Levy Laplacians and instantons on manifolds

TL;DR

This paper establishes a link between anti-selfduality Yang-Mills equations and Laplace equations for infinite dimensional Levy Laplacians, introducing a new class of modified Levy Laplacians parameterized by curves in SO(4).

Contribution

It proves the equivalence of instantons and solutions to Laplace equations for modified Levy Laplacians, and introduces a new class of these Laplacians based on curves in SO(4).

Findings

Anti-selfduality Yang-Mills equations are equivalent to Laplace equations for certain Levy Laplacians.

A new class of modified Levy Laplacians parameterized by curves in SO(4) is introduced.

Connections that are instantons correspond to solutions of Laplace equations for these modified Levy Laplacians.

Abstract

The equivalence of the anti-selfduality Yang-Mills equations on the 4-dimensional orientable Riemannian manifold and Laplace equations for some infinite dimensional Laplacians is proved. A class of modificated Levy Laplacians parameterized by the choice of a curve in the group $SO(4)$ is introduced. It is shown that a connection is an instanton (a solution of the anti-selfduality Yang-Mills equations) if and only if the parallel transport generalized by this connection is a solution of the Laplace equations for some three modificated Levy Laplacians from this class.