Non-commutative $AdS_2/CFT_1$ duality: the case of massless scalar fields

TL;DR

This paper constructs correlators for a $CFT_1$ dual to non-commutative $AdS_2$, demonstrating that the non-commutative geometry preserves key symmetries and yields boundary correlators similar to the commutative case.

Contribution

It explicitly develops the $AdS_2/CFT_1$ duality framework for non-commutative $AdS_2$, including representations, symmetries, and boundary correlators for massless scalar fields.

Findings

Correlators on non-commutative $AdS_2$ match commutative results up to a field redefinition.

Non-commutative $AdS_2$ preserves all isometries and is asymptotically $AdS_2$.

Boundary two-point functions are computed to leading order in noncommutativity.

Abstract

We show how to construct correlators for the $CFT_1$ which is dual to non-commutative $AdS_2$ ($ncAdS_2$). We do it explicitly for the example of the massless scalar field on Euclidean $ncAdS_2$. $ncAdS_2$ is the quantization of $AdS_2$ 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 $ncAdS_2$. Killing vectors can be constructed on $ncAdS_2$ which reduce to the $AdS_2$ Killing vectors near the boundary. We therefore conclude that $ncAdS_2$ is asymptotically $AdS_2$, and so the $AdS/CFT$ correspondence should apply. For the example of the massless scalar field on Euclidean $ncAdS_2$, 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.