Complex saddles of three-dimensional de Sitter gravity via holography

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, 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.

Automated summary

In this study, we applied holography to determine the complex saddles of three-dimensional gravity with a positive cosmological constant. Considering a complexified metric is sometimes useful in studying quantum gravity, as demonstrated by Hartle and Hawking's no-boundary proposal. However, there are too many saddles for complexified gravity, necessitating the determination of which ones to consider. By describing the gravity theory using three-dimensional SL(2, C) Chern-Simons theory, we found that its holographic dual at the leading order is Liouville theory with a large imaginary central charge. We analyzed geometries with conical defects, such as the de Sitter black hole, using Liouville two-point function, and also studied geometries with two conical defects, where the saddles were determined by the monodromy matrix of Liouville four-point function. We extended this analysis to the case of higher-spin gravity using the Chern-Simons description.