In this talk, I will discuss the construction of the Yang-Mills-Higgs measure on the two-dimensional torus. In the 1980s, Parisi and Wu proposed a dynamical approach to constructing such measure via the corresponding Langevin dynamics, aiming to sidestep the difficulty of fixing a global gauge. Our result carries out this program rigorously and provides the first construction of the measure in 2D. I will begin by introducing the Yang-Mills-Higgs model and explaining Parisi and Wu's idea and will then discuss the new ingredients needed to implement this approach rigorously, including a dynamical way of gauge fixing, covariant heat kernel estimates, and the exploitation of the instability of two-dimensional Yang-Mills connections.
This is based on ongoing joint work with Bjoern Bringmann, Sky Cao, and Martin Hairer.