Two-dimensional Markovian Holonomy Fields

— We define a notion of Markov process indexed by curves drawn on a compact surface and taking its values in a compact Lie group. We call such a process a two-dimensional Markovian holonomy field. The prototype of this class of processes, and the only one to have been constructed before the present work, is the canonical process under the Yang-Mills measure, first defined by Ambar Sengupta [32] and later by the author [21]. The Yang-Mills measure sits in the class of Markovian holonomy fields very much like the Brownian motion in the class of Lévy processes. We prove that every regular Markovian holonomy field determines a Lévy process of a certain class on the Lie group in which it takes its values, and construct, for each Lévy process in this class, a Markovian holonomy field to which it is associated. When the Lie group is in fact a finite group, we give an alternative construction of this Markovian holonomy field as the monodromy of a random ramified principal bundle. This is in heuristic agreement with the physical origin of the Yang-Mills measure as the holonomy of a random connection on a principal bundle.