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 infinitedimensional equations for parallel transport that contain the Levy 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.