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

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