sequal, we restrict ourselves to the case k 2 ) 0.
G.E. ArutyunoÕ, S.A. FroloÕr Physics Letters B 441 (1998) 173–177
ponents are related to each other. To find this relation, note that the components a 0 i can be directly found from Ž2.: a0 i s y
i 2m
x 0 ´ i jk l E j a k l .
Ž 11 .
Then substituting into Ž6. one obtains the constraint x0 Ž E E k Ž aqi k y ayi k . y E i E k Ž aqjk y ayjk . . m j m q s E 0 ai j q Ž 12 . Ž a y ayi j . , x0 i j which after projecting on its Žanti.self-dual part results into the following equations x0 k . " I a i"j q I a i.j q 2 Ž E i E k a . jk y E j E a i k . 2m m s E 0 a i"j " a i"j . Ž 13 . x0
ž
/
With the solution for a i j at hand one can compute the derivative E 0 a i j . By using the following properties of the Macdonald function Knq1 Ž z . y Kny1 Ž z . s
2n z
q
E 0 ay ij
Ž x0 , k . s
ž
ž
q x0
2m
/
2 m ´ K mq 1 Ž ´ k . q
x0
k 2 x0 2m
/
aq ij Ž k. ,
2 m ´ K my 1 Ž ´ k .
K my 1 Ž ´ k . K mq 1 Ž ´ k .
q2
ž
j sy
x 02 y ´ 2 2 1 2
1 ay ij
q
Ž k . . Ž 14 .
aq ij Ž k.
Ž k i aqjl Ž k . y k j aqi l Ž k . . k l k2
i
Ai
Ž Ai x 2 y 2 x iA k x k . Ž 17 .
k q 0 q d aq i js j E k a i jq j E 0 a i j
Finally, substituting Eqs. Ž10. and Ž14. into Ž13. qŽ . Ž . we find the relation between ay i j k and a i j k : ay ij Ž k. sy
j 0 s x0 Ž Ak x k q D . ,
where Ai , D, L ij , P i generate on the boundary special conformal transformations, dilatations, Lorentz transformations and shifts respectively. Since E i j 0 ; x 0 and a 0 i ,ay i j tend to zero when ´ ™ 0, in this limit one finds the following transformation law for the boundary value of aq i j:
ay i j Ž x0 , k .
k 2 x 02 K mq1 Ž x 0 k .
Ž 16 .
Note that the Killing vectors of the AdS background can be written as
/
aq i j Ž x0 , k .
k 2 x 02 K my1 Ž x 0 k .
m
y
k 2 x0
d a i j s j rEr a i j q a i r E j j r y h j r E i j r
qDx i q L ij x j q P i ,
one finds m
K mq1Ž ´ k .
for m G 1 and as Ž ´ k . 2 m for 0 - m - 1. Thus, y if we keep aq i j finite in the limit ´ ™ 0, then a i j y tends to zero. Otherwise, keeping of a i j finite leads q to divergency of aq i j . Therefore, only the a i j component can couple on the boundary with the CFT operator Oi j . This conclusion can be also verified by considering the conformal transformations of aq i j on the boundary induced by isometries of AdS. Denote by j a a Killing vector of the background metric. Under diffeomorphisms generated by j the antisymmetric tensor a i j transforms as follows
q y
Knq1 Ž z . q Kny1 Ž z . s y2 KnX Ž z .
E 0 aq i j Ž x0 , k . s y
In the sequal, we restrict ourselves to the case m ) 0. Ž . When ´ ™ 0 the ratio K my1 ´ k behaves as Ž ´ k . 2
ž
Kn ,
175
2
q ik
ž a ŽE j j
k
k k y E kj j . y aq jk Ž E i j y E j i .
qE k j k aq ij .
/
m q Ž . Recalling that E 0 aq i j s y x 0 a i j q O 1 and taking into account the explicit form of the Killing vectors we finally arrive at
k q k q q k d aq i js j E k a i jq Ž 2 y m . Ž A k x q D . a i jq a i k L j
/
.
Ž 15 .
k y aq jk L i ,
Ž 18 .
G.E. ArutyunoÕ, S.A. FroloÕr Physics Letters B 441 (1998) 173–177
176
where L i j s L i j q x iA j y x j Ai. Eq. Ž18. is nothing but the standard transformation law for an antisymmetric tensor with the conformal weight 2 y m under the conformal mappings. Thus, on the boundary aq ij couples to the operator of conformal dimension D s 2 q m. In particular, for m s 1 the antisymmetric Ž . tensor field aq i j transforms in 6 c irrep of SU 4 and couples on the boundary to the following YM operator w17x: Oi Aj B s c Asi j c B q 2 i f A B Fiqj that obviously has the conformal weight 3. It is clear from the discussion above that aq ij plays the role of the coordinate. Now rewriting action Ž1. in the form HŽ pq˙ y H Ž p,q .. we get
When ´ ™ 0 and for m integer one finds m
K my 1 Ž ´ k .
Ž y1. s 2 my1 K mq 1 Ž ´ k . 2 Ž m y 1 . !m! =Ž ´ k .
2m
log ´ k q ...,
while for non-integer m: K my 1 Ž ´ k .
G Ž 2 y m.
sy
K mq 1 Ž ´ k .
2
2m
Ž m y 1. G Ž m q 1.
=Ž ´ k .
2m
q ...,
where in both cases we indicated only the first non-analytical term. Hence, from Ž21. we deduce the two-point function of O in the boundary CFT: m
S s y d5 x
H
ž
Ž ayi j .
)
)
q y E 0 aq i j q E 0 Ž ai j . ai j
² O i j Ž k . O k l Ž q . : s yd Ž k q q .
qi ´ i jk l Ž a0)i E j a k l y a )i j E k a0 l . m q x0
Ž
a )i j a i j q 2 a0)i a 0 i
.
=Ž ´ k .
/ q2
)
yi j q d 5 x E 0 Ž aq . ij . a
H
ž
Ž 19 .
/
The last term in Ž19. is a total derivative, which is omitted in passing to the Hamiltonian formulation. Thus, the action one should use in computing the Green functions is given by S s y d5 x
H
žŽ
)
)
q q y ay i j . E 0 ai j q E 0 Ž ai j . ai j
qi ´ i jk l Ž a 0)i E j a k l y a )i j E k a0 l . m q x0
Ž a)i j ai j q 2 a0)i a0 i .
/
In the Lagrangian picture the total derivative term can be compensated by adding to action Ž19. the following boundary term I s d4 x Ž
H
) yi j aq a . ij
Ž 20 .
.
ž
log ´ k h i w kh l x j
Ž k ih j w k y k jh i w k . k l x k2
/
and a similar result for m non-integer. The last expression exhibits the structure of the correlation function for an antisymmetric tensor field of the conformal weight 2 q m in the D s 4, N s 4 SYM theory. Note that on shell instead of Ž20. one can use the following boundary term Is
.
2m
Ž y1. 2 my2 2 Ž m y 1 . !m!
1 2
Hd
4
xa )i j a i j .
Finally, we remark that in the case m - 0 the component ay i j should be regarded as the coordinate that leads to the change of the sign in the last formula. The authors thank L.O. Chekhov for valuable discussions. This work has been supported in part by the RFBI grants N96-01-00608 and N96-01-00551.
Thus, the on-shell value of Sis given by S s y d4 k
H
ž
K my1 Ž ´ k . K mq 1 Ž ´ k .
= aq ij q2
Ž aqi j .
Ž k i aqjl y k j aqi l . k l k2
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.
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