  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.
