Dimensional Reduction of the $S^3$/WZW Duality

TL;DR

This paper proposes a dimensional reduction of the $S^3$/WZW duality, connecting 3D gravity to 2D JT gravity and complex Liouville quantum mechanics, providing a new example of holography without boundary.

Contribution

It introduces a novel dimensional reduction of the $S^3$/WZW duality, linking 3D gravity to 2D theories with boundary terms and gauge symmetries, expanding the scope of holographic dualities.

Findings

2D JT gravity captures 3D gravity effects at low temperature.

The reduced CFT is a complex Liouville quantum mechanics with $SU(2)$ symmetry.

The approach offers a boundary-less holographic example.

Abstract

Recently proposed duality relates the critical level limit $\hat{k} \to -2$ of $SU(2)_{\hat{k}}$ WZW models to a classical three-dimensional Einstein gravity on a sphere. In this paper, we propose a dimensional reduced version of this duality. The gravity side is reduced to a Jackiw-Teitelboim (JT) gravity on $S^2$ with a non-standard boundary term, or a BF theory with $SU(2)$ gauge symmetry. At least in low temperature limit, these two-dimensional gravity theories completely capture the original three-dimensional gravity effect. The CFT side is reduced to a certain complex Liouville quantum mechanics (LQM) with $SU(2)$ gauge symmetry. Our proposal gives an interesting example of a holography without boundary. We also discuss a higher-spin generalization with $SU(N)$ gauge symmetry.