Wilson surface observables from equivariant cohomology

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 $G=SO(3)$.