The Yang–Mills gradient flow and loop spaces of compact Lie groups

Abstract

Abstract We study the L 2 ${L^{2}}$ gradient flow of the Yang–Mills functional on the space of connection 1-forms on a principal G-bundle over the sphere S 2 ${S^{2}}$ from the perspective of Morse theory. The resulting Morse homology is compared to the heat flow homology of the space Ω ⁢ G ${\Omega G}$ 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 L 2 ${L^{2}}$ gradient flow of the classical energy functional on loops. Our result gives a positive answer to a question due to Atiyah.