Higher Yang-Mills Theory

TL;DR

This paper extends Yang-Mills theory to higher dimensions by introducing Lie 2-groups and principal 2-bundles, deriving higher Yang-Mills equations, and demonstrating the existence of self-dual solutions in five dimensions.

Contribution

It formulates a higher-dimensional Yang-Mills theory using Lie 2-groups and principal 2-bundles, generalizing gauge theory to a new mathematical framework.

Findings

Derived higher Yang-Mills equations for 2-form connections.

Established a formulation of higher gauge theory using Lie 2-groups.

Found self-dual solutions in five-dimensional higher Yang-Mills equations.

Abstract

Electromagnetism can be generalized to Yang-Mills theory by replacing the group U(1)$ by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2-form electromagnetism to a kind of "higher-dimensional Yang-Mills theory". It turns out that to do this, one should replace the Lie group by a "Lie 2-group", which is a category C where the set of objects and the set of morphisms are Lie groups, and the source, target, identity and composition maps are homomorphisms. We show that this is the same as a "Lie crossed module": a pair of Lie groups G,H with a homomorphism t: H -> G and an action of G on H satisfying two compatibility conditions. Following Breen and Messing's ideas on the geometry of nonabelian gerbes, one can define "principal 2-bundles" for any Lie 2-group C and do gauge theory in this new context. Here we only consider trivial 2-bundles, where a connection consists of a Lie(G)-valued 1-form together with an Lie(H)-valued 2-form, and its curvature consists of a Lie(G)-valued 2-form together with a Lie(H)-valued 3-form. We generalize the Yang-Mills action for this sort of connection, and use this to derive "higher Yang-Mills equations". Finally, we show that in certain cases these equations admit self-dual solutions in five dimensions.