Quantum Wilson surfaces and topological interactions

TL;DR

This paper introduces Wilson surfaces as 2D topological quantum field theories, explores their interaction with 2D Yang-Mills and BF theories, and computes their partition functions for various gauge groups.

Contribution

It provides a novel description of Wilson surfaces as 2D TQFTs and analyzes their topological interactions with other theories, including explicit partition function calculations.

Findings

Wilson surfaces define topological invariants for principal G-bundles.

Interaction with 2D Yang-Mills modifies partition functions.

Wilson surfaces are nontrivial for non-simply connected gauge groups.

Abstract

We introduce the description of a Wilson surface as a 2-dimensional topological quantum field theory with a 1-dimensional Hilbert space. On a closed surface, the Wilson surface theory defines a topological invariant of the principal $G$-bundle $P \to \Sigma$. Interestingly, it can interact topologically with 2-dimensional Yang-Mills and BF theories modifying their partition functions. We compute explicitly the partition function of the 2-dimensional Yang-Mills theory with a Wilson surface. The Wilson surface turns out to be nontrivial for the gauge group $G$ non-simply connected (and trivial for $G$ simply connected). In particular we study in detail the cases $G=SU(N)/\mathbb{Z}_m$, $G=Spin(4l)/(\mathbb{Z}_2\oplus\mathbb{Z}_2)$ and obtain a general formula for any compact connected Lie group.