Ordered exponential and its features in Yang-Mills effective action

In this paper, we discuss some non-trivial relations for ordered exponentials on smooth Riemannian manifolds. As an example of application, we study the dependence of the four-dimensional quantum Yang-Mills effective action on the special gauge transformation with respect to the background field. Also, we formulate some open questions about a structure of divergences for a special type of regularization in the presence of the background field formalism.