L\'evy Laplacians, holonomy group and instantons on 4-manifolds

Abstract

The connection between Yang--Mills gauge fields on $4$-dimensional orientable compact Riemannian manifolds and modified L\'evy Laplacians is studied. A modified L\'evy Laplacian is obtained from the L\'evy Laplacian by the action of an infinite dimensional rotation. Under the assumption that the 4-manifold has a nontrivial restricted holonomy group of the bundle of self-dual 2-forms, the following is proved. There is a modified L\'evy Laplacian such that a parallel transport in some vector bundle over the 4-manifold is a solution of the Laplace equation for this modified L\'evy Laplacian if and only if the connection corresponding to the parallel transport satisfies the Yang--Mills self-duality (anti-self-duality) equations. An analogous connection between the Laplace equation for the L\'evy Laplacian and the Yang--Mills equations was previously known.