Elliptic Yang-Mills flow theory

We lay the foundations of a Morse homology on the space of connections A(P) on a principal G‐bundle over a compact manifold Y, based on a newly defined gauge‐invariant functional J on A(P). While the critical points of J correspond to Yang–Mills connections on P, its L2‐gradient gives rise to a nove...

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Bibliographic Details
Published in:Mathematische Nachrichten 2015-06, Vol.288 (8-9), p.935-967
Main Authors: Janner, Rémi, Swoboda, Jan
Format: Article
Language:English
Subjects:
35R01
58E15
58J30
58J30,35R01
Morse homology
nonlinear Cauchy-Riemann equation
spectral flow
Theory
Yang-Mills functional
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Summary:We lay the foundations of a Morse homology on the space of connections A(P) on a principal G‐bundle over a compact manifold Y, based on a newly defined gauge‐invariant functional J on A(P). While the critical points of 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 analytical details of our programme in the case of a compact two‐dimensional base manifold Y. We furthermore discuss its relation to the well‐developed parabolic Morse homology over closed surfaces. Finally, an application of our elliptic theory is given to three‐dimensional product manifolds Y=Σ×S1.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201400109