The R-Matrix in 3d Topological BF Theory

TL;DR

This paper demonstrates how Wilson line operators in a 3d topological BF theory, related to a twisted 3d N=4 theory, realize solutions to the quantum Yang-Baxter equation through their crossing expectations.

Contribution

It establishes a connection between Wilson line operators in a specific 3d topological gauge theory and solutions to the quantum Yang-Baxter equation, including all orders in perturbation theory.

Findings

Wilson line crossings encode solutions to the quantum Yang-Baxter equation.

The theory is a topologically twisted 3d N=4 gauge theory.

Perturbative calculations confirm the all-order correspondence.

Abstract

In this paper I study Wilson line operators in a certain type of split Chern-Simons theory on a manifold with boundaries. The resulting gauge theory is a 3d topological BF theory equivalent to a topologically twisted 3d $\mathcal N=4$ theory. I show that this theory realises solutions to the quantum Yang-Baxter equation all orders in perturbation theory as the expectation value of crossing Wilson lines.