The Yang-Mills measure, formally a probability measure on principal connections, is a quantum field theory which models particles carrying the fundamental forces. In this talk, I will present two recent approaches based on stochastic analysis to gauge-fixing the Yang-Mills measure in dimension 2, which is the only dimension in which a rigorous construction of the measure is currently known. The first approach is based on lattice approximations and an Uhlenbeck-type theorem which is applicable to distributional 1-forms. The second is based on stochastic quantisation, and relies on the theory of parabolic singular stochastic PDEs. I will discuss the link between these approaches, as well as the link to other constructions in the literature, and how a novel state space helps to make sense of the space of gauge orbits on which the measure is supported. Finally, I will discuss the difficulties in extending the construction to 3 dimensions.

Based partly on a joint work with A. Chandra, M. Hairer, and H. Shen.