# Kac-Moody and Virasoro Characters from the Perturbative Chern-Simons Path Integral

## TL;DR

This paper computes Chern-Simons partition functions using a one-loop functional integral approach, successfully deriving Kac-Moody and Virasoro characters, and extends the method to Euclidean AdS3 gravity, revealing the need for analytic continuations.

## Contribution

It introduces a covariant gauge fixing method for computing Chern-Simons partition functions and extends the approach to non-compact groups and 3D gravity, highlighting the importance of analytic continuations.

## Key findings

- Correctly computes Kac-Moody characters
- Extends to non-compact groups via analytic continuation
- Reveals the necessity of gauge field continuations in gravity

## Abstract

We evaluate to one loop the functional integral that computes the partition functions of Chern-Simons theories based on compact groups, using the background field method and a covariant gauge fixing. We compare our computation with the results of other, less direct methods. We find that our method correctly computes the characters of irreducible representations of Kac-Moody algebras. To extend the computation to non-compact groups we need to perform an appropriate analytic continuation of the partition function of the compact group. Non-vacuum characters are found by inserting a Wilson loop in the functional integral. We then extend our method to Euclidean Anti-de Sitter pure gravity in three dimensions. The explicit computation unveils several interesting features and lessons. The most important among them is that the very definition of gravity in the first-order Chern-Simons formalism requires non-trivial analytic continuations of the gauge fields outside their original domains of definition.