(2+1) Dimensional Black Hole and (1+1) Dimensional Quantum Gravity

TL;DR

This paper explores the boundary actions of a (2+1)D black hole in Chern-Simons theory, revealing connections to (1+1)D quantum gravity through WZW and Liouville models, highlighting a deep relationship between higher and lower-dimensional gravity.

Contribution

It demonstrates that the boundary actions of a (2+1)D black hole correspond to well-studied (1+1)D models, establishing a link between (2+1)D black hole physics and (1+1)D quantum gravity.

Findings

Boundary actions are gauged SL(2,R)/U(1) WZW and SL(2,R) WZW models.

The boundary at infinity is equivalent to the Liouville model.

The (2+1)D black hole relates to (1+1)D quantum gravity models.

Abstract

In the Chern-Simons gauge theory formulation of the spinning (2+1) dimensional black hole, we may treat the horizon and the spatial infinity as boundaries. We obtain the actions induced on both boundaries, applying the Faddeev and Shatashvili procedure. The action induced on the boundary of the horizon is precisely the gauged $SL(2,R)/U(1)$ Wess-Zumino-Witten (WZW) model, which has been studied previously in connection with a Lorentz signature black hole in (1+1) dimensions. The action induced on the boundary of spatial infinity is also found to be a gauged $SL(2,R)$ WZW model, which is equivalent to the Liouville model, the covariant action for the (1+1) dimensional quantum gravity. Thus, the (2+1) dimensional black hole is intimately related to the quantum gravity in (1+1) dimensions.