Massive symmetric tensor field on AdS

Massive symmetric tensor field on AdS Aleksey Polishchuk Steklov Mathematical Institute, Gubkin str.8, GSP-1, 117966, Moscow, Russia arxiv:hep-th/99548v4 1 Oct 1999 Abstract The two-point Green function of a local operator in CFT corresponding to a massive symmetric tensor field on the AdS background is computed in the framework of the AdS/CF T correspondence. The obtained two-point function is shown to coincide with the two-point function of the graviton in the limit when the AdS mass vanishes. 1 Introduction The AdS/CFT correspondence conjectured in [1] states that in the large N limit and at large t Hooft coupling λ = gy MN the classical supergravity/m-theory on the Anti-de Sitter space AdS times a compact manifold is dual to a certain SUN conformal gauge theory CFT defined on the boundary of AdS. One notable example of this correspondence being the duality between D = 4, N = 4 supersymmetric Yang-Mills theory and D = 1 Type IIB supergravity theory on AdS 5 S 5. The precise formulation of the conjecture was given in [], [3] where it was proposed to identify the generating functional for connected Green functions of local operators in CFT with the on-shell value of the supergravity action under the restriction that the supergravity fields satisfy the Dirichlet conditions on the boundary of AdS. Recall that in the standard representation of AdS as an upper-half space x >, x k R, k = 1,...d with the metric ds = g µν dx µ dx ν = 1 dx dx + dx i dx i, x the boundary includes the plane x = as well as the point x =. Since the boundary is located infinitely far away from any point in the interior the supergravity action is infrared divergent and must be regularized. As was pointed out in [4] the consistent regularization procedure with respect to Ward identities requires one to shift the boundary of AdS to the surface in the interior defined by x = ε. Then the Dirichlet boundary value problem for supergravity fields is properly defined and one can compute the on-shell value of the supergravity action as a functional of the boundary fields. With the account of this regularization procedure the standard formulation of the AdS/CFT correspondence assumes the form: Ox 1 Ox n = lim ε alexey@mi.ras.ru δ δφ 1 x 1 δ δφ n x n S on shell φ 1 x 1,..., φ n x n φi x i =φ i x =ε,x i, 1

where x i are some points on the boundary of AdS d+1, O i x i are gauge invariant composite operators in CFT and φ i x i are the corresponding supergravity fields. Here we used the convention in which the coordinates x µ of AdS d+1 are split according to: x = x,x, so that x R d. The action S on shell is the sum of the bulk supergravity action and the boundary terms necessary to make the AdS/CFT correspondence complete. The origin of these boundary terms was elucidated in [5] where it was shown that they appear in passing from the Hamiltonian description of the bulk action to the Lagrangian one. The AdS/CFT correspondence has been tested by computing various two- and three-point functions of local operators in D = 4, N = 4 supersymmetric Yang-Mills theory on AdS 5. In particular two-point functions corresponding to scalars [3] - [8], vectors [3], [4], [9], [1], spinors [1] - [1], the Rarita-Schwinger field [13] - [16], antisymmetric form fields [17] - [] and the graviton [5], [1], [] were computed on the AdS background. The only field in the supergravity spectrum, found in [3], that has evaded the attention is the massive symmetric second-rank tensor field. In this paper we fill in the gap by computing the remaining two-point Green function. We note that the Dirichlet boundary value problem for the massive symmetric tensor field is nontrivial due to the fact that the equations of motion for various components are coupled. Furthermore, in computing the two-point Green function we have to stick with the regularization procedure described above in order to obtain the consistent result. At the end of our computation we find that in the limit when the AdS mass vanishes the correlation function reduces to that of the massless symmetric tensor field, i.e. the graviton. Equations of motion The starting point in the calculation is the action for the symmetric second-rank tensor φ µν on AdS d+1 [4] 1 : S [φ µν ] = 1 1 d d+1 x g κ AdS 4 λφ λ φ 1 4 λφ µν λ φ µν 1 µ φ ν µ ν φ + 1 µφ µν λ +φ λν d φ µνφ µν + d 4 φ 1 4 m φ µν φ µν 1 4 m φ,.1 where g is the determinant of the AdS metric g µν. The action.1 leads to the following equations of motion [3], [4] λ λ φ µ ν + m φ µ ν =,. φ µ µ =, µφ µ ν =..3 The massive terms in.1 destroy the standard symmetry: δφ µν = µ ξ ν + ν ξ µ, which is present in the case of the massless symmetric tensor field. As a result of this symmetry breaking one can no longer perform gauge fixing. Since φ µ ν is traceless, we can eliminate the component φ from the equations of motion by using the constraint φ +φ i i =. Let us introduce a concise notation: φ = φ i i, ϕi = φ i, ϕ = iϕ i. 1 Note that the AdS background satisfies restriction 1 of [4] and also to simplify our calculations we set the parameter ξ of [4] equal to 1.

Then starting from.,.3 one can obtain the following system of coupled differential equations ϕ = d + 1 φ,.4 x d + 1 ϕ i + k φ i k =,.5 x d + 3 d + + + m φ =..6 x x x Here eq..6 corresponds to µ = ν = in. while eqs..4 and.5 follow from.3. To simplify the notation, we chose the convention in which indices are raised and lowered with the flat metric δ µν = x g µν so that, in particular, = δ µν µ ν. The Fourier mode for the field φ vanishing at x is given by: Here k = k, and K is a modified Bessel functions [5]: K z = π I z I z sin π φx,k = Aε,kx d + K kx..7 d = + m,, I z = k= 1 z +k. k!γ + k + 1 Recall that the modified Bessel function satisfies the recurrence relations [5]: K +1 z K 1 z = z K z, K +1 z + K 1 z = d dz K z..8 By differentiating.5 with respect to x i and using.4 and.6 one gets: d 1 i j φ i j = + x d 1d + + m φ..9 x From.7 and.9 we can easily deduce the Fourier mode solution for the field πx,k k i k j φ i j x,k = = Aε,k x d d 1x kk +1 x k + 1 + x k K x k, where recurrence formulae.8 were used. Here we introduced a concise notation = d, + = d +. 3

Taking the ratio of φ and π at x = ε results in the following relation πk = ε d 1εkK +1εk + 1 + εk K εk K εk φk, where πk = πε,k and, similarly for other fields. In the limit ε, πk ε φk and therefore if we keep πk finite as ε, then φk will tend to zero. On the other hand, keeping φk finite leads to the divergence of πk. Consequently, we ought to fix π at x = ε. Thus we have: φx,k = x d + ε d πx,k = x d ε d K x k d 1εkK +1 εk + 1 + εk K εk πk,.1 d 1x kk +1 x k + 1 + x k K x k πk. d 1εkK +1 εk + 1 + εk K εk.11 The Fourier mode solution for the field ϕ is found by substituting.1 into.4: ϕx,k = x d +1 1 K x k x kk +1 x k πk,.1 ε d d 1εkK +1 εk + 1 + εk K εk where once again recurrence formulae.8 were used. Next we turn our attention to the field ϕ i. Here we need to use equation m x ϕ i + x i φ = k φ i k,.13 which follows from i-component of. by taking into account.5. Now we decompose ϕ i into the transversal and longitudinal parts: ϕ i = ϕ i i ϕ. Rewriting.5 and.13 for the transversal part of ϕ i leads to: ϕ m i = x k φ k i i l m φ l m,.14 d + 1 ϕ i = k φ k i x i l m φ l m..15 By differentiating.15 with respect to x and taking into account.14 we obtain an homogeneous differential equation for the field ϕ i : d + 1 + d + 1 + m x x x 4 ϕ i =..16

The Fourier mode solution is given by: ϕ i x,k = B i ε,k x d +1 K x k..17 To find B i ε,k substitute.17 into.15 to obtain the following formula k l φ l i x,k k i k πx,k = ib i ε,k x d x kk +1 x k + K x k..18 Taking the ratio of.18 and.17 at x = ε we find: k l φ i l ki εkk +1 εk k kπk = iε 1 K εk + K εk ϕ i k. Using the same arguments as before one finds that in order to avoid the divergence at ε the solutions for the field ϕ i and k l φ l i k i π should take the following form k ϕ i x,k = iε x d +1 K x k k l φ i ε d εkk +1 εk + K εk lk ki k πk,.19 k l φ i l x,k ki k πx,k = x d x kk +1 x k + K x k k l φ i ki ε d l k εkk +1 εk + K εk k πk.. Setting µ = i, ν = j in. and taking into account.3 we arrive at the following equation d 1 + m x x Let us introduce the transversal traceless part of φ i j: φ i j = j ϕ i + i δ ϕ j + j i φ..1 x x φ i j = φi j 1 i k φ k j 1 j k φ i k + 1 i j k m φ k m + 1 i j d 1 δi j φ k l φk l.. Rewriting.1 for the transversal traceless part yields: d 1 + m x x φ i j =..3 The Fourier mode solution of eq..3 is φ i jx,k = x d ε d Taking into account.,.4,.1,.11, and. we obtain: K x k K εk φ i jk..4 [ φ i j x,k = x K x k ε K εk φ i j k + K +1x k + K x k k i k l K +1 εk + K εk k φl j k + k jk l k φi l k 5

ki k j k πk + d 1K x k + 1 + x k K x k 4 d 1K εk + 1 + εk K εk ki k j k πk 1 k i k j δ i 4 d 1 k j x k K x k d 1εkK +1 εk + 1 + εk K εk d 1K x k + 1 + x k ] K x k πk,.5 d 1K εk + 1 + εk K εk k while taking into account.19 and.1 gives: [ x 1+ K x k ϕ i x,k = iε k l φ l i ε K +1 εk + K εk k k i k πk ] 1 K x k K +1 x k k i d 1K +1 εk + 1 + εk K εk k πk,.6 where we introduced a concise notation: K z = z K z. 3 Two-point Green function To compute the Green function in the framework of the AdS/CF T correspondence we need to evaluate the on-shell value of the action. Taking into account equations of motion. and.3 one finds that the on-shell value of.1 is S on shell = ε d+1 8κ x =ε d d x φ i j φ j i φ φ + φϕ ϕ k i φ i k + d + 1 ε φ + ϕ k ϕ k. Let us first consider the contribution to S on shell that depends locally on boundary fields, i.e. does not contain the normal derivative. We expect that such terms do not contribute to the non-local part of S on shell. So we need to consider the behavior of φ and ϕ k on the boundary of AdS. To this end we note that according to.1 the field φk is local since K εk d 1εkK +1 εk + 1 + εk K εk = 1 + + 1 + Oε k, while expanding solution.6 in ε gives: [ 1 ϕ i k = iε k l φ i ki l k k πk + + 1 k i ] + k πk + local. 3.1 Here we included only non-local terms. In deriving 3.1 use was made of the power series expansion of the modified Bessel function [5]: K z = 1 z Γ 1 + 41 +..., 3. 6

where ellipsis indicate terms of order z 4 and higher which evidently lead to local expressions as well as terms of order z + which will become negligible in the limit ε. From expression 3.1 it follows that ϕ i k is a local field. Consequently, the terms in 3.1 that depend only on the value of fields at the boundary do not contribute to the non-local part of S on shell as we expected. Next we consider terms with the normal derivative. In evaluating such terms it is useful to employ the following identity d dx x ε γ Fx k Fεk = γ x ε + k ε =ε d ln Fkε. 3.3 dk Taking into account solution.1, identity 3.3 and expansion 3. we find that φε,x is equal to d d k + φε,x = + Oεk ε 1 π de ikx ε + + 1 + Oε k πk. Clearly φ is a local expression and therefore does not contribute to the non-local part of S on shell. Consequently, the non-local part is entirely determined by the following expression ε d+1 d d xφ j d d k 8κ ix x =ε π de ikx φ i j x,k. 3.4 Taking into account solution.5, identity 3.3 and expansion 3. we see that the expression for φ i j gets three different contributions: the first contribution comes from differentiating the ratio x ε raised to the power or + ; the second contribution comes from Oεk terms in the power expansion of the logarithmic derivative of various functions in.5 and the third contribution comes from Oε 1 k terms in the same expansion. From.5 we can easily read off the first contribution which is equal to ε φ i jk + k i k l ε k φl jk + k jk l k φi lk ki k j k πk + k i k j 4 ε k πk 4 1 k i k j + δ i d 1 k j εk 1 ε + + 1 πk ε k = k i k j ε πk + local, 3.5 d 1 k + + 1 where we took into account the expansion of φ i j into the transversal and longitudinal parts given by.. Next, Oεk terms from the power series expansion of the logarithmic derivative give: εk [ 1 1 φ i j k + + k i k l 1 + k φl j k + k jk l k φi l k k j ki k πk 4 + + + 3 k i k j 1 + + 1 k πk 1 k i k j δ i εk 4 d 1 k j 1 + + 1 ] + 3 + πk 1 + + 1 k = k i k j ε πk + local, 3.6 d 1 k + + 1 7

where we used 3. and.. Adding together 3.6 and 3.5, the non-local part cancels leading to a purely local expression which does not contribute to the non-local part of S on shell. Finally, Oε 1 k terms in the expansion of the logarithmic derivative give: Here, η ε 1 k [ φ i j k + = η ε 1 k [ + 1 d 1 + φ i j k + k i k l k φl j k + k jk l k φi l k k j ki k πk 4 k i ] k j δ i 1 πk k j + + 1 k k i k l k φl j k+ +k jk l k φi l k η = Γ1 1 Γ1 +. + 1 k i k j + + 1 k πk 4 4 1 k i ] k j + + 1 k πk. 3.7 4 In the process of deriving 3.7 we dropped all terms containing δj i since such terms are negligible in the limit ε. To understand why this is so, note that when δj i is contracted with φj ix from 3.4 it gives the trace of φ i j x which according to.1 is order Oε at x = ε. Now putting together 3.4 and 3.7 we arrive at the following formula for S on shell S on shell [φ i j ] = ηε d d d xd d y Φ j 8κ ixφ r s y d d k k π de ikx y [ 1 δi r δs j + 1 δis δ jr k i k r + k δs j + ki k s k δ jr + k jk s k δi r + k jk r k δis + 1 ki k j k r k s + k rk s + + 1 k 4 k δi j + ki k j k δs r + ] d δi jδr s, 3.8 where we introduced the traceless part of φ i j: Φ i j x = φi j x δi j d φx. In order to complete the calculation of the two-point Green function we need to evaluate the integral over k in 3.8. To this end, we employ the standard formula for the Fourier transformation of generalized functions [6]: d d k de ikxki1 k i k in k δ = δ n Γ π k n d+δ n π d/ Γ δ+n i n 1 i1 i in x d+δ n. 3.9 With the help of 3.9 we find that the non-local part of the on shell value of the action is equal to: S on shell [φ i j] = C d, ε d d d xd d y Φj ixφ r s y [ 1 x y +d Ji rx yjj s x y ] + 1 Jis x yj jr x y 1 d δi j δr s, 8

where we introduced J i jx = δ i j xi x j x and C d, = + + 1Γ + π d κ + 1Γ. 3.1 From this we can easily deduce the two-point function of local operators in the boundary CFT corresponding to the massive symmetric traceless rank two tensor field Φ i j : [ < Oj i xos r y >= C d, 1 x y +d Ji r x yjs j x y + 1 Jis x yj jr x y 1 ] d δi j δs r. Here we performed rescaling in order to remove the regularization parameter ε. Note that in the limit m or, equivalently, d, the obtained expression correctly reproduces the two-point function corresponding to the massless symmetric tensor field graviton [5], [1], []. ACKNOWLEDGEMENT The author would like to thank S.Frolov and G.Arutyunov for valuable discussions and A.Slavnov for the support during the preparation of the manuscript. References [1] J. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, Adv. Theor. Math. Phys. 1998 31, hep-th/9711. [] S. S. Gubser,I. R. Klebanov and A. M. Polyakov, Gauge Theory Correlators from Non-critical String Theory, Phys. Lett. B48 1998 15, hep-th/9819. [3] E. Witten, Anti De Sitter Space and Holography, Adv. Theor. Math. Phys. 1998 53, hepth/9815. [4] D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT d /AdS d+1 correspondence, hep-th/98458. [5] G.Arutyunov and S.Frolov, On the origin of supergravity boundary terms in the AdS/CF T correspondence, hep-th/98616. [6] W. Mück and K. S. Viswanathan, Conformal Field Theory Correlators from Classical Scalar Field Theory on AdS d+1, Phys. Rev. D58 1998 4191, hep-th/98435. [7] L. Chekhov, AdS/CFT correspondence on torus, hep-th/9811146. [8] I.Ya. Arefeva and I.V. Volovich, On large N conformal field theories, field theories in anti-de Sitter space and singletons, hep-th/9838. [9] G. Chalmers, H. Nastase, K. Schalm and R. Siebelink, R-Current Correlators in N = 4 Super Yang-Mills Theory from Anti-de Sitter Supergravity, hep-th/98515. [1] W. Mück and K. S. Viswanathan, Conformal Field Theory Correlators from Classical Field Theory on Anti-de Sitter Space II. Vector and Spinor Fields, Phys. Rev. D58 1998 166, hep-th/985145. 9

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