Langevin Dynamics of Yang-Mills in 2D and 3D

We will discuss stochastic quantization i.e. Langevin dynamic of the Yang-Mills model on two and three dimensional torus. This is described by a Lie algebra valued stochastic PDE driven by space-time white noise. We construct (local) solution to this singular SPDE via a limit of smooth regularizations; gauge invariance is broken in the regularizations but is restored in the limit. This then defines a Markov process on the space of gauge orbits. We also explain what it means by an orbit space in this singular setting, namely we construct new state and orbit spaces on which holonomies and Wilson loop observables are well-defined. We also compare these constructions in 2D and 3D. These are based on joint work with Ajay Chandra, Ilya Chevyrev, and Martin Hairer. Time permitting, we will also briefly discuss some progress on Langevin dynamic of lattice Yang-Mills, based on joint work with Ilya Chevyrev, Scott Smith, Rongchan Zhu and Xiangchan Zhu.