Morse Homology for the Yang-Mills Gradient Flow

TL;DR

This paper develops a Morse homology framework for the Yang-Mills gradient flow on Riemann surfaces, constructing a chain complex based on Yang-Mills connections and flow lines.

Contribution

It introduces a Morse-Bott chain complex for Yang-Mills theory using gradient flow lines, linking gauge theory with Morse homology.

Findings

Constructed a Morse-Bott chain complex for Yang-Mills connections.

Defined boundary operators via moduli spaces of flow lines.

Established a new homological approach to Yang-Mills theory.

Abstract

We use the Yang-Mills gradient flow on the space of connections over a closed Riemann surface to construct a Morse-Bott chain complex. The chain groups are generated by Yang-Mills connections. The boundary operator is defined by counting the elements of appropriately defined moduli spaces of Yang-Mills gradient flow lines that converge asymptotically to Yang-Mills connections.