Levy Laplacian on a Four-Dimensional Riemannian Manifold

Abstract

It is known that the Yang-Mills equations for a connection are equivalent to the Levy-Laplace equation for the parallel transport associated with this connection (see the paper [4] by L. Accardi, P. Gibilisco and I. V. Volovich). The Levy-Laplace equation is the Laplace equation for Levy Laplacian, which can be defined as the Cesaro mean of the second-order directional derivatives along the vectors from the orthonormal basis of some Hilbert space. The author’s work [11] has proved for the case of a four-dimensional Euclidean space, that with a certain choice of the orthonormal basis, the Laplace-Levy equation for a parallel transport becomes equivalent to the self-duality equations for a connection. A connection that is a solution to the self-duality equations is called an instanton and is a solution to the Yang-Mills equations. In the paper we define the Levy Laplacian for the case of a four-dimensional Riemannian manifold. This operator is a generalization of both the Levy Laplacian, introduced by the author in [11], and the Levy Laplacian, introduced in [3] by L. Accardi and О. G. Smolyanov for a Riemannian manifold. In the paper we consider the case of a line bundle over a four-dimensional Riemannian manifold and Maxwell's equations (the commutative case of the Yang-Mills equations). We find the conditions, which imply that from the fact that the parallel transport is a harmonic functional for the introduced Levy Laplacian follows that the appropriate connection is a solution to the self-duality equations. In addition, the paper considers the relationship of the introduced Levy Laplacian and the Laplace-Beltrami operator on a manifold. It can be expected that the results of the paper can be generalized to the non-commutative case of Yang-Mills fields.