Lévy Laplacian on manifold and Yang-Mills heat flow (Q2284235)

The Lévy Laplacian on an infinite dimensional manifold \(M\) is introduced in this article in terms of covariant derivatives, namely, as the composition of an infinite dimensional divergence operator and the \(H^{0}\)-gradient on the space of sections of the vector bundle over the base Hilbert manifold of all curves in \(M\). The author calls this the \textit{covariant Lévy Laplacian} and conjectures that on its domain of definition it coincides with the Lévy Laplacian introduced by Léandre-Volovich, as well as with the Levy Laplacian by Accardi-Smolyanov. The Main Theorem of the article states that a flow of connections on a finite dimensional vector bundle solves the Yang-Mills heat equations if and only if the associated flow of the parallel transports solves the heat equation for the covariant Lévy Laplacian over the infinite dimensional manifold \(M\). This generalizes a result by Accardi-Gibilisco-Volovich relating the Yang-Mills equations to the Laplace equation for the Lévy Laplacian.