General Solutions of Einstein Gravity at $D\rightarrow 2$

TL;DR

This paper explores the general solutions of Einstein gravity as the dimension approaches two, connecting it to JT gravity and Liouville CFT, and analyzing black holes, wormholes, and boundary dynamics.

Contribution

It provides a comprehensive analysis of solutions in the $D ightarrow 2$ limit, including black holes, wormholes, and boundary actions, bridging key 2D models.

Findings

Scalar-tensor theory reduces to JT gravity or Liouville CFT at large central charge.

The on-shell action of nearly AdS$_2$ is derived, with Schwarzian dynamics at finite boundary cutoff.

Scalar field remains well-defined on the 2-sphere for positive cosmological constant.

Abstract

Einstein gravity at $D\rightarrow 2$ limit can be obtained from the Kaluza-Klein procedure by taking the dimensions of the internal space to zero while keeping only the breathing mode. The resulting scalar-tensor theory can be further reduced to JT gravity or the Liouville CFT at the large central charge limit, bridging the two important 2d models. We study the general solutions of the theory, including black holes and wormholes for both positive and negative cosmological constants. We obtain the on-shell action of the nearly AdS$_2$ and show that for suitable boundary slices, the Schwarzian action governs the leading-order dynamics at the finite boundary cutoff in later time. For positive cosmological constant, we find that the scalar is well defined on the 2-sphere.