Wilson surface observables from equivariant cohomology

TL;DR

This paper develops a new path integral formulation for Wilson surface observables in gauge theories using equivariant cohomology, extending their definition to arbitrary surfaces and analyzing specific cases like U(1) and SO(3).

Contribution

It introduces a novel equivariant cohomology-based approach to define Wilson surface observables on arbitrary 2D surfaces, including closed surfaces, and connects them with Poisson sigma-models.

Findings

Wilson surface observable is nontrivial for non-simply connected groups.

New path integral representation of Wilson lines via Poisson sigma-models.

Explicit analysis of Wilson surfaces for G=U(1) and G=SO(3).

Abstract

Wilson lines in gauge theories admit several path integral descriptions. The first one (due to Alekseev-Faddeev-Shatashvili) uses path integrals over coadjoint orbits. The second one (due to Diakonov-Petrov) replaces a 1-dimensional path integral with a 2-dimensional topological $\sigma$-model. We show that this $\sigma$-model is defined by the equivariant extension of the Kirillov symplectic form on the coadjoint orbit. This allows to define the corresponding observable on arbitrary 2-dimensional surfaces, including closed surfaces. We give a new path integral presentation of Wilson lines in terms of Poisson $\sigma$-models, and we test this presentation in the framework of the 2-dimensional Yang-Mills theory. On a closed surface, our Wilson surface observable turns out to be nontrivial for $G$ non-simply connected (and trivial for $G$ simply connected), in particular we study in detail the cases $G=U(1)$ and $G=SO(3)$.