We study the $L^2$ gradient flow of the Yang--Mills functional on the space of connection 1-forms on a principal $G$-bundle over the sphere $S^2$ from the perspective of Morse theory. The resulting Morse homology is compared to the heat flow homology of the space $\Omega G$ 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 $L^2$ gradient flow of the classical action functional on loops. Our result gives a positive answer to a question due to Atiyah.Comment: 50 page
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