I'm a physicist by trade, and I've recently been working with the dimensional reduction of 4d supersymmetric gauge theories. In particular, the GL-twisted $\mathcal{N}=4$ gauge theory with a simple gauge group $G$ reduces on two compact dimensions to a 2d sigma model with its target being the moduli space of Hitchin's equations (See Kapustin-Witten for more information). Being simple, it means that we can't have any abelian gauge groups. This got me thinking about what it means if we get $G=U(1)$.
So my question is this: Is there an abelian version of Hitchin's equations?