Elliptic Yang-Mills Flow Theory

TL;DR

This paper develops a new elliptic Morse homology framework for Yang-Mills connections on principal bundles over compact manifolds, contrasting with traditional parabolic approaches, and applies it to 2D and 3D cases.

Contribution

It introduces a gauge-invariant functional leading to elliptic equations for Yang-Mills theory, providing a novel analytical foundation and connecting to existing Morse homology methods.

Findings

Established elliptic Morse homology for Yang-Mills connections.

Analyzed the case of 2D base manifolds with complete analytical details.

Applied the theory to 3D product manifolds  .

Abstract

We lay the foundations of a Morse homology on the space of connections on a principal $G$-bundle over a compact manifold $Y$, based on a newly defined gauge-invariant functional $\mathcal J$. While the critical points of $\mathcal J$ correspond to Yang-Mills connections on $P$, its $L^2$-gradient gives rise to a novel system of elliptic equations. This contrasts previous approaches to a study of the Yang-Mills functional via a parabolic gradient flow. We carry out the complete analytical details of our program in the case of a compact two-dimensional base manifold $Y$. We furthermore discuss its relation to the well-developed parabolic Morse homology of Riemannian surfaces. Finally, an application of our elliptic theory is given to three-dimensional product manifolds $Y=\Sigma\times S^1$.