Noncommutative tori, Yang-Mills and string theory

TL;DR

This paper reviews the mathematical and physical aspects of noncommutative tori, focusing on Yang-Mills theory, Morita equivalence, and their connections to string theory and T-duality.

Contribution

It provides an overview of differential geometry, Yang-Mills theory, and dualities related to noncommutative tori, highlighting recent developments and the noncommutative Nahm transform.

Findings

Link between T-duality and Morita equivalence

Overview of noncommutative Yang-Mills theory

Introduction of noncommutative Nahm transform

Abstract

Noncommutative tori are among historically the oldest and by now the most developed examples of noncommutative spaces. Noncommutative Yang-Mills theory can be obtained from string theory. This connection led to a cross-fertilization of research in physics and mathematics on Yang-Mills theory on noncommutative tori. One important result stemming from that work is the link between T-duality in string theory and Morita equivalence of associative algebras. In this article we give an overview of the basic results in differential geometry of noncommutative tori. Yang-Mills theory on noncommutative tori, the duality induced by Morita equivalence and its link with the T-duality are discussed. Noncommutative Nahm transform for instantons is introduced.