Aspects of Gauge Theory on Commutative and Noncommutative Tori

TL;DR

This paper explores gauge theories on commutative and noncommutative tori, focusing on Morita equivalence and the behavior of Wilson line correlators across rational Theta values, with implications for gauge theory relations.

Contribution

It investigates the constraints on Wilson line correlators necessary for smooth Theta dependence and verifies these constraints in specific gauge theories at leading order.

Findings

Constraints on Wilson line correlators are satisfied at leading order in 1/N expansion.

Relations between small and large N gauge theories are derived from these constraints.

Behavior of gauge theories on noncommutative tori is linked to commutative descriptions via Morita equivalence.

Abstract

We study aspects of gauge theory on tori which are a consequences of Morita equivalence. In particular we study the behavior of gauge theory on noncommutative tori for arbitrarily close rational values of Theta. For such values of Theta, there are Morita equivalent descriptions in terms of Yang-Mills theories on commutative tori with very different magnetic fluxes and rank. In order for the correlators of open Wilson lines to depend smoothly on Theta, the correlators of closed Wilson lines in the commutative Yang-Mills theory must satisfy strong constraints. If exactly satisfied, these constraints give relations between small and large N gauge theories. We verify that these constraints are obeyed at leading order in the 1/N expansion of pure 2-d QCD and of strongly coupled N=4 super Yang-Mills theory.