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

Abstract

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 $AdS/CFT$ 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.

arXiv comment25 pages

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• Spectral Properties of the Symmetry Generators of Conformal Quantum Mechanics: A Path-Integral Approach
• Asymptotic commutativity of quantized spaces: the case of ${\mathbb{CP}}^{p,q}$
• Quantized Noncommutative Geometry from Multitrace Matrix Models
• Exact solutions for scalars and spinors on quantized (Euclidean) $Ad{S}_2$ and the correspondence principle
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• QFT on Fuzzy AdS Spaces: Classical Limit and Boundary Correlation Functions
• The AdS^2_θ/CFT_1 Correspondence and Noncommutative Geometry II: Noncommutative Quantum Black Holes