Lévy differential operators and Gauge invariant equations for Dirac and Higgs fields

Abstract

We study the Lévy infinite-dimensional differential operators (differential operators defined by the analogy with the Lévy Laplacian) and their relationship to the Yang-Mills equations. We consider the parallel transport on the space of curves as an infinite-dimensional analogue of chiral fields and show that it is a solution to the system of differential equations if and only if the associated connection is a solution to the Yang-Mills equations. This system is an analogue of the equation of motion of chiral fields and contains the Lévy divergence. The systems of infinite-dimensional equations containing Lévy differential operators, that are equivalent to the Yang-Mills-Higgs equations and the Yang-Mills-Dirac equations (the equations of quantum chromodynamics), are obtained. The equivalence of two ways to define Lévy differential operators is shown.

Lévy Laplacians, holonomy group and instantons on 4-manifolds

“…Let H 1 0,0 denote the sub-bundle of H 1 0 such that the fiber of H 1 0,0 over γ ∈ Ω m is the space {X ∈ H 1 γ (T M ) : X(1) = 0}. By canonical isomorphism (12) its fiber is isomorphic to the Hilbert space…”

“…and Q S (•) are called the Volterra integral kernel, the Lévy integral kernel and the singular integral kernel respectively. It was proved in [2] that these kernels are defined in a unique way (see also [12]). This operator was introduced by Accardi, Gibilisco and Volovich in [2] for the flat case and by Leandre and Volovich in [3] for the case of Riemannian manifold.…”

“…where the Lévy trace tr AGV L is a special linear integral functional on some space of infinitedimensional bilinear forms. Also the Lévy Laplacian ∆ AGV L can been defined as a Cesàro mean of the second-order directional derivatives (see [10,11,12]). In this paper, this approach is not used.…”

“…On the contrary, a parallel transport generated by the Yang-Mills field can been considered as a chiral field with an infinite-dimensional base manifold and a finite-dimensional target space (see [20,21,22]). In [23], the equations of the motion of the chiral fields on the parallel transport with the divergence associated with the Lévy Laplacian were considered (see also [11,12]). The approach to the Yang-Mills fields based on the Lévy Laplacian goes back to the paper [23].…”

Levy Laplacian on manifold and Yang-Mills heat flow

“…The relationship of this Levy Laplacian and the Yang-Mills equations was studied in [40]. The relationship between the Yang-Mills equations and different Levy Laplacians was also studied in [41,42,43,44,45].…”