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

2011/04/28 by Swoboda, Jan
#35K55 #58E05 #58E15 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1104.5514

Abstract

We study the L² gradient flow of the Yang--Mills functional on the space of connection 1-forms on a principal G-bundle over the sphere S² from the perspective of Morse theory. The resulting Morse homology is compared to the heat flow homology of the space ΩG 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 L² gradient flow of the classical action functional on loops. Our result gives a positive answer to a question due to Atiyah.

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