Elliptic Yang-Mills Flow Theory

2013/03/06 by Rémi Janner, Janner, Remi, Jan Swoboda +1
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1303.1401

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 L²-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=Σ× S¹.

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