Rough analysis meets 2D Yang–Mills theory

Abstract

We study probabilistic two-dimensional Yang–Mills theory with tools from rough analysis across three different projects. The first project develops tools for the Yang–Mills–Higgs measure: we define a Polish state space that carries Wilson loop observables for the gauge fields and string observables for the Higgs fields. We also provide conditions under which a separable Banach space with a continuous action of a topological group admits a Polish quotient space—conditions that find application in both the first and third projects.

The second project is the main contribution of the thesis and is concerned with rough Uhlenbeck compactness, which provides a gauge fixing for the Yang–Mills measure. In this project, we introduce the concept of rough additive functions, extending rough paths theory to line integrals of distributional 1-forms. In the context of gauge theory, this allows us to define controlled gauge transformations and holonomies via rough differential equations. The metric for rough additive functions serves as a gauge-invariant quantity used to prove a rough Uhlenbeck compactness result on the unit square. The main ingredient is a singular elliptic stochastic partial differential equation used 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.

In the third and final project we initiate an operator approach to Yang–Mills theory. We study Dirac operators induced by the covariant derivative associated to a connection form. This operator is viewed as an unbounded operator on an Lp-space of differential forms. We characterise the domain via paraproducts, prove resolvent estimates and further analytic properties of this operator. Using spectral theory we define gauge-invariant observables that characterise the underlying forms up to gauge equivalence. This is a new result even in the smooth setting. Finally, we prove closedness of orbits under gauge transformations of a Hölder–Besov space of connection forms of certain regularity using resolvent estimates. This is a crucial step which we use to prove that a certain quotient space is Polish. That space is important as the Yang–Mills measure is supported on such a space under a suitable viewpoint.