Intertwiners for D=3 Gauge Theories

TL;DR

This work develops a general BRST intertwiner framework for D=3 gauge theories and gravity by grafting a finite-filtering construction onto Q = Q_0 + Q_I and solving for an operator Omega that maps between simple and full BRST cohomologies. It applies the method across a spectrum of models—Maxwell theory with and without scalars, non-Abelian Chern-Simons theory, BF theory with a cosmological term, and 2+1 gravity in canonical and time-gauge formulations, including cosmological constant effects—and derives explicit intertwiners in both canonical and holomorphic quantization, as well as nonlocal field representations that reproduce Wilson loops as BRST-invariant objects. A key outcome is the explicit construction of Omega via Gamma = [Q_1, R]_+ and its variants, along with invariant fields such as φ^{inv}(x) = e^{i e t Φ(x)} φ(x) and gravity analogues built from nonlocal Φ^a, Φ^{ab}, enabling a unified treatment of quantization schemes and observables in 3D topological theories. The results furnish a versatile toolkit for relating different quantization pictures, clarifying the role of BRST cohomology in topological theories, and paving the way to extending these ideas to higher dimensions and more intricate observables, including Wilson loops and their higher-dimensional analogues.

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.