Applications of Lévy Differential Operators in the Theory of Gauge Fields

This paper is a survey of results on the relationship between gauge fields and infinite-dimensional equations for parallel transport that contain the Lévy Laplacian or the divergence associated with this Laplacian. Also we analyze the deterministic case where parallel transports are operator-valued functionals on the space of curves and the case of the Malliavin calculus where (stochastic) parallel transports are operator-valued Wiener functionals.