Quantum Holonomy Fields

Abstract

We investigate lattice and continuous quantum gauge theories on the Euclidean plane with a structure group that is replaced by a H-algebra. H-algebras are non-commutative analogues of groups and contain the class of Voiculescu's dual groups. We are interested in non-commutative analogues of random gauge fields, which we describe through the random Holonomy that they induce. We propose a general definition of a Holonomy Field with symmetries displaying the structure of a H-algebra and construct such a field starting from a quantum Lévy process on a H-algebra in the category of probability spaces. We call them Quantum Holonomy Fields. We also consider the more abstract case of an H-algebra in a given algebraic category. This yields the notion of a Categorical Holonomy Field. As an application, we define higher dimensional generalizations of the so-called Master Field on the plane.