Abelian Duality at Higher Genus

In three dimensions, a free, periodic scalar field is related by duality to an abelian gauge field. Here I explore aspects of this duality when both theories are quantized on a Riemann surface of genus g. At higher genus, duality involves an identification of winding with momentum on the Jacobian variety of the Riemann surface. I also consider duality for monopole and loop operators on the surface and exhibit the operator algebra, a refinement of the Wilson-'t Hooft algebra.