Non-commutative $Ad{S}_2\mathrm{/}CF{T}_1$ duality: the case of massless scalar fields

We show how to construct correlators for the $CF{T}_1$ which is dual to non-commutative $Ad{S}_2$ ($ncAd{S}_2$). We do it explicitly for the example of the massless scalar field on Euclidean $ncAd{S}_2$. $ncAd{S}_2$ is the quantization of $Ad{S}_2$ that preserves all the isometries. It is described in terms of the unitary irreducible representations, more specifically discrete series representations, of $so(2,1)$. We write down symmetric differential representations for the discrete series, and then map them to functions on the Moyal-Weyl plane. The Moyal-Weyl plane has a large distance limit which can be identified with the boundary of $ncAd{S}_2$. Killing vectors can be constructed on $ncAd{S}_2$ which reduce to the $Ad{S}_2$ Killing vectors near the boundary. We therefore conclude that $ncAd{S}_2$ is asymptotically $Ad{S}_2$, and so the $AdS\mathrm{/}CFT$ correspondence should apply. For the example of the massless scalar field on Euclidean $ncAd{S}_2$, the on-shell action, and resulting two-point function for the boundary theory, are computed to leading order in the noncommutativity parameter. The results agree with those of the commutative scalar field theory, up to a field redefinition.