Lévy Laplacians and instantons

We describe dual and antidual solutions of the Yang–Mills equations by means of Lévy Laplacians. To this end, we introduce a class of Lévy Laplacians parameterized by the choice of a curve in the group $\mathrm {SO}(4)$. Two approaches are used to define such Laplacians: (i) the Lévy Laplacian can be defined as an integral functional defined by a curve in $\mathrm {SO}(4)$ and a special form of the second-order derivative, or (ii) the Lévy Laplacian can be defined as the Cesàro mean of second-order derivatives along vectors from the orthonormal basis constructed by such a curve. We prove that under some conditions imposed on the curve generating the Lévy Laplacian, a connection in the trivial vector bundle with base $\mathbb R^4$ is an instanton (or an anti-instanton) if and only if the parallel transport generated by the connection is harmonic for such a Lévy Laplacian.