Non-commutative $AdS_2/CFT_1$ duality: the case of massive and interacting scalar fields

We continue the study of the nocommutative $AdS_2 / CFT_1$ correspondence. We extend our previous results obtained for a free massless scalar field to the case of a massive scalar field. Both the free and interacting cases are considered. For both cases it is confirmed that to the leading order in noncommutative corrections the 2- and 3-point correlation functions have the form that is assumed by some (yet unspecified) dual $CFT$. We also argue that there does not exist a map which connects the commutative model to its non-commutative counterpart, and therefore the conformal behaviour of the noncommutative correlators is a non-trivial result.


Introduction
In a recent paper [1] aspects of the AdS 2 /CF T 1 correspondence were studied in a noncommutative setting, namely when the geometry on the gravity side of the correspondence is replaced by the non-commutative version of two-dimensional anti-de Sitter space (AdS 2 ). The motivation for making the AdS 2 space non-commutative is to include some quantum gravitational corrections, since there is a general belief (supported by multiple arguments) [2] that the quasiclassical regime of quantum gravity should appear as a quantum field theory on some non-commutative background. The introduction of noncommutativity on the AdS 2 background can be made unique by demanding that it preserves the SO(2, 1) isometry group. [3], [4], [5], [6], [7], [8] In [1], analogues of AdS 2 Killing vectors generating SO (2,1) were constructed on the noncommutative space, denoted by ncAdS 2 . Such ncAdS 2 Killing vectors are deformations of the AdS 2 Killing vectors K µ , yet, due to the unique introduction of noncommutativity, they generate the undeformed isometry group, i.e., SO(2, 1). These deformations were shown to vanish in the boundary limit, and so the ncAdS 2 Killing vectors asymptotically tend towards K µ . In other words, ncAdS 2 , is asymptotically AdS 2 . The AdS/CF T correspondence principle posits a weak/strong duality between the quantum gravity in the bulk of an asymptotically AdS space and a conformal field theory (CFT) on the boundary of this space. Baring the known difficulties of the correspondence principle for two dimensional anti-de Sitter space (see for example, [9], [10]), the result that ncAdS 2 , is asymptotically AdS 2 opens up the possibility of a dual conformal field theory on the boundary.
In the usual prescription for the AdS/CF T correspondence principle, the connected correlation functions for operators O spanning the CF T are generated by the on-shell field theory action on the corresponding asymptotically AdS space, and the boundary values φ 0 of the fields are sources associated with O. The article [1] specialized to the case of a single "free" massless scalar field on ncAdS 2 . This provided a particularly simple example, in part because of the fact that solutions to the field equation on AdS 2 are regular at the boundary, i.e., |φ 0 | < ∞. Perturbative theory was utilized to compute the leading corrections to twopoint correlation function of operators on the boundary induced by the bulk-to-boundary and bulk-to-bulk propagators. It was found that these corrections have the same form as the commutative result, and thus preserve the SO(2, 1) conformal symmetry at the boundary. This result is consistent with the isometry preserving construction of noncommutativity. 1 In this article we extend the work of [1] to include the case of both a free and interacting massive scalar field on ncAdS 2 . We obtain an exact result for the leading noncommutative correction to the two-point correlation function on boundary. As with the massless limit, it preserves the SO(2, 1) conformal symmetry. In the interacting case, we add a cubic term to the action, and from it we obtain an integral expression for the leading order noncommutative correction to the three point correlation function on the conformal boundary. It too is seen to be consistent with SO(2, 1) conformal symmetry. In this article we also argue that there does not exist a map connecting the commutative Killing vectors to their noncommutative analogues, an thus that the ncAdS 2 theory cannot be trivially related to its commutative counterpart.
The outline of this article is the following: After briefly reviewing the correspondence principle for free massive scalar field theory on AdS 2 in section two, we quantize the background space to get ncAdS 2 , and derive the leading order noncommutative correction to the two point correlation function on the conformal boundary. We introduce a cubic interaction in the scalar field action on ncAdS 2 , and compute corrections to the three point correlation function in section three. In the conclusion (section four), we briefly summarize the obtained results and discuss some possible directions for future developments. Some technical results are collected in the four appendices. In appendix A, we give some useful expressions for the asymptotic form of the relevant Green functions. In appendix B, we give the map between the noncommutative versions of canonical coordinates and Fefferman-Graham coordinates. In appendix C we argue that there does not exist a trivial map connecting the noncommutative and commutative field theory. In appendix D we show that the on-shell boundary action does not pick up noncommutative corrections.
2 Free massive scalar field

Commutative case
We start by briefly reviewing the definition of the 2-dimensional anti-deSitter space, AdS 2 . One should distinguish between Lorentzian and Euclidean AdS. Lorentzian AdS, which is standardly denoted by AdS 2 , can be defined using the global embedding coordinates, X µ , in R 2,1 with the signature (−, +, −) upon imposing the constraint: To avoid the closed time-like curves, one passes to a covering space. (See for example [12].) The Euclidean anti-deSitter space, which is standardly denoted by EAdS 2 , is defined by the analogous embedding in three-dimensional Minkowski space R 2,1 , but now with the signature (+, +, −), 2 with indices raised and lowered using the metric tensor g E = diag(1, 1, −1). In both cases ℓ 0 is the scale parameter. Equivalently, (2.1) [(2. 2)] represents EAdS 2 [AdS 2 ] for ℓ 0 imaginary. In this article we restrict to the case of the Euclidean version and will often refer to it as just AdS 2 . As it is clear from (2.2), in this case the constraint describes a double-sheeted hyperboloid.
There are two convenient choices for the coordinate charts of the lower hyperboloid, X 2 < 0, (and the same for the upper one with some sign changes). They are the canonical coordinates, (x, y), and the Fefferman-Graham (FG) coordinates, (t, z). The canonical coordinatization was defined in [1] by The Fefferman-Graham coordinatization [13] of the lower hyperboloid is given by where the coordinates span the half-plane, z > 0, −∞ < t < ∞ and z → +0 corresponds to approaching the boundary. The relation between two coordinate systems is given by (2.5) We will see that the canonical coordinates are essential to quantize the geometry. On the other hand, for the case of the 'classical' geometry we can work directly in terms of the Fefferman-Graham coordinates. The metric tensor when expressed in FG coordinates is given by while the Euclidean Laplace operator is L (0) can be expressed in terms of the three Killing vectors K µ , µ = 0, 1, 2, for AdS 2 , satisfying so(2, 1) Lie algebra commutation relations [K µ , K ν ] = ǫ µνρ K ρ . In terms of the FG coordinates they are given by where K ± = K 2 ± K 1 . (For the expressions in terms of the canonical coordinates, see (C.1).) It can be checked that (2.7) is the quadratic Casimir of the so(2, 1) Lie algebra As we will see, this has a straightforward generalization in the non-commutative theory. When expressed in terms of Fefferman-Graham coordinates the action for a free real massive scalar field Φ (0) in a Euclidean AdS 2 background is where m 0 is the mass. The (0) superscript on the field denotes the commutative theory. The scalar field equation resulting from (2.10) is Near the boundary, which is located at z = +0, 3 the dominant solution behave as (for a general discussion of asymptotic behavior, see for example [14].) where [15,16] This bound comes about from the requirement of the absence of the normalizable negative energy states, i.e. instabilities of the theory. We shall assume that the boundary function φ 0 (t) = lim nonvanishing. Then from (2.12), Φ (0) is singular in the boundary limit when (m 0 ℓ 0 ) 2 > 0.
Away from the boundary, regular solutions to (2.11) can be expressed in terms of φ 0 (t) using the boundary-to-bulk propagator (A.1) where . We denote such solutions by Φ sol [φ 0 ]. They are regular for all z ≥ 0.
Substituting the solutions (2.13) back into (2.10) gives the on-shell action, which is a boundary term: where we used the boundary value (A.5) and the result is valid for any t ′ = t ′′ on the boundary. The standard prescription for the AdS/CF T correspondence is to identify the on-shell action with the generating functional of the n−point connected correlation functions for operators O defined on the boundary. [17] Here φ 0 is treated as the source for O. For a 2-dimensional theory in the bulk, both O and φ 0 are functions of only one coordinate, the time t. The n−point connected correlation functions are thus the (0) superscript again indicating that this is the commutative theory. The two-point function for the example of the massive scalar is then This is exactly the form expected for the 2-point function in conformal theory, the conformal dimension being ∆ + .

Non-commutative case
Now we want to adopt the previous construction to the suitable non-commutative deformation of the Euclidean AdS 2 . In so doing, we set the stage for the much less trivial case of an interacting field (see the section 3.2), as well as generalize the treatment in [1] where we restricted to the case of a massless field. If we are to have any hope of preserving the conformal symmetry at the boundary, we should quantize the AdS geometry in the way that preserves the isometries of the commutative counterpart. An analogous example of such a deformation is the fuzzy sphere, where the SO(3) rotation symmetry is preserved upon quantization. [18], [19], [20], [21], [22], [23] (see [21] for the discussion of the coherent states and the recovery of the commutative limit). The preservation of the symmetries (as opposed to their breaking or deformation) allows for many results to be obtained on a purely algebraic level. So, in our case we are aiming at the non-commutative version of AdS 2 (ncAdS 2 ), which preserves the full so(2, 1) isometry of AdS 2 . It is defined similarly to the case of fuzzy sphere by three hermitian operators,X µ , here satisfying, [3], [4], [5], [6], [7], [8] X µX µ = −ℓ 2 1 , which is analogous to the AdS constraint (2.2). As before, raising/lowering of the indices is performed using the metric g E (2.2). In addition to (2.19),X µ is a basis for the so(2, 1) where α and ℓ are two real parameters with units of length. The former is the noncomutativity parameter, while the latter can be interpreted as the length scale of ncAdS 2 . 1 denotes the identity operator. The commutative limit corresponds to (α, ℓ) → (0, ℓ 0 ). It is easy to see that (2.19) is the quadratic Casimir of the algebra generated byX µ subject to the relations (2.20). But now, in contrast with the finite dimensional representations of the algebra of a fuzzy sphere, due to the non-compactness of the SO(2, 1) group, we have to consider infinite unitary representations. A detailed study of these representations, as well as the interplay between the Lorentzian and Euclidean cases, was performed in [1]. The non-commutative generalization of the action (2.10) for a free massive field is given by the standard form whereΦ is non-commutative field on ncAdS, Tr denotes a trace, and m is the mass of the scalar field. We assume that m → m 0 in the commutative limit in order to recover the previous scalar field dynamics. As was discussed in detail in [1], the canonical coordinates (2.3) upon quantization satisfy the standard canonical relation [x, y] = iα1 . (Here for convenience we use the same letters for the commutative and non-commutative coordinates.) Then one can pass from the operator algebra generated byX µ to the algebra of their symbols X µ (x, y), with the operator product replaced by the Moyal-Weyl star product, ⋆. (for a nice review, see [24]) The Moyal-Weyl star product of any two symbols F and G is defined in the standard way by while the trace on the algebra, Tr, becomes 1 α 2 R 2 dxdy. It was shown in [1] that the symmetric ordering in the deformation of the canonical coordinatization (2.3) y ⋆ e −x ⋆ y + ℓ sinh x , leads to the correct star product product realization of the defining relations, (2.19) and (2.20), for ncAdS 2 , is the star-commutator of any two functions F (x, y) and G(x, y) on the Moyal-Weyl plane spanned by coordinates (x, y). Using this, the action (2.21) can be trivially mapped to a functional on the Moyal-Weyl plane where Φ = Φ(x, y) is the symbol of the fieldΦ. Varying the action with respect to Φ, one gets the corresponding equation of motion where L is the noncommutative Laplace operator. Alternatively, the action (2.27) and the equation of motion (2.28) can be written explicitly in terms of the canonical coordinates as [1]. For the former we get up to boundary terms, where ∆ y and S y are the nonlocal operators while for the latter We search for a perturbative solution to the field equations, with α 2 ℓ 2 being the perturbative parameter. For this we will use the method discussed in the Appendix C. Namely, we will map the full non-commutative equation (2.28) into the commutative one (2.11) plus some corrections. To achieve this, we construct an operator U, given explicitly up to order α 2 in (C.18), which maps the set of the commutative Killing vectors (C.1) to the non-commutative ones (C.2) modulo an additional term in K + ⋆ . Taking into account that in both cases the Laplacian can be written in terms of their respective Killing vectors, (2.9) and (2.33), the necessary map between the Laplacians is trivially found up to order α 2 to be , which when expressed in terms of canonical coordinates is Actually, there is a slightly simpler construction. Instead of mapping L to L (0) , we can map it to L (0) ℓ , which is given by exactly the same expression as in (2.35), but with ℓ 0 → ℓ. For this we just need to set ℓ 1 = 0 and ℓ 0 = ℓ in (C.18), so then U simplifies to (we keep on using the same letter U for the operator) This leads to The difference between ℓ and ℓ 0 in the second term is of the next order in α.
As discussed in Appendix B, the quantization in terms of the Fefferman-Graham coordinates is equivalent to making the commutative change of variables (2.5) (with ℓ 0 replaced by ℓ) in any non-commutative expression written in canonical coordinates. Then transforming (2.38) to FG coordinates we get ℓ takes exactly the same form as in (2.7), but this was only possible only because of the use of the "quantum" change of variables (B.4). Below we will use the notation which with the help of (2.37) and (2.39) can be written as In order to find the leading noncommutative corrections to Φ (0) we can first solve (2.40) for Φ U and then apply the inverse map to get Φ, We note that the leading order correction to U −1 vanishes as one approaches the boundary z → 0, 4 and so 44) 4 Strictly speaking, this is true except for the term 3 32 α 2 ℓ 2 z∂ z , which only changes U −1 by a constant factor when acting on any power of z, but the result (2.44) can also be seen from (2.41), which in the limit z → 0 goes to the commutative equation (2.11).
while ∂ z Φ near the boundary gets an additional correction compared to ∂ z Φ U : Using standard techniques, [25] one can write down a perturbative solution to (2.40) in terms of a field on the boundary, which we again denote by φ 0 . Because noncommutativity vanishes at the boundary (see the further comments in this section and Appendix D), we can assume that φ 0 is independent of the perturbation parameter α/ℓ. Then we introduce a noncommutative version of the boundary to bulk propagator, denoted by K U nc (z, t; t ′ ), which is defined in analogy to (2.13), Because we search for perturbative solutions, we expand Φ U in even powers of α/ℓ about the commutative solution Φ (0) : From (2.40), Φ (0) satisfies the free commutative equation (2.11), which is again solved by (2.13) (with ℓ 0 → ℓ), while Φ (1) satisfies (2.13) gives Next apply the bulk-to-bulk propagator G(z, t; z ′ , t ′ ) to obtain an integral expression for Φ (1) . Using the conventions in [25], and its explicit form in terms of the hypergeometric function is given in Appendix A (A.2). Then the solution of (2.49) is given by Combining (2.13) and (2.51) we obtain an expression for the noncommutative boundary-tobulk propagator up to first order in α 2 /ℓ 2 : From the asymptotic behavior of the commutative Green functions (A.3) and (A.7) it follows From the solution (2.46) for Φ U , we then get the solution (2.42) for Φ. They are functionals of φ 0 and we denote the latter solution by Φ sol [φ 0 ]. We next need to substitute Φ sol [φ 0 ] back into (2.27) to compute the on-shell action. For this purpose it is convenient to re-write the action (2.27) as From the field equation, (2.28), the quantity in parenthesis · · · vanishes on-shell. The remaining term is only defined on the boundary z = 0, since the Moyal star-commutator is a total divergence. We argue in the Appendix D that it contains no non-commutative corrections, and so, up to an overall factor, has the same form, i.e., (2.14), as in the commutative theory. Thus (2.55) Therefore the non-commutative effects are only due the corrections to the solution of the field equation.
Using the near-boundary behavior (2.44) and (2.45), one can express the on-shell action up to order α 2 ℓ 2 in terms of Φ U , and then using (2.46), it can be written in terms of the non-commutative boundary to bulk Green function In the commutative AdS/CFT correspondence it is assumed that the same relation (2.17) between the on-shell bulk action and the generating functional for the boundary theory holds for all asymptotically AdS spaces. We assume that this continues to be a valid assumption even when the bulk does not correspond to a commutative geometry. Furthermore, after examining the asymptotic behavior of the Killing vectors, it was argued in [1] that our formulation of noncommutative AdS 2 coincides with "commutative" AdS 2 as one approaches the boundary. So applying (2.17) the resulting expression for the two-point correlator on the boundary is Expanding this in powers of (α/ℓ) 2 : and using (2.52) in (2.57) one recovers (2.18) at zeroth order, while for the next order we get The integrals can be evaluated using the asymptotic obtained in (A.8), leading to Then the first order correction to the two-point correlation function becomes where and we used the fact that K(z, t; t ′′ ) is only a function of z and t − t ′ . From the formula (22) of [26], which is analytic for ∆ + > 2. Then using properties of Gamma function, Substituting into (2.61) then gives Therefore the leading non-commutative correction to the two-point correlator is just a rescaling of the commutative two-point correlator.
Note that though the zero order correlator, < O(t)O(t ′ ) > (0) was calculated for the free commutative theory defined by m and ℓ, rather than m 0 and ℓ 0 , it actually depends on those parameters only through ∆ + , and only in the case when m 0 ℓ 0 = mℓ does the conformal weight ∆ + not receive a leading order non-commutative correction.
For the special case of a massless scalar field, i.e. when ∆ + = 1, (2.65) reduces to reproducing the main result of [1]. 5 3 Interacting scalar field

Commutative case
We first review the commutative theory. Upon adding a cubic term to the free scalar field action (2.10), we get where λ is a real parameter, and the (0) superscript again indicates that this is the commutative system. The resulting field equation is now with the same L (0) as in (2.7). We again assume the asymptotic behavior (2.12). Then (3.2) can be solved perturbatively in λ using the boundary-to-bulk and bulk-to-bulk propagators, K(z, t; t ′ ) and G(z, t; z ′ , t ′ ), (A.1) and (A.2) respectively. Of course, at zeroth order in λ the solution is (2.13). Up to first order one has We again denote the solution by Φ . The on-shell action now includes a bulk term, as well as a boundary term (which is the same as in (2.14)) Evaluating the boundary term for where we used the asymptotic expressions (A.5) and (A.8). While the first term is exactly (2.16) and will lead to the same 2-point function (2.18), the second term will give a nontrivial contribution to the 3-point function. This should be combined with the bulk term (3.5), which after substitution of Combining this with (3.6) and using the definition (2.17), the three-point function is The dependence on t 1 , t 2 and t 3 is determined from conformal invariance (3.9) The coefficient a ∆ + was computed in [26] (3.10)

Non-commutative case
The natural generalization of (3.1) to the noncommutative case is given by This action can again be mapped to an integral on the Moyal-Weyl plane The field equation following from (3.12) is where L is the noncommutative Laplace operator, defined by (2.28), (2.31) or (2.33). The free theory, λ = 0, is solved by (2.42), (2.46) and (2.52) where with the differential operator D z,t defined in (2.43). For small λ, (3.13) can be solved perturbatively in analogy with (3.3), by replacing the commutative source λΦ (0) 2 by its non-commutative analogue λ Φ ⋆ Φ, and by replacing the commutative Green functions by their non-commutative analogues. The non-commutative analogue of the boundary to bulk Green function is given by (3.15). We denote the noncommutative analogue of the bulk to bulk Green function by G U nc (z, t; z ′ , t ′ ). We require it to satisfy the analog of the commutative equation (2.50) so G U nc (z, t; z ′ , t ′ ) → G(z, t; z ′ , t ′ ) in the commutative limit. Upon expanding in λ, the solution to (3.13) is where K (n) nc (z, t) denotes the function K nc (z, t; t n ) and the star-product is with respect to the explicitly shown variables. The solution to G U nc (z, t; z ′ , t ′ ) can be computed perturbatively in powers of α 2 . If we write then the leading order non-commutative correction G (1) (z, t; z ′ , t ′ ) satisfies where we used (2.41). The solution is We denote the solution (3.17) by Φ sol [φ 0 ] and substitute it back in (3.12) to get the on-shell action. The latter can be split into two terms as in the commutative case where S bdy nc [Φ] was defined in (2.54) and Substituting Φ sol [φ 0 ] into the boundary term S bdy nc [Φ], which again reduces to (2.55), and collecting the third order terms in φ 0 , we get In the commutative limit, we recover the commutative boundary term (3.6). This follows from K nc (z, t; t ′ ) → K(z, t; t ′ ), G U nc (z, t; z ′ , t ′ ) → G(z, t; z ′ , t ′ ) and U z,t → 1, as α → 0 along with the identity (A.8).
The sum of (3.24) and (3.23) gives all the φ 3 0 terms in S nc Φ sol [φ 0 ] . So the expression for the three-point function is nc ](z, t) + all permutations of (t 1 , t 2 , t 3 ) .

(3.25)
Near the boundary the first term in the integrand can be expanded in α 2 using the results of the appendix A. Using the relevant asymptotics in the definitions (2.52), (3.15) and (3.18) one easily establishes the following asymptotic formulas: which leads to the z → +0 value for the relevant term in (3.25) Substituting into (3.25) gives Next we will analyze the result (3.28) and demonstrate that it has the same scaling and translational transformation properties as the commutative 3−point function (3.9) (at least up to leading order in α 2 ). 6 First, we will establish the behaviour of (3.28) under the simultaneous scaling of t i , i = 1, 2, 3: K U nc (µz, µt; µt ′ ) = µ −∆ + K U nc (z, t; t ′ ) , K nc (µz, µt; µt ′ ) = µ −∆ + K nc (z, t; t ′ ) , (3.29) at least up to order α 2 . Noticing that the star-product (2.23) is constructed from the derivatives ∂ ∂x = −z ∂ ∂z − t ∂ ∂t and ∂ ∂y = z ℓ ∂ ∂t , which are invariant under a simultaneous rescaling of z and t, we see that the star-product is scale invariant to all orders. Using this, along with (3.29) in (3.28), we find that the non-commutative 3−point function scales just like the commutative 3−point function (3.9) (3.30) Now we would like to demonstrate the invariance of (3.28) under simultaneous translations t i → t i + a. This will effectively guarantee that the non-commutative 3-point function has the same functional dependence on t i as its commutative counterpart (3.9). Because both commutative propagators, (A.1) and (A.2), depend only on the time translational invariant combination, they are time translationally invariant. Using this in (2.52), we see that this is also true for K U nc (z, t; t ′ ) (at least up to order α 2 ). The first non-trivial effect due to translations appears in the transformation of U −1 (µz, µt). Under t → t + a it transforms as This leads to a non-trivial transformation of K nc (z, t; t ′ ): The second non-trivial contribution comes from the star-product. Expanding (2.23) up to the second order in α gives Then up to order α 2 the translated star product gets an extra term: Using the definitions (3.32) and (3.36), one can easily verify the following useful relation between J z,t (a) and S z,t (a) for any two arbitrary functions F (z, t) and G(z, t). Using this relation we get (J z,t ≡ J z,t (a)) K nc (z, t + a; t 2 + a) ⋆ z,t+a K nc (z, t + a; t 3 + a) = where in non-invariant terms we used K(z, t; t i ) instead K nc (z, t; t i ) because these terms are already of the order of α 2 . Combining (3.31) and (3.38) and using the translational invariance of K U nc (z, t; t ′ ), we see that the contribution of the boundary term to the 3-point function (3.28) is explicitly translationally invariant. The bulk contribution to the 3-point function is also translationally invariant due to the fact that the non-invariant term coming from (3.39) is given by integral of a total derivative, which will drop out of (3.28) even without symmetrization.
We have demonstrated that the non-commutative 3-point function (3.28) shares the symmetries of its commutative counterpart and, as a consequence, should have the following form is given in (3.9) and the coefficient c should be calculated by an explicit evaluation of the terms in (3.28) up to the order α 2 .

Conclusion
In this paper we have tested the possibility of extending the AdS/CF T correspondence to the case of a non-commutative bulk. Making the domain on the gravity-side of the correspondence non-commutative is physically well motivated for, at least, three reasons: 1. As was mentioned in the introduction, non-commutative space-time can be interpreted as a quasi-classical regime of essentially any theory of quantum gravity. Because the strong form of the AdS/CF T correspondence assumes a duality between full quantum gravity (and not just classical gravity) and CF T living on the conformal boundary, it is reasonable to assume that the duality can be extended to the quasi-classical regime as well.
2. It is believed that the correspondence should work for any asymptotically AdS space. As we demonstrated in [1], ncAdS has this property -the non-commutativity effectively vanishes near the boundary.
3. The deformation leading to ncAdS used in this paper preserves the undeformed SO(2, 1) isometry of the bulk, which supports the notion that a dual theory will have undeformed SO(2, 1) conformal symmetry.
To test whether the AdS/CF T correspondence can be extended to a non-commutative setting, we have calculated the leading non-commutative corrections to 2-and 3-point correlation functions for a scalar field on the non-commutative background. To get a non-vanishing 3-point function, the field has to be interacting. This created serious technical complications compared to the free case. Still we were able to show that for both correlators the overall effect of non-commutativity is in re-scaling of their commutative counterparts, supporting and generalizing the conclusions of [1] for the free massless case. For the 2-point function we were able to compute the answer in a closed form (2.65), explicitly demonstrating its conformal behaviour (and greatly simplifying the analysis of [1]). On the other hand, due to the aforementioned technical complications, we were not able to obtain a corresponding closed form expression for the 3-point function. There instead the analysis was done implicitly by studying the transformation properties of the correlator under the conformal transformations, which also confirmed its conformal behaviour.
This result prompts several immediate questions. The first one is about the triviality of the result. Is it possible that the commutative theory can be mapped to the non-commutative one, analogous to the Seiberg-Witten map for gauge theories? More explicitly, can the noncommutative Killing vectors be obtained from the commutative ones by some similarity transformation? If true, this would give a trivial solution to the non-commutative field equations. In this paper we gave a perturbative argument to show that this is not the case, and so the field equation (and as the result, the solution) receive a genuinely non-commutative correction, proving that our result is really non-trivial.
The second question one can ask is whether or not the whole effect of the non-commutativity (associated with a quantum gravitational bulk) on the n-point functions of a dual CF T is due to some kind of non-commutative (or quantum gravitational) renormalization of the corresponding boundary operator O. To address this issue, one has to compare the factors in front of the non-commutative 2-and 3-point correlators. The question then is are the corrections to the correlators related by a factor of 2/3? While we have an explicit form (2.65) for the leading correction to the 2-point correlator, due to technical complications we were unable to do the integration in (3.40) to get the explicit form for the 3-point correlator. This is a very important technical problem, whose solution would give a better understanding of the correspondence. We are planning to report on the progress in this direction elsewhere.
An obvious question is whether or not our perturbative analysis can be generalized beyond the first non-trivial order, and furthermore if the conclusion we found at leading order survives? What gives us hope that the conclusion does indeed survive is the presence of the full undeformed symmetry in the non-commutative theory, which might lead to the construction of some exact non-perturbative results. It is important to have this analysis done to ensure that our findings are not an artifact of the first order approximation. We are planning to look at this problem as one of the natural continuations of the project.
The final question concerns the possibility of generalizing of the noncommutative construction presented in this paper to higher dimensions. This is particularly relevant because the (commutative) AdS d+1 /CF T d correspondence is best understood for d ≥ 2. It should be possible to apply the arguments presented in this paper to a non-commutative AdS d+1 /CF T d , d ≥ 2 correspondence provided that one first succeeds to construct an isometry preserving non-commutative deformation of AdS d+1 . The problem, however, is that in more then two dimensions, due to the lack of a natural Poisson structure, this is a much more difficult, both technically and conceptually, task. One possible exception is for AdS 4 . [27] We hope to report some preliminary results on this case of in the near future.

A Useful asymptotic expressions
Here we calculate some asymptotics of the Green's functions used in the main text. The definitions of the boundary-to-bulk and bulk-to-bulk propagators, respectively, are as follows: and we defined ∆ + = ∆, ∆ − = 1 − ∆, while F (a, b; c; z) ≡ 2 F 1 (a, b; c; z) is the standard hypergeometric function normalized as F (a, b; c; 0) = 1. Taking the derivative of (A.1) with respect to z, one finds Combining this with the asymptotics for K(z, t; t ′ ), which trivially follows from (2.12) and (2.13) one gets One should be careful in taking the z → 0 limit for K(z, t; t ′ )∂ z K(z, t; t ′′ ). First assume that |t ′ − t ′′ | ≫ ǫ > 0. From (A.1) one gets Using (A.3) one more time and taking into account that |t ′ − t ′′ | ≫ ǫ > 0 we obtain which is now valid for any t ′ = t ′′ on the boundary. To get the analogous results for the bulk-to-bulk propagator we just use the expansion of the hypergeometric function in the definition (A.2) Then taking into account we immediately get We also need to evaluate z → 0 behaviour of ∂ z (K(z, t; t ′ )G(z, t; z ′′ , t ′′ )) which is now trivially found using (A.3) and (A.7)

B Noncommutative canonical coordinates versus noncommutative Fefferman-Graham coordinates
As was discussed in detail in [1], the canonical coordinates (2.3) have the canonical Poisson structure (hence the name). Via (2.5) this induces the following Poisson brackets for the FG coordinates appearing in (2.4) While the quantization of the canonical structure leads to the Moyal-Weyl star product (2.23), the quantization of (B.1) is not that obvious due to the non-triviality of the Poisson structure in terms of FG coordinates. Here we argue that the most natural choice of the quantization, the symmetric one, leads to the minimal quantization of (B.1), i.e. does not introduce the higher-order in α terms. Towards this end, we upgrade t and z to operators using the definitions where here x and y are also interpreted as operators. Using the canonical commutation relation (2.22) for x and y, one readily finds the commutator between t and z So, this is really the minimal quantization of (B.1). Re-written in terms of the star-product (2.23), it will become the star-commutator between the corresponding symbols, which are, as trivially verified, given by the commutative expression (2.5) (without any star-product!) but now with Now we are regarding x, y, t and z as symbols of the corresponding operators. Then the star-product in terms of the Fefferman-Graham coordinates will be given just by the standard Moyal-Weyl product (2.23) in which one performs the commutative change of variables (B.4). One could also explicitly verify this by calculating the star-product and observing that z ⋆ z = e −x ⋆ e −x ≡ z 2 and 1 2 (y ⋆ e −x + e −x ⋆ y) = ye −x .

C Similarity transformation and no-triviality
Both sets of Killing vectors, commutative and noncommutative, satisfy the same undeformed so(2, 1) algebra: [K µ , K ν ] = ǫ µνρ K ρ . The expressions for the Killing vectors in terms of the canonical coordinates was given in [1]. The commutative Killing vectors are while the non-commutative Killing vectors are where ∆ y and S y were defined in (2.30) and, in general, ℓ = ℓ(α), such that ℓ(0) = ℓ 0 .
Here we want to ask whether or not these two sets of vectors can be mapped to each other. More specifically, we ask if there exists a non-degenerate map U such that If this were to be the case, then the two theories, commutative and noncommutative, would essentially be equivalent, as one would be able to map all the solutions of one theory to the ones of the other, and it would be easy to see that this map would preserve the conformal structure of the n-point functions. The absence of such a map would tell us that the conformal behaviour of the noncommutative theory is really a non-trivial result. Below we will see that the latter is true. Because the addition to K + of a term proportional to ℓ 0 e −x ∂ y ≡ −e −2x K − does not spoil the so(2, 1) algebra, 7 it makes sense to study the effect of this term separately. Towards this end introduce the "shifted" Killing vectors: .

(C.4)
We will look for a similarity transformation that takes the "shifted" commutative generators (C.4) to "shifted" non-commutative ones. Here by "shifted" non-commutative generators we mean the following Note that adding const×e −x ∆ y to K + ⋆ also does not effect the algebra (so, (C.2) corresponds to const = ℓ + α 2 4ℓ ). Of course, later we also will be interested in the effect of such a transformation (if exists) on the shift term.
To deal with the difference between ℓ 0 in (C.4) and ℓ in (C.5), one can perform the trivial similarity transformation, U 0 = exp − ln( ℓ ℓ 0 )∂ x , which changes ℓ 0 in (C.4) to ℓ (It is important to note that this is not the case for the original, non-shifted, generators (C.1) because ℓ 0 enters K + not only via a common factor.) We will assume that this trivial similarity transformation has been done and will keep on using the same notation for (C.4) but now with ℓ 0 → ℓ.
Because in this paper we are interested in leading order perturbations in the noncommutativity parameter, we will look for the similarity transformation to the α 2 -order. To this order we have .
Here the term proportional to the unknown constant κ is exactly the possible shift term. More specifically, the constant in (C.5) is equal to − α 2 4ℓ κ. We see that there is no α-linear term, so it is natural to suggest the following expansion for the mapŨ fromK µ toK µ ⋆ = U −1K µŨŨ Then we have the following conditions on G: x , (C.9) Let us analyze these conditions one by one.
1) The condition (C.8) requires that G does not depend on x.
2) The condition (C.9) is [e x ∂ y , G] = − 1 24 e x ∂ 3 x . We note that there is a trivial solution to it: G 0 = − 1 24 y∂ 3 y , so, writing G = G 0 +G, this is equivalent to e x ∂ y ,G = 0. Because the full form of the non-commutative Killing vectors depends on derivatives with respect to x only up to ∂ 2 x , it is possible to argue thatG also does not involve terms with ∂ (k) x with k ≥ 3, i.e. G takes the following form G = g 2 (y, ∂ y )∂ 2 x + g 1 (y, ∂ y )∂ x + g 0 (y, ∂ y ) . (C.11) Taking into account the independence of ∂ (k) x for different k, after some trivial calculation, we arrive at the following most general form for the candidate for the infinitesimal similarity transformation: where g 2 andg i are some arbitrary functions of the argument ∂ y .
3) Using the result of the previous step (C.12) in (C.10) and requiring that the term proportional to ∂ 3 x is absent on the RHS of (C.10) we immediately conclude that g 2 (∂ y ) is actually a constant, i.e. the whole dependence on it drops out. Continuing to compare the coefficients of ∂ (k) x for different k = 0, 1, 2, we arrive at the following result where a and b are some arbitrary constants, which do not contribute at this level. We still keep the dependence on a and b explicit to study the transformation of the shift term (see below). This completes the prove of the perturbative (up to α 2 -terms) equivalence ofK µ andK µ ⋆ (with the very precise form of the generated shift term) (C.14) wherẽ U = 1 + α 2 G(∂ x , y, ∂ y ) + O(α 4 ) = 1 + α 2 − 1 24 y∂ 3 y + a(∂ 2 x + 2y 2 ∂ 2 y + 2y∂ y ∂ x + y∂ y )+ + 1 32 (2y∂ y + 2∂ x + 1)∂ 2 y + b(y∂ y + ∂ x ) + O(α 4 ) . (C. 15) Note, that as it was stressed above, (C.14) does not depend on the arbitrary a and b.
For the future use, we consider a more general choice forg 0 :g 0 = 1 2 (1 + λ)g 1 . Of course this will produce some extra terms on the right hand side, but we will see how they are cancelled by the shift term. 8 So, we have (including the contribution from U 0 ) U = 1 + α 2 G(∂ x , y, ∂ y ) + O(α 4 ) = 1 + α 2 − 1 24 y∂ 3 y + a(∂ 2 x + 2y 2 ∂ 2 y + 2y∂ y ∂ x + y∂ y )+ The problem with the shift term, const × e −x ∂ y , is immediately clear from the fact that neither expression in front of the constants a and b in (C.16) commutes with this term. So, as the consequence, we will produce terms explicitly depending on these constants. It is easy to obtain the perturbative form of the transformation of the shift term (we expand ℓ = ℓ 0 + α 2 While the first term has a correct form (which, of course, remains correct after the expansion in α is done), the rest presents a correction (the difference between ℓ and ℓ 0 is of the next order in α.) Combining (C. 16) and (C.17) one can easily see that the choice λ = 32aℓ 2 0 and b − a = 1 ℓ 2 0 ℓ 1 ℓ 0 + 3 32 (the separate values of a and b turn out to be irrelevant) almost does the mapping between the two sets of Killing vectors, (C.1) and (C.2) (we return to the "untilded" notation for U, because this is a map between K µ and K µ ⋆ as in (C.3)) U = 1 + α 2 G(∂ x , y, ∂ y ) + O(α 4 ) = = 1 + α 2 1 96 (2y∂ y + 6∂ x + 3)∂ 2 y + 1 To conclude, we can almost map the commutative Killing vectors to the noncommutative ones. The obstruction is the shift term, the presence of which leads to the appearance of the extra ∂ 3 y -term. This is not only the proof of the "non-triviality" of ncAdS 2 but also serves as very convenient technical tool to simplify the perturbative analysis of the sections 2.2 and 3.2.

D Noncommutative boundary term
Here we want to show that the commutative expression for the on-shell action (2.55) is valid in the non-commutative case to all orders in α. Towards this end, let us re-write (2.54) in terms of the non-commutative Killing vectors (C.2) where K µ ⋆ are defined as αK µ ⋆ Φ := i[X µ , Φ] ⋆ . In two dimensions, the Stokes theorem takes the form (ω = ω µ dx µ is an arbitrary 1-form) V dxdy(∂ x ω y − ∂ y ω x ) = ∂V ω x dx + ω y dy . (D.2) Because the boundary of our space is located at z = 0 it is convenient to pass to Fefferman-Graham coordinates (B.4). Then z = const corresponds to x = const with dy = ℓ z dt and for our case the Stokes formula takes the form This means that while studying the integrand of (D.1) we need to keep track only of the term of the form ∂ x (· · · ). Also, because ℓ z ω y is evaluated at z = 0, we only need to keep terms in ω y up to O(z 2 ). This will allow us to arrive at the exact result. Using where ∆ y and S y are defined in (2.30), the Killings (C.2) take the form ℓ e −x y S y + ∆ y 1 ℓ e −x y 2 + ℓ 2 + α 2 2 − α 2 4 ∂ 2 x , (D.5)