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

TL;DR

This work computes one-loop partition functions for CS theories with compact and non-compact gauge groups using covariant gauge fixing on thermal AdS$_3$, showing that the results reproduce vacuum and non-vacuum characters of Kac–Moody algebras and, via gravity CS duality, Virasoro characters. A resonance-pole method is developed to evaluate functional determinants, and Wilson loops are used to access non-vacuum sectors, matching the full character decompositions after appropriate analytic continuation $k o k+2$ and boundary twists. The analysis demonstrates that holomorphic boundary data and a carefully chosen steepest-descent contour yield partition functions that are exactly the (vacuum) Kac–Moody characters, while gravity requires Brown–Henneaux boundary constraints to recover Virasoro characters. Overall, the paper clarifies how perturbative CS path integrals encode representation-theoretic data of both Kac–Moody and Virasoro algebras and provides explicit prescriptions for analytic continuation to non-compact groups and gravity contexts. The results reinforce the CS/WZW/AdS$_3$ correspondence and illuminate how boundary conditions and topological counterterms shape the quantum spectrum.

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.

Figures (4)

• Figure 1: Steepest Descent Paths of the $L_-$ path integral \ref{['eq:SD1eqn']}.
• Figure 2: Framing in thermal $AdS_3$. Blue arrows denote coordinate frames $\{\partial_\mu\}$ of $T(M)$ and red arrows denote local frames $\{f_a\}$ of $V$. The Dreibein is an isomorphism between them.
• Figure 3: Deformation of Steepest Descent Paths of $Z_H(t)$ wrt. $t$, \ref{['eq:ZHtdef']}.
• Figure 4: Steepest Descent Paths of $Z_{H}(t=|k|e^{i\pi})$ VS. $\tilde{Z}_{H}(k>0)$.