On boundary correlators in Liouville theory on AdS2

We consider the Liouville theory in fixed Euclidean AdS2 background. Expanded near the minimum of the potential the elementary field has mass squared 2 and (assuming the standard Dirichlet b.c.) corresponds to a dimension 2 operator at the boundary. We provide strong evidence for the conjecture that the boundary correlators of the Liouville field are the same as the correlators of the holomorphic stress tensor (or the Virasoro generator with the same central charge) on a half-plane or a disc restricted to the boundary. This relation was first observed at the leading semiclassical order (tree-level Witten diagrams in AdS2) in [19] and here we demonstrate its validity also at the one-loop level. We also discuss arguments that may lead to its general proof.


Introduction
Study of quantum field theories in AdS 2 background is of interest from several points of view (see, e.g., [1][2][3][4][5][6] and references there). A recent example appeared in the investigation of correlators of operators inserted on a straight or circular Wilson line in the context of AdS 5 /CFT 4 correspondence [7][8][9][10][11][12][13][14]. As discussed in [12], starting with the string action in AdS 5 × S 5 and expanding it near the corresponding minimal surface one gets a 2d field theory action in AdS 2 background. The boundary correlators of elementary excitations that should match the strong coupling limit of the Wilson loop "defect" correlators in N = 4 super Yang-Mills theory are constrained by the isometry of AdS 2 or the 1d conformal group SO (2,1) SL(2, R) providing a non-trivial example of a non-gravitational AdS 2 /CFT 1 duality. One technical problem in the approach of [12,14] is how to systematically compute loop corrections to boundary correlators of 2d quantum fields in AdS 2 .
Using this as a partial motivation, it is interesting to study loop corrections to boundary correlators in some simple 2d QFTs defined on AdS 2 background. Here we shall consider the familiar example of the Liouville theory [15][16][17] S = 1 4π d 2 x √ g ∂ a ϕ∂ a ϕ + µ e 2 b ϕ + Q R ϕ , (1.1)

JHEP07(2019)008
The tree-level relation (iii) was checked in [19] while the non-trivial one-loop relation (iv) will be confirmed below. 5 Our one-loop results will thus provide a strong evidence for the validity of the conjecture (1.3). One may wonder how it can be proved in general. The conformal symmetry of the Liouville theory in the bulk (on half-plane) viewed as an infinite global symmetry extending the SL(2, R) isometry of the AdS 2 implies strong constraints on the boundary correlators which are effectively equivalent to the Virasoro symmetry constraints on the form of the stress tensor correlators (1.4), (1.5). This suggests that here we get an effective CFT 1 which is simply a restriction of a CFT 2 defined on a half-plane to its real-line boundary so that in this case the AdS 2 /CFT 1 becomes effectively "AdS 2 /(CF T 2 ) 1/2 duality". 6 In the semiclassical limit of small b the relation (1.3) may then be argued for as follows. Since the theory (1.1) is Weyl-covariant theory in the bulk, in this limit we may eliminate the conformal factor of the AdS 2 metric by redefining the Liouville field as ϕ(t, z) → φ(t, z) + b −1 log z, i.e. transform the action in (1.1) into the flat-space one (cf. [2,5]). This gives the Liouville CFT for field φ defined on a flat upper half-plane (w = t + i z, z > 0) with the (classical) stress tensor ). The shifted field has then the boundary asymptotics φ(t, z) z→0 = z 2 Φ(t) − b −1 log z + . . . . Taking the boundary limit in T (w → t) gives This is precisely the operator relation which is required for the validity of (1.3) with κ(b) = − 2 3 b + · · · . The same value of the leading term in κ(b) in (1.3), (1.7) will be found by an explicit computation below (see (3.22)).
This argument should admit an extension to the full quantum level, i.e. to all orders in b, allowing to determine an exact expression for κ(b). One important point is the role of the boundary limit: the relation (1.3) between the elementary Liouville field correlators and the stress tensor correlators may hold only at the boundary. A general proof should use the boundary conformal field theory considerations. In particular, the completeness of the OPE in the boundary CFT (cf. appendix D) implies that it contains only the boundary stress tensor and its descendants; since the boundary operator dual to the m 2 = 2 Liouville field in AdS 2 should have dimension 2 it can only be (proportional to) the stress tensor itself. Considering the OPE of the stress tensor in the bulk and the Liouville field at the 5 As a technical remark, let us note that the relations in (1.9) and in (1.15) are consequences of the main identification (1.3) only. As such they are independent of a particular regularization scheme that one is adopting, provided conformal invariance is respected. In particular, (1.9) and (1.15) will be valid for both schemes that we shall propose in the main text. 6 This may be compared to the discussion of the conjectured duality between a gravitational theory in AdS2 and a chiral half of a 2d CFT in [20]. While here we assume that the AdS2 background is fixed and the Virasoro symmetry is not gauged, in the gravitational context the 2d diffeomorphisms is a gauge symmetry and the Virasoro symmetry (with c = 0 due to ghosts, etc.) was assumed to be an asymptotic symmetry (also, the Liouville field in [20] was a gravitational mode). Whether this symmetry is spontaneously broken to SL(2, R) appears to depend on a particular model in question (cf. [21][22][23]).

JHEP07(2019)008
boundary should then fix the value of the proportionality constant κ(b) or C 2 (b) in the two-point function in (1.6). 7 A possible generalization of the above semiclassical argument is as follows. Starting with the Liouville theory on a complex plane, the (normal ordered) operator V a = e 2 a φ is a primary with the conformal dimension ∆ = a (Q − a), i.e. we have On the upper half-plane (w = t + iz, z > 0) the boundary conformal Ward identity takes into account also the poles at w = w . Including them in the OPE of T and φ and assuming the exact form of the boundary asymptotics to be 8 (1.18) one finds that the leading 1 (t−t ) 2 term in T (t)φ(t, 0) cancels and we get 9 (1.19) Taking the expectation value of (1. 19) and comparing the result with (1.4), (1.6), (1.7) we get the following prediction for the exact value of the coefficient κ(b) between Φ and T : (1.21) The first two terms in this expansion match precisely with the above semiclassical value and the one-loop value that we shall find below in (3.24). Given (1.20) we get from (1.7) 22) which are in agreement with (3.25). A natural generalization of the above duality (1.3) for the Liouville theory is to all conformal Toda theories for finite Lie algebras [24][25][26] defined on the AdS 2 background. For instance, in the A n case, expanding near the minimum of the Toda potential in AdS 2 one finds n scalar fields ϕ ∆ with masses m 2 = ∆(∆ − 1) corresponding to ∆ = 2, . . . , n + 1 [19]. The duality relation extending (1.3) is then 7 We are grateful to Simone Giombi and Xi Yin for useful suggestions on this problem. 8 This generalizes the classical asymptotics where Q → b −1 in a way which is consistent with the value of the one-loop tadpole (see [5] and (2.16)). 9 In more details, eq. (1.19) is the coefficient of z 2 term in the z → 0 expansion of c where the second term comes from the −Q log z term in (1.18) and the complex conjugate contribution comes from poles at w = w .

JHEP07(2019)008
where Q ∆ = {Q 2 ≡ T, Q 3 , . . . , Q n+1 } are the generators of the chiral W n+1 algebra generalizing the Virasoro symmetry and having the same central charge as the Toda theory in the bulk. The relation (1.23) was noticed at the tree level in [19] by computing few sample 4-point functions in the Toda theories associated to some rank-2 algebras with two scalar fields (one dual to the stress tensor T and the other dual to a higher spin current Q s ). It is natural to expect (1.23) to hold also at the quantum level as should be possible to check by the methods used in the present paper. Another interesting generalization is to the super-Liouville theory on AdS 2 , cf. [27]. In this case one will need to compute loop corrections for both the bosonic and fermionic fields.
The structure of the rest of this paper is as follows. In section 2.1 we will discuss two alternative formulations of the Liouville theory on AdS 2 . One starts with the flat space action expanded around the classical solution corresponding effectively to the AdS 2 background while the other starts directly with the Liouville theory defined on fixed AdS 2 . The two formulations differ in the choice of the regularization of the short distance propagator and also in the coefficient in front of the potential. They turn out to give the equivalent results for the physically relevant connected one-loop correlators.
In section 2.2 we will compute the two-point function of the Liouville field at the one-loop order demonstrating the consistency of its boundary limit with (1.6). The oneloop corrections to the boundary three-point and four-point functions will be computed in sections 3 and 4 respectively finding again the agreement with (1.6) and (1.13). As a result, we will provide a strong support for the validity of the duality (1.3) at the loop level. Appendix A will present some details of the diagram computations on the Poincaré disc. Comments on tadpole diagrams (that lead to different predictions for the 1-point functions in the two formulations) will be made in appendix B. In appendix D we will review the structure of the 1d conformal block expansion of the four-point function in (1.5).

Liouville theory on Euclidean AdS background
The correlators in the Liouville theory (1.1) in AdS 2 background may be computed using two alternative approaches. In the first (the "ZZ formulation" [5]) one starts with the Liouville action on a flat upper half plane (or flat disc) and expands it near a non-trivial non-constant solution [1,2] preserving the SL(2, R) symmetry. In the second (the "AdS formulation") one starts directly with the Liouville action (1.1) in the AdS 2 background and expands near the constant minimum of the curved-space potential. 10 The two approaches are classically equivalent by a field redefinition but imply the use of different regularizations at the quantum level.

Two alternative formulations
The flat-space action 11 admits a particular solution Expanding near this "AdS 2 vacuum", ϕ = ϕ (0) + χ, we find the following action for the fluctuation χ This action is the same as the classical action ( ). An alternative choice is the half-plane parametrization with coordinates t ∈ R, z ≥ 0 and the metric ds 2 = 1 z 2 (dt 2 + dz 2 ). The two are related by the standard conformal map The analog of the action (2.3) on the Poincaré plane reads As follows from (2.3) or (2.5) the field χ has the mass m 2 = ∆(∆ − 1) = 2. Assuming the standard Dirichlet boundary condition (corresponding to the ∆ = 2 choice) the canonically normalized AdS 2 bulk-to-bulk propagator is given by and u is the invariant chordal distance. Equivalently (taking into account the factor 1 2π in front of the action in (2.3), (2.5)) the free two-point function of the field χ is given by

JHEP07(2019)008
Taking one point to the boundary we get the corresponding bulk-to-boundary propagator g b written in either disc or plane parametrization 12 where the relation between the Poincaré plane and disc boundary coordinates t and θ is (2.10) The starting point of the AdS formulation is the action (1.1) in unit-radius AdS 2 background (R = −2). Shifting the field by a constant as ϕ = ϕ (0) + χ where µe 2bϕ (0) = Q b we get the following analog of (2.3) The two coincide in the semiclassical limit of small b when The two formulations will differ also by the required choice of the UV regularization of the propagator (2.7) at the coinciding points but the two differences will happen to compensate each other leading to the equivalent results for the relevant physical correlators (but not for the tadpole values). The ZZ formulation that starts with the flat-space action requires the use of the simple flat-space regularization z − z → z − z + ε. Omitting the singular log ε term this is equivalent to the subtraction of the −[log(z − z )] z→z term from (2.7) [5]. At the same time, the regularization in the AdS formulation should be the AdS 2 covariant one with η → η + ε or u → u + ε in (2.7), (2.8). In general, the value of the propagator at the coinciding points may be parametrized as Here in (2.14) we assumed that the logarithmic UV divergences are cancelled by the mass renormalization with the "minimal" subtraction (more generally, one may keep the constant q AdS 2 arbitrary). Starting with (2.3) or (2.11) one may compute the corresponding quantum corrections to bulk correlators in perturbation theory in b. The two actions differ in the coefficient in front of the potential term implying that in the AdS formulation one has extra higher order terms in the χ 2 , χ 3 , . . . vertices. In addition, there is a difference in the regularization choices in (2.13), (2.14). 12 The factor 4 3 is the same as 2 π C2, where C2 appeared in (2.6).

JHEP07(2019)008
While the values of the one-point functions χ n (z) computed in the ZZ and AdS formulations will differ, the connected correlators at separated points are expected to be same; we shall find evidence for that below.

One-point and two-point correlation functions
Let us consider the leading order corrections to the tadpole or the one-point function χ(z) .
Computing it for generic choice of the regularization (2.12) one finds (see (A.7), The leading-order tadpole thus vanishes in the AdS formulation and is constant in general, as required by the AdS symmetry (under which χ transforms as a scalar). Explicitly, one finds Note that combining the one-loop tadpole with the classical solution (2.2) the z-dependent part of the tadpole in the ZZ case gets coefficient in agreement with the results of [5] and [29] (see also appendix B). 13 Next, let us consider the one-loop order b 2 correction to the connected part of the two-point function χ(z)χ(z ) . In the AdS formulation one needs to take into account an extra contribution coming from the b 2 χ 2 term in the potential in (2.15). The final result in both formulations can be written as where (see (A.11), (A.15), (A.17)) 20) and the coefficient q 1 = q 1 − 1 in the ZZ case and q 1 = q 1 in the AdS case. As follows from (2.13), (2.14) in both cases and so we get the equivalent result for (2.18) given by the sum of the free propagator in (2.7) and the "self-energy" correction in (2.20). 13 The tadpole (2.17) is the simplest cumulant that can be extracted from the expectation value of the exponential vertex operator e α χ(z) determined in [5]. For a perturbative discussion of the O(b 3 ) corrections to higher cumulants and a comparison with the bootstrap predictions see [29].
(2.23) Thus there are no b 2 log(t − t ) corrections, i.e. the boundary operator dual to χ has no anomalous dimension confirming the expected behaviour (1.6). We conclude that the coefficients in the perturbative expansion of C 2 introduced in (1.8) are thus . (2.24) Let us note that it is only the boundary limit of the 2-point function (2.18) that takes the simple rational form in (2.23) matching the stress tensor two-point function (1.4) according to (1.3). The same will apply to higher-point correlation functions.

Three-point boundary correlation function
Next, let us compute the tree-level and one-loop contributions to the connected boundary three-point function. We shall use mainly the ZZ formulation, but will comment on its equivalence with the AdS one. The relevant terms in the expansion of the potential term in (2.3) will be (3.1) We will need the tree boundary-to-bulk propagator in the disc parametrization given in (2.9) and also will use the following notation for the conformal prefactor in the threepoint function (1.6) (cf. (2.10)) We shall divide the relevant Witten diagrams in AdS 2 by the overall conformal factor K 3 , i.e. the resulting expressions will represent the contributions to the coefficients in C 3 (b) in (1.6). The tree level contribution to the three-point function is given by the diagram (3.3)

JHEP07(2019)008
The same result in the Poincaré plane coordinates follows from (see, e.g., [30] Thus the value of the leading coefficient in C 3 in (1.6), (1.8) is which together with C 2,0 in (2.24) is in agreement with the first relation in (1.9). The value (3.5) combined with (2.24) and the second relation in (1.9) gives the following prediction for the one-loop coefficient C 3,1 in C 3 (b) in (1.8) This coefficient should be given by the sum of the contributions of the four different types of diagrams that we shall consider below.
Diagrams with dressed propagators. There are three tree-like diagrams with the one-loop self-energy correction to one of the propagators 3,1 b 3 = . (3.11) In the AdS formulation we get the same result (by a mechanism similar to the one in the case of the two-point function (2.18)-(2.21)). Here the tadpole contributions vanish (g(z, z) = 0 in (2.14)) but the cubic vertex ∼ b χ 3 in (3.1) is rescaled by b Q = 1 + b 2 (see (2.15)), i.e. −8bχ 3 → −8bχ 3 − 8b 3 χ 3 resulting in an additional one-loop correction. Thus (3.14) where 1 2 is the leg symmetry factor. Part of this diagram (the loop with upper leg and integration over the cubic vertex point) is the same as the function B(z 2 , z 1 ) = d 2 z g(z , z 1 ) g(z 2 , z ) 2 in (A.11), (A.14). Taking the point z 1 to the boundary then gives, according to (A.14), the boundary-to-bulk propagator (2.9) times the 1 8 factor This should be multiplied by the coupling constants from the two vertices (−8b)(−16b 2 ), by a symmetry factor 1 2 and by the diagram multiplicity factor 3. Compared to the tree diagram contribution this gives an extra −3 b 2 factor in total, i.e. Diagram with one-loop vertex correction. The most complicated diagram is the one with the three χ 3 vertices and thus with the three bulk-to-bulk and the three bulk-toboundary propagators: (3.17) As this is the triple AdS 2 integral its analytic computation is non-trivial. According to (3.6), (3.7) and the results in (3.9), (3.12), (3.16) the expected value of C 3,1 it is sufficient to evaluate it numerically. We did this by using the Suave routine of the Cuba library [31]. One important point is to check that C (4) 3,1 given by (3.17) is indeed independent of the boundary coordinates θ 1 , θ 2 , θ 3 . From a practical perspective, one is interested in the θ i dependence of the systematic error in the numerical estimate at fixed computational effort (the total number of integrand evaluations). In figure 1 we fixed the two points at θ 1 = 2π 3 , θ 2 = − 2π 3 and varied θ 3 showing the resulting numerical estimate for C 3,1 . The horizontal line is the expected value − 56 27 in (3.18). One finds that the integral value is more "flat" when the three external points θ i are well separated. This is expected as the integrand has a more singular behaviour when at least two external points get close. Considering the equally spaced points and increasing statistics we found the best estimate to be 16 C (4) 3,1 = −2.08 (2) . (3.19) This is in good agreement with (3.18) or − 56 27 ≈ −2.0741. 16 The number in the parenthesis is the statistical uncertainty (the least significant digits of a given numerical value for the average). Here the uncertainty is the standard deviation of the Monte-Carlo evaluation of the loop integral. Explicitly, 2.08(2) ≡ 2.08 ± 0.02.

JHEP07(2019)008
The expected value (3.6) and thus (3.18) was found from (1.9) which was based on matching the two-point and three-point correlators (1.2) with the stress tensor ones (1.4) for the Liouville central charge value c = 1 + 6(b + b −1 ) 2 . Alternatively, we may determine the values of c and κ in (1.7) from the explicit tree plus one-loop results for C 2 and C 3 found above. Omitting the overall conformally covariant factors we may symbolically summarize the results for the two-point and three-point correlators in (2.23) and (3.5), (3.9), (3.10), (3.16), (3.19) as follows implies that Using the numerical estimate for C 3,1 found in (3.19) gives which is in good agreement with the value of the Liouville central charge c = 6 b −2 +13+6 b 2 . Note also that for C 3,1 = − 56 27 corresponding to the exact value of c we get 24) and thus the following expressions for C 2 (2.23) and C 3 in (1.6), (1.8)

Four-point boundary correlation function
The four-point function (1.11) receives contributions from the disconnected and connected diagrams in AdS 2 . The disconnected part is given by Φ(t 1 ) Φ(t 2 ) Φ(t 3 ) Φ(t 4 ) (+ permutations) corresponding to (1.12). From (3.21), (4.1) the prediction for the non-trivial connected part is , (4.1) implying that in (1.14) one should have These values are in agreement with the relations in (1.15) for the coefficients C 2,0 = 4 3 , C 2,1 = − 2 9 , C 3,0 = − 16 9 , C 3,1 = 64 27 found above. The value of the leading coefficient C 4,0 follows from the sum of the tree-level contact diagram and the three exchange diagrams as in (1.10) (see appendix C) The one-loop coefficient C 4,1 is determined from the sum of the contributions of the five classes of diagrams To compute them we shall use the AdS formulation in which there are no tadpoles but the potential has an extra factor (1 + b 2 ) in (2.15) rescaling the vertices. C 4,1 will denote the resulting contribution originating, due to this rescaling, from the contact and exchange tree diagrams in (4.4). Since the contact 4-vertex is rescaled by 1 + b 2 while the exchange diagrams with two 3-vertices by (1 + b 2 ) 2 the two are combined now with the relative factor of 2 (cf. (4.4)) The contributions C (i) 4,1 with i = 1, 2, 3, 4 in (4.5) come from the genuine one-loop diagrams described below.
In writing (4.5) and (4.6) we formally assumed that t i -dependence of each individual diagram contribution is given by the same simple conformal factor as in (4.1) (cf. , t ij = t(θ i ) − t(θ j ), (4.7) However, this need not be true in general. Let us first recall that the tree level fourpoint boundary correlator (1.10) has the canonical form in (1.12), (1.13) without any logarithmic terms depending on the 1d invariant cross-ratio χ = t 12 t 34 t 13 t 24 . Such terms cancel between the contact and exchange diagram contributions [19] (see appendix C) but they survive if the contact and exchange contributions are combined with a different relative coefficient like in (4.6) (see (C.2)). These extra terms should still not appear in the total expression (4.1) for the one-loop corrected correlator, i.e. they should cancel against similar "extra" contributions of other diagrams. Checking this directly would require the analytic computation of all the diagrams discussed below which we will not attempt here.
Here we will compute the total value of the coefficient C 4,1 numerically by fixing particular values of the boundary coordinates t i . We will also give the values of the individual diagram contributions C (k) 4,1 defined formally as the corresponding integrals divided by the factor K 4 in (4.7).

JHEP07(2019)008
Diagrams with dressed external propagators. These diagrams are analogous to the ones in (3.8) (their contribution is actually proportional to K 4 (θ)). As in (3.8), (3.9) the one-loop dressing of the bulk-to-boundary propagators gives the contribution Diagrams with dressed internal propagator.
Here the grey circle stands for the one-loop self energy correction Σ(z, z ) in (2.20) or explicitly Diagrams with one-loop 1PI cubic vertex correction.

JHEP07(2019)008
The integrals corresponding to these diagrams are ×g(z ,z ) g(z ,z ) g(z ,z ), (4.14) We evaluated C (i) 4,1 by the numerical integration for the boundary point configurations where the four θ i are almost equally spaced. As was discussed above in the case of the threepoint function, such choice is expected to minimize the systematic numerical error. The integrator routine is again chosen to be Suave. The most challenging is the computation of C  963 . . .. The relative error is thus below the 1% level and within the statistical uncertainty of the multidimensional integration. This very good agreement with the prediction (4.3) effectively confirms our assumption that like the tree-level one, the total one-loop correlator has simple dependence on t i as in (4.1).

JHEP07(2019)008
Acknowledgments We are grateful to Simone Giombi, Erik Tonni and Xi Yin for very useful discussions. AAT was supported by the STFC grant ST/P000762/1. AAT also thanks the Galileo Galilei Institute for Theoretical Physics for hospitality and INFN for partial support during the completion of this work.

A AdS 2 integrals in disc parametrization
Below we collect some results about the SU(1, 1) covariant integrals over the unit disc that were used in the main text. We shall use the invariant measure and the propagator g(z, z ) defined in (2.7) with g(z, z) parametrized as in (2.12). A very useful representation of the propagator is the Fourier expansion derived in [29]. Denoting by ϑ(z, z ) the angle between the disc points z, z , one has where θ(y − x) in (A.3) is the step function.

A.1 Tadpole reduction identity
Let us consider the integral (using (2.12)) Substituting (A.2) the only contributing term is n = 0. After integrating over the angle and setting |z | = r we have (A.6)

JHEP07(2019)008
Integrating over x = r 2 gives Combining (A.5) and (A.7) we get a simple relation (which follows also from the defining equation for the propagator) It may be used to perform the following reduction of the tadpole diagrams (special cases of which are mentioned in [29]) Here dots stand for N external lines and the crossed vertex in the r.h.s. corresponds to the interaction vertex ∼ χ N . Its precise coefficient may be determined as follows. The explicit form of the l.h.s. of (A.9) is where we used (A.8) and took into account the 1 2 symmetry factor due to the tadpoles. In the ZZ formulation we have q ZZ 1 = 1 (see (2.13)) and thus the r.h.s. corresponds to (2b) N χ N N ! , i.e. b 2 times the interaction term in the ZZ Lagrangian (2.3). Thus the one-loop tadpoles in the ZZ formulation give the same contribution as the extra couplings in (2.15) in the AdS formulation.

D Global conformal block decomposition of the four-point correlator
The four-point correlator in (1.11)-(1.13) has a simple dependence on the boundary points without any logarithmic terms of the 1d cross ratio χ. This means that the 1d conformal operators appearing in the OPE of the boundary correlator (1.11) will have no anomalous dimensions (cf. [12,14]). The same structure is fixed by the Virasoro symmetry in the case of the stress tensor correlator (1.5). In view of this connection it is of some interest to discuss the equivalent OPE decomposition of the stress tensor correlator in terms of the conformal blocks of the global part of the 2d conformal symmetry (which is effectively equivalent to 1d conformal block decomposition).

JHEP07(2019)008
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.