Canonical gauges in higher gauge theory

TL;DR

This paper develops canonical gauge choices for higher gauge theories, simplifying their structure and revealing that certain selfdual connections inherently satisfy higher Yang-Mills equations in specific dimensions.

Contribution

It introduces natural, essentially unique Coulomb gauges for 2- and 3-connections, making higher gauge theories more linear and tractable.

Findings

Existence of canonical Coulomb gauges in higher gauge theories.

Selfdual 2- and 3-connections are automatically solutions to higher Yang-Mills equations in specific dimensions.

Gauges simplify the structure of higher gauge theories, facilitating their analysis.

Abstract

We study the problem of finding good gauges for connections in higher gauge theories. We find that, for $2$-connections in strict $2$-gauge theory and $3$-connections in $3$-gauge theory, there are local "Coulomb gauges" that are more canonical than in classical gauge theory. In particular, they are essentially unique, and no smallness of curvature is needed in the critical dimensions. We give natural definitions of $2$-Yang-Mills and $3$-Yang-Mills theory and find that the choice of good gauges makes them essentially linear. As an application, (anti-)selfdual $2$-connections over $B^6$ are always $2$-Yang-Mills, and (anti-)selfdual $3$-connections over $B^8$ are always $3$-Yang-Mills.