Partition Functions of Chern-Simons Theory on Handlebodies by Radial Quantization

TL;DR

This paper employs radial quantization to compute Chern-Simons partition functions on handlebodies of any genus, revealing a method to obtain these functions through transition amplitudes between specific quantum states.

Contribution

It introduces a novel radial quantization approach to compute Chern-Simons partition functions on arbitrary genus handlebodies, including non-Abelian cases.

Findings

Partition functions derived for Abelian theories on arbitrary genus handlebodies.

Reproduction of known genus one non-Abelian partition functions.

Identification of a unique Hilbert space state via Kähler quantization.

Abstract

We use radial quantization to compute Chern-Simons partition functions on handlebodies of arbitrary genus. The partition function is given by a particular transition amplitude between two states which are defined on the Riemann surfaces that define the (singular) foliation of the handlebody. The final state is a coherent state while on the initial state the holonomy operator has zero eigenvalue. The latter choice encodes the constraint that the gauge fields must be regular everywhere inside the handlebody. By requiring that the only singularities of the gauge field inside the handlebody must be compatible with Wilson loop insertions, we find that the Wilson loop shifts the holonomy of the initial state. Together with an appropriate choice of normalization, this procedure selects a unique state in the Hilbert space obtained from a K\"ahler quantization of the theory on the constant-radius Riemann surfaces. Radial quantization allows us to find the partition functions of Abelian Chern-Simons theories for handlebodies of arbitrary genus. For non-Abelian compact gauge groups, we show that our method reproduces the known partition function at genus one.