Lévy Laplacians and instantons on manifolds

Abstract

The equivalence of the anti-selfduality Yang–Mills equations on the four-dimensional orientable Riemannian manifold and the Laplace equations for some infinite-dimensional Laplacians is proved. A class of modified Lévy Laplacians parameterized by the choice of a curve in the group [Formula: see text] 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 modified Lévy Laplacians from this class.

Lévy Laplacians, holonomy group and instantons on 4-manifolds

“…It turns out that, in the case of a Riemannian manifold, the situation is influenced by the holonomy group. In this paper, we strengthen the results of [16]. We prove that an analog of Theorem B holds for orientable compact Riemannian manifolds with nontrivial restricted holonomy group Hol 0 m (Λ 2 + (T * M )) of the bundle of self-dual 2-forms.…”

“…The group of 4-dimensional rotations SO(4) has two normal subgroups S 3 L and S 3 R . In the previous works [15,16], a connection between Laplace equations for the modified Lévy Laplacians ∆ W L for W ∈ C 1 ([0, 1], S 3 L ) (for W ∈ C 1 ([0, 1], S 3 R )) and the Yang-Mills anti-self-duality (self-duality) equations was discovered. In this paper, we have strengthened the results of these works.…”

“…The form of the Volterra kernel in (15) does not play a role in the present paper. The exact expression of this kernel can been found in [16].…”

“…In the proof of Theorem B, the fact that R 4 is not compact was essentially used. In [16] by author, the following theorem was proved for instantons on an orientable Riemannian 4-manifold M .…”

Modified Lévy Laplacian on manifold

“…Note the Lévy Laplacian in [2] was defined in a different way from (1), namely as a special integral functional defined by a special form of the second derivative. But it was proved in [3,4] that this Lévy Laplacian can been defined by (1) if we choose the Sobolev space H 1 ([0, 1], R d ) as E, the Hilbert space L 2 ([0, 1], R d ) as H and some natural basis {e n } in L 2 ([0, 1], R d ) (see also [5]). The theorem on equivalence of the Laplace equation for the Lévy Laplacian and the Yang-Mills equations was generalized for manifolds (see [6,5]).…”