The Atiyah-Singer index theorem and the gauge field copy problem

TL;DR

This paper explores how K-theory and the Atiyah-Singer index theorem can identify false gauge field copies, proposing extensions and future research directions in gauge theory.

Contribution

It introduces a novel analytical condition for false gauge copies using K-theory and the Atiyah-Singer index theorem, and discusses potential extensions beyond the theorem.

Findings

K-theory provides a way to detect false gauge copies

The Atiyah-Singer index theorem links topology and gauge field properties

Proposes possible extensions without the index theorem

Abstract

K-theory allows us to define an analytical condition for the existence of `false' gauge field copies through the use of the Atiyah-Singer index theorem. After establishing that result we discuss a possible extension of the same result without the help of the index theorem and suggest possible related lines of work.