The Yang-Mills Gradient Flow and Loop Spaces of Compact Lie Groups

TL;DR

This paper establishes an isomorphism between Morse homologies derived from the Yang-Mills gradient flow on connections over S^2 and the heat flow homology of based loops in a compact Lie group G, connecting gauge theory and loop space topology.

Contribution

It introduces a novel coupling of Yang-Mills gradient flow with loop space flow to prove an isomorphism between their Morse homologies, answering a question posed by Atiyah.

Findings

Morse homology of Yang-Mills flow is isomorphic to heat flow homology of loop space

Coupling of gradient flows provides a new method for comparing different Morse homologies

Results confirm a conjecture relating gauge theory and loop space topology

Abstract

We study the $L^2$ gradient flow of the Yang--Mills functional on the space of connection 1-forms on a principal $G$-bundle over the sphere $S^2$ from the perspective of Morse theory. The resulting Morse homology is compared to the heat flow homology of the space $\Omega G$ of based loops in the compact Lie group $G$. An isomorphism between these two Morse homologies is obtained by coupling a perturbed version of the Yang--Mills gradient flow with the $L^2$ gradient flow of the classical action functional on loops. Our result gives a positive answer to a question due to Atiyah.