Rough Uhlenbeck Compactness with Application to 2D Yang-Mills Measure

We introduce the concept of rough additive functions, which extends rough paths theory to line integrals of distributional 1-forms. In the context of gauge theory, we use this notion to define controlled gauge transformations and holonomies via RDEs. The metric for rough additive functions can be used as a gauge-invariant quantity to prove a rough Uhlenbeck compactness result on the unit square. This compactness result is the focus of the talk. The main ingredient is a singular elliptic SPDE to obtain a Coulomb gauge, which we solve using regularity structures. Surprisingly, we manage to define the model on the singular terms occurring in the regularity structure deterministically via the given rough additive function. This leads to a phenomenon where the underlying model is determined by a much simpler—and geometrically more natural—object. Consequently, our result can be seen as the first gauge-fixed representation of the Yang-Mills measure on the unit square using PDE techniques. This is joint work with Ilya Chevyrev and Tom Klose.