Levy Laplacian and instantons on manifold

One of the main causes for the interest in the Levy Laplacian is its connection with the gauge fields. We introduce the Levy Laplacian parameterized by the curve in $SO(4)$. This operator acts on the space of sections of the fiber bundle over the space of paths in a 4-dimensional Riemannian manifold. The connection of the Laplace equation for this Laplacian with the instantons on the 4- dimensional Riemannian manifold is studied. Such a relationship was previously considered by the author in the flat case.