A Note on Ads/CFT Dual of SL(2,Z)Action on 3D Conformal ﬁeld Theories with U(1) Symmetry

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Physics Letters B 598 (2004) 139–148 www.elsevier.com/locate/physletb

A note on AdS/CFT dual of SL(2,Z)action on 3D conformal ﬁeld theories with U(1) symmetry

Ho-Ung Yee

School of Physics, Korea Institute for Advanced Study, 207-43 Cheongryangri-Dong, Dongdaemun-Gu, Seoul 130-722, South Korea Received 4 March 2004; received in revised form 28 May 2004; accepted 28 May 2004

Editor: L. Alvarez-Gaumé

Abstract In this Letter, we elaborate on the SL(2,Z) action on three-dimensional conformal field theories with U(1) symmetry introduced by Witten, by trying to give an explicit verification of the claim regarding holographic dual of the S operation in AdS/CFT correspondence. A consistency check with the recently proposed prescription on boundary condition of bulk fields when we deform the boundary CFT in a non-standard manner is also discussed.  2004 Elsevier B.V. Open access under CC BY license.

PACS: 11.25.Hf; 11.25.Tq

1. Introduction field. This symmetry is supposed to be the symmetry of U(1) phase rotation in the latter. Because magnetic Mirror symmetry found in three-dimensional theo- vortices break the shift symmetry of the dual photon, ries with extended supersymmetry [7] gives us much they can be identified to elementary excitations in the insight about non-trivial duality in quantum field the- free theory side. ory. For the cases with Abelian gauge groups, it was Recently, it was observed in Ref. [1] that the above shown [8] that many aspects of duality may be derived simplest duality between vortex and particle may be by assuming a single ‘elementary’ duality, that is, the seen as an invariance under certain transformation on duality (in IR) between the N = 4 SQED with sin- three-dimensional CFTs. Specifically, given a CFT gle flavor hypermultiplet and the free theory of single with global U(1) symmetry, this transformation is de- hypermultiplet. The former has a global U(1) symme- fined by gauging the U(1) symmetry without intro- try that shifts the dual photon scalar of U(1) gauge ducing gauge kinetic term. Although the above ex- ample is in the context of supersymmetric version of this transformation, there is no problem in defining E-mail address: [email protected] (H.-U. Yee). this transformation in non-supersymmetric cases, in

0370-2693  2004 Elsevier B.V. Open access under CC BY license. doi:10.1016/j.physletb.2004.05.082 140 H.-U. Yee / Physics Letters B 598 (2004) 139–148 general. The intriguing fact shown in Ref. [1] is the A basic ingredient used in the discussion of Ref. [1] possibility of extending this transformation into a set is the equation, of transformations forming the group SL(2,Z).The above transformation corresponds to S with S2 =−1, DA exp iI(A,B) while the transformation T with (ST )3 = 1 was introduced. i 3 ij k = DA exp d x Ai∂j Bk The meaning of this SL(2,Z) in the space of 3D 2π CFTs has been studied in Refs. [1,4–6] for theories in Y = which Gaussian approximation is valid in calculating δ(B), (2.1) correlation functions [12,13].(See[10] for implica- where A and B are connections of line bundles on an tions on QHE.) These analysis identified the SL(2,Z) oriented base three-manifold Y . The delta function on as certain transformations of basic correlation func- the right-hand side means that B is zero, that is, its tions of the theory. While we may be almost convinced field strength vanishes and there is no non-vanishing that the transformations of correlation functions found Wilson line. The path integral DA in the left-hand in these analysis hold true in general, its proof is cur- side includes summing over topologically distinct line rently limited to the theories with Gaussian approxi- bundles as well as integral over trivial connections. For mation. topologically non-trivial connections, especially when As suggested in Ref. [1], another way of interpret- a quantized magnetic flux on a 2-dimensional cycle Σ ing the SL(2,Z) transformations may be provided by does not vanish, AdS/CFT correspondence [9]. According to AdS/CFT, 1 a global U(1) symmetry in the CFT corresponds to F =0, (2.2) having a U(1) gauge theory in the bulk, whose asymp- 2π totic value on the boundary couples to the U(1) cur- Σ rent of the CFT. The U(1) gauge theory in the bulk has it is not possible to define a global connection A such a natural SL(2,Z)duality [2]. While it is easy to iden- that dA = F . In this case, we need to understand tify the T operation in the CFT as the usual 2π shift of I(A,B) as follows. Pick up a compact-oriented four the bulk θ parameter [1], describing holographic dual manifold X whose boundary is Y , and extend connec- of the S operation turns out to be much more subtle. It tions (and line bundles) A, B on Y to connections A, was suggested that the S-transformed CFT is dual to B on X.ThenI(A,B) is defined to be the same gauge theory in the bulk, but its U(1) current 1 couples to the S-dualized gauge field. Note that the re- FA ∧ FB, (2.3) 2π sulting CFT with different coupling to the bulk field is X not equivalent to the original CFT [14,18]. where FA, FB are the field strengths of A, B. Because Although a compelling discussion on holographic ¯ 1 for any closed four manifold X, ¯ FA ∧ FB is an dual of the S operation was provided in Ref. [1] using 4π2 X integer Chern number, the above definition of I(A,B) various aspects of AdS/CFT [17–19], and was further is easily shown to be independent of extensions mod- supported in Ref. [6] by explicitly calculating cer- ulo 2π. This is fine as long as we are concerned only tain correlation functions, a rigorous verification of the with eiI(A,B). claim is missing. In this Letter, we propose a rigorous In Ref. [1], several ways of showing (2.1) were argument that fills this gap. given. In simple terms, we split A = Atriv + A ,where Atriv is a globally defined trivial connection, and A is a representative of a given topologically non-trivial 2. Setting up the stage line bundle (which does not have a global definition). Note that we can write This section is intended to give a brief review of rel- evant facts in Ref. [1] on SL(2,Z) transformations of 1 3 ij k triv I(A,B)= d x A ∂j Bk + I(A ,B), 3D CFTs, as a necessary preparation for the discussion 2π i in next section. Y (2.4) H.-U. Yee / Physics Letters B 598 (2004) 139–148 141

ij k 1 ij k because ∂j Bk = Fjk is well-defined on Y . A theory is thus specified by 2 The path integral over Atriv gives us delta function 3 i setting the field strength of B zero. Then, only re- exp i d xAiJ , (2.9) maining component of B is its possible Wilson line Y in H 1(Y, U(1)). With vanishing field strength of B, where ···means to evaluate expectation value in the it is clear as in (2.4) that I(A ,B) is invariant under given CFT. The above generating functional can pro- adding trivial connection to A ,thatis,I(A ,B) de- duce all correlation functions of U(1) current J i .The pends only on the cohomology of the field strength of S operation is defined by letting A be dynamical and A . This cohomology (characteristic class) belongs to i introducing a background gauge field B with a cou- H 2(Y, Z) as a consequence of Dirac quantization (a i pling Chern’s theorem), or more specifically, 1 3 ij k 1 I(A,B)= d x Ai∂j Bk, (2.10) F ∈ Z, (2.5) 2π 2π Y Σ that is, the transformed theory is now specified by where Σ is any integer coefficient 2-cycle. Thus, we see that I(A ,B)is a kind of bilinear form, D 3 i A exp i d xAiJ exp iI(A,B) , (2.11) 2 1 H (Y, Z) × H Y,U(1) → R. (2.6) Y In Ref. [1], this bilinear form was identified and sum- where ··· means expectation value in the origi- ming over H 2(Y, Z) was shown to give the remaining nal conformal field theory. Noting that I(A,B) ∼ ∧ delta function setting Wilson line of B zero. A possible Y B FA, we see that the U(1) current of the S-trans- ˜i = 1 i = intuitive picture on this may be the following. Con- formed theory that B couples is J 2π (FA) sider a non-zero 1-cycle γ on which there is a Wilson 1 ij k 4π (FA)jk.TheU(1) symmetry corresponding to i B line e γ ∈ U(1). We roughly consider Y as a prod- this current is the shift symmetry of dual photon scalar uct of γ and two-dimensional transverse space Σ,and of Ai. write I(A,B)as The definition of T operation is a little subtle, because it involves modifying a theory in a way which 1 I(A ,B)∼ B ∧ FA is not manifest in low energy action that is supposed 2π Y to define the theory. Concretely, the T operation is de- i 1 fined to shift the 2-point function of J by a contact ∼ FA · B ∼ n · B, (2.7) term, 2π γ γ Σ J i(x)J j (y) → J i(x)J j (y) 2 where n ∈ Z. Hence, summing over FA ∈ H (Y, Z) i ∂ + ij k δ3(x − y). (2.12) involves something like 2π ∂xk in· B e γ , (2.8) Because the above contact term has mass dimension 4, which is the right dimension of JJ correlation, this n∈Z term does not introduce any dimensionful coupling. i B which imposes vanishing Wilson line, e γ = 1. This Moreover, it does not conflict with any symmetry of argument is intended to be just illustrative, and we re- the theory (in some cases [16], we need this term to fer to Ref. [1] for rigorous derivation. preserve gauge invariance). In fact, whenever there is Now, we are ready to describe the SL(2,Z)actions freedom to add local contact terms that are consistent defined in Ref. [1] on three-dimensional conformal with the symmetry of a theory, this signals the intrinsic field theories with global U(1) symmetry. The defini- inability of our low energy action in predicting them, tion of a conformal field theory here means to specify andwehavetorenormalize them. In other words, they the global U(1) current J i and introduce a background must be treated as input parameters rather than out- i gauge field Ai without kinetic term that couples to J . puts. Note that this is not an unusual thing; it is an 142 H.-U. Yee / Physics Letters B 598 (2004) 139–148 essential concept of renormalization in quantum field Let X denote the bulk AdS, and ∂X = Y be our theory. The effect of the modification (2.12) on our space–time. Let A be the U(1) gauge field in the bulk generating functional (2.9) is whose boundary value couples to the global U(1) current J i in the CFT side. According to AdS/CFT, we exp i d3xA J i have i Y 3 i exp i d xAiJ → exp i d3xA J i i Y Y = DAexp iS(A) , (3.16) × i 3 ij k exp d x Ai∂j Ak , (2.13) Ai →Ai 4π Y 1 where S(A) = FA ∧∗FA +··· is the action of e2 X which can be shown by first expanding the exponent the bulk gauge field and we omitted other bulk fields in series of J and re-exponentiating the effects of T for simplicity. Before considering holographic dual operation on J correlation functions. of S operation, it is easy to identify from (3.16) the Another fact in Ref. [1], which is needed to show holographic dual of T operation as in Ref. [1].The the SL(2,Z) group structure of the above transforma- iI(A) T operation simply multiplies e in both sides of tions is, = 1 F ∧ F (3.16). But, note that I(A) 4π X A A modulo 2π irrespective of the bulk extension A as long as its DA iI(A) = , exp 1 (2.14) boundary value is fixed, hence in the right-hand side, iI(A) up to possible phase factor [2,3]. This equation should multiplying e is equivalent to shifting the bulk θ term, be understood as a statement that the theory has only one physical state and trivial [15]. Here, θ S(A) ⊃ FA ∧ FA, (3.17) 8π2 1 3 ij k I(A)= d x Ai∂j Ak X 4π by θ → θ + 2π. Y Now, using (3.16), we want to show that (2.11) is 1 4 ij kl ≡ d x Fij Fkl 16π nothing but the bulk path integral of the same bulk the- X ory, but with the boundary condition that the ‘dual’ 1 field B has the specified boundary value Bi .Interms = F ∧ F, (2.15) 4π of the original field A, this corresponds to specifying X electric field on the boundary, instead of specifying defined with some extension over X similarly as be- magnetic field. (When Bi = 0, the boundary condition fore [11]. This is well-defined modulo 2π for a spin in terms of A is that the electric field vanishes on the manifold Y .Using(2.1) and (2.14), it is readily shown boundary, as given in Ref. [1].) = 1 × that S and T satisfy the SL(2,Z) generating algebra, Using AdS/CFT and the fact that I(A,B) 2π 3 = 2 =− − 3 ij k (ST ) 1andS 1, where 1 is the transforma- Y d x Ai∂j Bk can be written as a bulk integral tion J i →−J i commuting with everything. (up to mod2π) 1 I(A,B)= FB ∧ FA, (3.18) 2π 3. Holographic dual of the S operation in X AdS/CFT where B and A are ‘arbitrary’ extensions of Bi and Ai, we have We now try to elaborate on the claim in Ref. [1] and to give an explicit proof that the S operation on 3 i i 3 ij k exp i d xAiJ exp d x Ai∂j Bk CFTs is dual to the Abelian S-duality in the bulk AdS 2π in AdS/CFT correspondence. Y Y H.-U. Yee / Physics Letters B 598 (2004) 139–148 143 i = DAexp iS(A) + FB ∧ FA , 2π A →A X i i (3.19) where B is some fixed extension of Bi . (2.11) is the integral of this quantity over the boundary value Ai , hence (2.11) is equal to the r.h.s. of (3.19) without any boundary conditions on A, 3 i Fig. 1. DA exp i d xAiJ i 3 ij k Thus, when expressed in terms of G, it gives the de- × exp d x Ai∂j Bk 2π sired delta function (up to a constant factor) that says G should be a field strength on X. Now, we can split = DA A + i F ∧ F exp iS( ) B A . V¯ on X¯ into a connection V on X and a connection V 2π X on X , and we have (3.20) We now perform a dualizing procedure in the bulk i ¯ i i F ¯ ∧ G = FV ∧ G + FV ∧ G X, which is similar to the one in Ref. [2], but appropri- 2π V 2π 2π ¯ ately taking care of the fact that our space–time now X X X has a boundary ∂X = Y . First we want to argue that, i i = FV ∧ G − FV ∧ G for a bulk 2-form field G, the integral 2π 2π X X i DV exp FV ∧ G (3.21) i 2π = (FV − FV ) ∧ G, (3.23) V→0 X 2π X over all possible connections V (and also sum over line where in the last line, we consider V as a connection bundles) in X with boundary condition that V vanishes on X, but with the minus sign in the integral due to on Y (up to gauge transformations), gives a delta func- orientation reversal. Note that V and V should agree tion on G that precisely says G is a field strength of on the boundary Y , as they are from a common V¯ on some connection of a line bundle. To show this, we ¯ ¯ ¯ X, hence we can rewrite the path integral over V into consider a “closed” 4-manifold X which is obtained from X by attaching on ∂X = Y a orientation reversed DV¯ = DV DV copy of X which we call X,asinFig. 1.Wealso . (3.24) consider a 2-form field G¯ on X¯ , whose value on X (V−V)→0 is the identical copy of G on X. It is clear that G is From the above two observations, we have a field strength of some connection on X if and only ¯ ¯ if G is a field strength of some connection on X.As ¯ i ¯ DV exp F ¯ ∧ G X¯ is closed, we can use the well-known procedure of 2π V ¯ ¯ requiring G to be a field strength [2]; the integral X i ¯ i ¯ = DV DV exp (FV − FV ) ∧ G DV exp FV¯ ∧ G (3.22) 2π 2π (V−V)→0 X X¯ i over connections V¯ on X¯ gives a delta function im- = DV · DV exp FV ∧ G , ¯ 2π posing that G is a field strength of some connection V→0 X on X¯ . Simply put, the integration over trivial part in V¯ (3.25) imposes that G¯ be a closed 2-form, while the remain- where we have changed the variable (V − V) → V in ing sum over line bundles requires G¯ to satisfy Dirac the last line. Thus, (3.21) indeed gives a desired delta quantization, G¯ ∈ H 2(X,¯ Z). function (up to a constant factor). 144 H.-U. Yee / Physics Letters B 598 (2004) 139–148 i Now, we are ready to perform the duality pro- × exp iS(G)+ (FB + FV ) ∧ G cedure in a space–time with boundary. Introduce a 2π X 2-form field G and replace every FA in the action F + V i with A G. Also introduce a connection with the = DV DG exp iS(G)+ FV ∧ G boundary condition that V vanishes on Y , and add the 2π V→B X coupling i = DV F i exp iSD( V) , (3.30) FV ∧ G. (3.26) V→ 2π Bi X where in the fourth line, we changed the variable B + The resulting action is invariant under the extended V → V with the new boundary condition that V goes gauge transform, to the specified Bi on Y . In the last line, integrating over G gives the dual bulk action SD(FV ) in terms A → A + C,G→ G − FC, (3.27) of the dual gauge field V with the coupling constant where C is an arbitrary connection in X. Precisely be- −1/τ , and we have the desired boundary condition for cause V vanishes on Y , (3.26) is invariant under (3.27) V on Y . modulo 2πi. Let us explain this fact in some detail. At this point, it would be clarifying to see explic- The vanishing connection on Y can be extended to a itly the relation between the boundary condition for trivial (globally defined one form) connection on X, the dual field V that we derived above, and the bound- say V. We also know that ary condition in terms of the original field A [1].Inthe bulk AdS, the dual field V is nothing but a non-local i i change of variable from the original variable A.Inthe FV ∧ FC = FV ∧ FC, (3.28) 2π 2π case of vanishing θ angle,1 they are related by X X 2 modulo 2πi because V and V agree on Y .Beingtriv- F = e F αβ ( A)µν µναβ ( V ) . (3.31) ial, FV can be written as FV = dV globally on X, 8π and performing partial integration, we have This is easily seen in a naive dualization procedure of making the field strength of A as a fundamental inte- 1 1 FV ∧ FC = V ∧ FC = 0, (3.29) gration variable by imposing the Bianchi identity. The 2π 2π V X Y dual field is introduced as a Lagrange multiplier V √ because vanishes on Y . 4 1 µν DV DFA exp i d x g (FA)µν(FA) We then consider G and V as dynamical, and mod 2e2 out the theory with gauge equivalence. If we integrate 1 µν αβ over V first, it gives a constraint that G is a field + µναβ (FV ) (FA) . 8π strength of some connection C by the discussion in the previous paragraphs. Then, by gauge fixing, we (3.32) can set G = 0 and recover the original theory of A. Integrating out FA gives the dual description. The The equivalent dual theory in terms of V is obtained equation of motion of FA is (3.31). A = by first gauge fixing 0, and integrating over G. Now, in Poincaré coordinate (x0, x) , (with the Applying this to (3.20),weget boundary at x0 = 0) i 2 2 DA F + F ∧ F dx + dx exp iS( A) B A ds2 = 0 , (3.33) 2π x2 X 0 = DV DG 1 This is just for simplicity. The case with non-vanishing θ angle V→ 0 is similar [2]. H.-U. Yee / Physics Letters B 598 (2004) 139–148 145 the usual boundary condition specifying the value of S and T action on boundary conditions as gauge field on the boundary corresponds to spec- D 0 −1 D ifying the (gauge invariant) ‘magnetic’ component S: i → i , = 1 jk = Mi 10 Mi Mi 2 ij k F , i, j, k 1, 2, 3. Because (3.31) interchanges the ‘magnetic’ component of V with the D 11 D T : i → i , (4.35) ‘electric’ component of A, Ei = ∂0Ai (in the gauge Mi 01 Mi A = 0), we see that in terms of the original field A, 0 where we take the usual ‘magnetic’ boundary condi- the S-transformed CFT is mapped to the bulk AdS tion in terms of transformed variable. This gives a nat- theory with ‘electric’ boundary condition. Note that ural correspondence between the SL(2,Z) action on ‘magnetic’ and ‘electric’ boundary conditions are nat- 3D CFTs with the SL(2,Z)action on boundary condi- ural counterparts of Dirichlet and Neumann bound- tion (or bulk gauge field). ary conditions for scalar field, and they are natu- In this section, we give another concrete evidence rally expected to be conjugate with each other in of this picture in the context of the recently proposed AdS/CFT. We will come to this point in the next sec- prescription [17] on boundary conditions when we de- tion. In fact, we need to look at the T -transformation form the boundary CFT in a non-standard manner. The more carefully in this respect. The AdS dual of the proposed prescription in Ref. [17] is for scalar fields T -transformation of 3D CFT was identified as a 2π in the bulk, and it goes as follows. Suppose we have a shift of the bulk θ-angle, while the ‘magnetic’ bound- scalar field φ in the bulk, whose asymptotic behavior ary condition is unchanged. In the presence of θ-angle, near the boundary x = 0is the ‘electric’ component naturally conjugate to the 0 ‘magnetic’ component (or more precisely, the value A ∆+ ∆− i φ(x0, x) ∼ A(x)x + B(x)x . (4.36) on the boundary) has a term proportional to θ-angle. 0 0 This is most easily seen from the fact that the natural We consider the CFT on x0 = 0 defined by the bound- = conjugate variable to Ai is obtained by varying the ac- ary condition A(x) 0. By standard AdS/CFT dictio- 2 O tion w.r.t. ∂0Ai . Denoting this as Di ,wehave nary, A(x) couples to a scalar operator of dimension ∆+ on the boundary, that is, a boundary condition with 1 θ non-vanishing A(x) maps to a deformation of CFT D = ∂ A + M , (3.34) i 2 0 i 2 i side by e 8π and shifting θ results in shifting of Di by a unit of SCFT → SCFT + dxA( x) O(x). (4.37) ‘magnetic’ component Mi . The expectation value of O in this deformed CFT is given by B(x) ,

4. In view of boundary deformations O ∼ (x) A B(x). (4.38) In the last section, we observed that the AdS dual They are natural conjugate pair of source and expec- of S-operation on 3D CFT interchanges the ‘magnetic’ tation value. For specific range of φ mass, it is pos- and ‘electric’ boundary conditions, while T -operation sible also to consider the boundary CFT defined by B(x) = 0. In this CFT, the roles of A(x) and B(x) are corresponds to shifting the ‘electric’ component Di by a unit of ‘magnetic’ component. Though T -operation reversed, and the partition function is just a Legendre does not really change the boundary condition by it- transform of the previous CFT [18]. It has been ar- self, it has a non-trivial effect when combined with S. gued that this CFT is the IR fixed point of the previous O With appropriate normalization, we can represent the CFT deformed by a term which is quadratic in (x). The question is what would be the boundary condition when we deform the boundary CFT (defined by 2 A(x) = 0) in a more general manner, If we consider x0 as a time variable, this is the Witten effect. This should be true even in Euclidean case when considering ‘natu- → + O rally’ conjugate boundary variables on x0 = 0. SCFT SCFT W( ), (4.39) 146 H.-U. Yee / Physics Letters B 598 (2004) 139–148 where W is an arbitrary (possibly non-local) function Having this in mind, let us go back to our SL(2,Z) of O(x) . The proposal in Ref. [17] is to take the fol- actions on 3D CFT and consider the action given by lowing boundary condition on A(x) and B(x) , ST n, which corresponds to the matrix, δW(O) 0 −1 1 n 0 −1 A(x) = . = . (4.46) O (4.40) δ (x) O(x) →B(x) 10 01 1 n

The situation with bulk gauge field in AdS4/CFT3 By definition, the partition function of the transformed correspondence might look similar to the case of bulk CFT is given by scalar field with its mass such that two CFTs are 3 i possible. We have two naturally conjugate variables Zafter = DBi exp i d xBi J (Ai,Di ) in the boundary, and two different bound- i 3 ij k ary conditions are possible. However, contrary to the + n · d x Bi ∂j Bk , scalar field case, these two boundary CFTs cannot pos- 4π sibly be related by an RG flow, because acting S twice (4.47) gives us the original theory and the degrees of freedom where ··· and J i are expectation values and U(1) are not lost. current, respectively, of the original CFT, and Bi is the However, the proposal (4.40) can be naturally ex- intermediate connection variable in defining the S op- tended to include the case of gauge fields, and we will eration. Because the exponent is quadratic in Bi ,we show that this extension indeed reproduces the results can perform the path integral over Bi explicitly. We 3 2 in the previous sections, providing a compelling check introduce the gauge fixing term i d xξ(∂i Bi) ,and for the proposal applied to gauge fields. We start with the Bi propagator is the CFT defined by the usual ‘magnetic’ boundary ≡ − condition specifying gauge field components tangen- Sij (p) Bi (p)Bj ( p) tial to the boundary as x0 → 0, pi pj 2π k = i + ij k p . (4.48) 2ξ(p2)2 np 2 Ai → Ai,i= 1, 2, 3. (4.41) Using this, (4.47) is given by As usual, this corresponds to deforming the CFT by 3 adding the coupling, 1 d p i j Z = exp − J (−p)Sij (p)J (p) after 2 (2π)3 = 3 i 3 k δSCFT d xAiJ , (4.42) π d p ij k p = exp − J i(−p) J j (p) , n (2π)3 p2 where J i is the 3D U(1) current. Now, we ask the question of what boundary condition we take when we (4.49) − deform the CFT with an arbitrary function of J i(x) , where J i(p) ≡ d3xe ipx J i(x), and we have used i the Ward identity pi J (p)...=0. Now, looking at = i δSCFT W J , (4.43) the last expression, it is clear that the transformed the- instead of a linear one (4.42). Recalling that ory is nothing but the original CFT with the deformation W[J ] given by δSbulk Di (x) = (4.44) 3 k A iπ d p i ij k p j δ∂0 i (x) δSCFT = W[J ]= J (−p) J (p) n (2π)3 p2 is the “electric” field that is canonically conjugate A iπ d3p to i (x), a direct analogy with the case of scalar = d3x n (2π)3 fields (4.40) suggests the following prescription on the boundary condition as x0 → 0, pk × d3yeipx e−ipy ij k J i(x)J j (y). δW[J ] p2 = (4.50) Ai(x) i , (4.45) δJ (x) i i J (x) →D (x) Hence, according to our proposal (4.45),thebulkAdS where Ai → Ai . gauge theory of the transformed CFT has a modified H.-U. Yee / Physics Letters B 598 (2004) 139–148 147 boundary condition, before taking the usual ‘magnetic’ boundary condition M = 0. 2πi d3p i Ai(x) = n (2π)3 k − ij k p Acknowledgements × d3yeipx e ipy J j (y) . 2 p J j (y)→Dj (y) We would like to thank Edward Witten for help- (4.51) ful comments, Kimyeong Lee, Kyungho Oh, Jae-Suk To see clearly what this means, take the x-derivative, Park and Piljin Yi for helpful discussions. This work is 1 mni ∂ 2 n, on both sides. The left-hand side gives partly supported by grant No. R01-2003-000-10391-0 m = 1 mni the ‘magnetic’ component M (x) 2 ∂nAi(x), from the Basic Research Program of the Korea Sci- while the right-hand side becomes ence & Engineering Foundation. 3 π d p 3 ipx −ipy mni − d ye e ij k n (2π)3 References p pk × n j 2 J (y) p j → j [1] E. Witten, hep-th/0307041. J (y) D (y) 3 [2] E. Witten, Selecta Math. 1 (1995) 383, hep-th/9505186. =−π d p 3 ipx −ipy m n − m n [3] C. Vafa, E. Witten, Nucl. Phys. 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