One loop partition function in AdS_3/CFT_2

TL;DR

This paper derives the 1-loop partition function in AdS_3 gravity using CFT techniques, confirming the gravitational result and extending to higher spin theories, highlighting the connection between Schottky groups and partition functions.

Contribution

It provides a CFT-based derivation of the AdS_3 1-loop partition function and generalizes the approach to higher spin gravity theories.

Findings

Exact match between CFT multi-point functions and gravitational partition functions.

Simplification of multi-point functions to two-point functions in large central charge limit.

Extension of the method to higher spin gravity and dual CFTs.

Abstract

The 1-loop partition function of the handle-body solutions in the AdS$_3$ gravity have been derived some years ago using the heat-kernel and the method of images. In the semiclassical limit, such partition function should correspond to the order $O (c^0)$ part in the partition function of dual conformal field theory on the boundary Riemann surface. The higher genus partition function could be computed by the multi-point functions in the Riemann sphere via sewing prescription. In the large central charge limit, to the leading order of $c$, the multi-point function is further simplified to be a summation over the product of two-point functions, which may form links. Each link is in one-to-one correspondence with the conjugacy class of the Schottky group of the Riemann surface. Moreover, the value of a link is determined by the eigenvalue of the element in the conjugate class. This allows us to reproduce exactly the gravitational 1-loop partition function. The proof can be generalized to the higher spin gravity and its dual CFT.