Intertwiners for D=3 Gauge Theories

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.