Dualities in Field Theories and the Role of K-Theory

TL;DR

This paper explores dualities in quantum field theories through the lens of K-theory and noncommutative geometry, highlighting mathematical structures underlying physical equivalences.

Contribution

It provides a mathematical perspective on field theory dualities, emphasizing the role of K-theory and extending previous work with new insights in the final section.

Findings

K-theory captures charge conservation in dualities

Noncommutative geometry provides a framework for understanding equivalences

New results on the mathematical structure of dualities

Abstract

It is now known (or in some cases just believed) that many quantum field theories exhibit dualities, equivalences with the same or a different theory in which things appear very different, but the overall physical implications are the same. We will discuss some of these dualities from the point of view of a mathematician, focusing on "charge conservation" and the role played by K-theory and noncommutative geometry. Some of the work described here is joint with Mathai Varghese and Stefan Mendez-Diez; the last section is new.