Noncommutative ${\mathrm{AdS}}_2/{\mathrm{CFT}}_1$ duality: The case of massless scalar fields

We show how to construct correlators for the ${\mathrm{CFT}}_1$ which is dual to noncommutative ${\mathrm{AdS}}_2$ ($nc{\mathrm{AdS}}_2$). We do it explicitly for the example of the massless scalar field on Euclidean $nc{\mathrm{AdS}}_2$. $nc{\mathrm{AdS}}_2$ is the quantization of ${\mathrm{AdS}}_2$ 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 $nc{\mathrm{AdS}}_2$. Killing vectors can be constructed on $nc{\mathrm{AdS}}_2$ which reduce to the ${\mathrm{AdS}}_2$ Killing vectors near the boundary. We, therefore, conclude that $nc{\mathrm{AdS}}_2$ is asymptotically ${\mathrm{AdS}}_2$, and so the $\mathrm{AdS}/\mathrm{CFT}$ correspondence should apply. For the example of the massless scalar field on Euclidean $nc{\mathrm{AdS}}_2$, the on-shell action, and resulting two-point function for the boundary theory, are computed to leading order in the noncommutativity parameter. The computation is nontrivial because nonlocal interactions appear in the Moyal-Weyl description. Nevertheless, the result is remarkably simple and agrees with that of the commutative scalar field theory, up to a rescaling.