Intertwiners for D=3 Gauge Theories

TL;DR

This paper extends the intertwiner operator method to 3D topological field theories, constructing operators on foliated manifolds and analyzing their role in quantization and Wilson loop path ordering.

Contribution

It introduces a unified approach to constructing intertwiners for 3D gauge theories, including gravity, in both covariant and canonical frameworks.

Findings

Constructed intertwiners for BF, Chern-Simons, and 3D gravity theories.

Compared canonical and holomorphic quantization methods.

Derived the path ordering of Wilson loops in Chern-Simons theory.

Abstract

We apply the intertwiner operator method of arXiv:2411.08865 to topological field theories, including BF theories, Chern-Simons theory, and three-dimensional gravity. We construct the operator on foliated manifolds while preserving covariance on the Cauchy surface, and compare canonical and holomorphic quantization, providing the intertwiner in both frameworks. For three-dimensional gravity, we present both covariant and time-gauge formulations, analyze the constraints, and construct the corresponding intertwiner. As an application, we derive the path ordering of Wilson loops in Chern-Simons theory. The study of observables is left for future work.