A Wilson Line Realisation of Quantum Groups

TL;DR

This paper demonstrates how Wilson line operators in 3D Chern-Simons theory can realize quantum groups, specifically showing that merging Wilson lines reproduces the quantum group coproduct, linking to moduli spaces of local systems.

Contribution

It provides a direct Feynman integral calculation showing Wilson line merging reproduces quantum group coproducts, connecting gauge theory with quantum algebra structures.

Findings

Merging Wilson lines reproduces quantum group coproducts.

Direct Feynman integral calculations confirm the theoretical link.

Connection established with moduli spaces of local systems.

Abstract

The study of this paper is Wilson line operators in 3-dimensional Chern-Simons theory on a manifold with boundaries. We prove to leading order through a direct calculation of Feynman integrals that the merging of parallel Wilson lines reproduces the coproduct on the quantum group $U_h(\mathfrak g)$. We outline a connection of this theory with the moduli spaces of local systems defined by Goncharov and Shen.