Complex saddles of three-dimensional de Sitter gravity via holography

TL;DR

This paper identifies complex saddle points in three-dimensional de Sitter gravity using holography, involving complexified metrics, Chern-Simons theory, and Liouville theory, to better understand quantum gravity and higher-spin extensions.

Contribution

It introduces a method to determine relevant complex saddles in 3D de Sitter gravity via holography, including higher-spin cases, reducing the complexity of saddle selection.

Findings

Complex saddles are characterized using Liouville theory correlators.

Monodromy matrices determine saddles with multiple conical defects.

Extension of analysis to higher-spin gravity cases.

Abstract

We determine complex saddles of three-dimensional gravity with a positive cosmological constant by applying the recently proposed holography. It is sometimes useful to consider a complexified metric to study quantum gravity as in the case of the no-boundary proposal by Hartle and Hawking. However, there would be too many saddles for complexified gravity, and we should determine which saddles to take. We describe the gravity theory by three-dimensional SL$(2,\mathbb{C})$ Chern-Simons theory. At the leading order in the Newton constant, its holographic dual is given by Liouville theory with a large imaginary central charge. We examine geometry with a conical defect, called a de Sitter black hole, from a Liouville two-point function. We also consider geometry with two conical defects, whose saddles are determined by the monodromy matrix of Liouville four-point function. Utilizing Chern-Simons description, we extend the similar analysis to the case with higher-spin gravity.