Abelian Duality at Higher Genus

TL;DR

This paper investigates the duality between scalar fields and abelian gauge fields on higher genus Riemann surfaces, revealing how winding and momentum are identified and analyzing operator algebra structures.

Contribution

It extends abelian duality to higher genus surfaces, detailing the operator algebra and the interplay between winding and momentum modes.

Findings

Duality involves identification of winding and momentum on the Jacobian.

Operator algebra refines Wilson-'t Hooft algebra at higher genus.

Analysis of monopole and loop operators on Riemann surfaces.

Abstract

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.