Yang–Mills theory is one of the cornerstones of modern physics: electromagnetism is its simplest instance, and its non-abelian generalisations underlie our description of the strong and weak nuclear forces. At the heart of the quantum theory sits a beautiful but delicate object — a probability measure on spaces of connections (or, physically, on gauge fields), formally written as a Feynman path integral. In two dimensions this measure can be made completely rigorous, and understanding it is a lovely meeting point of probability, geometry and stochastic analysis.
This talk will start from the beginning: recasting electromagnetism as the prototype of a gauge theory, before turning to general gauge theories such as Yang–Mills. I will explain why the associated Yang–Mills measure is hard to define directly, and survey how probabilists have approached it over the years through random holonomies and lattice approximations. With this background in place, I will describe a new, direct construction of the Yang–Mills measure on surfaces of arbitrary genus, obtained jointly with N. V. Dang, Y. Guedes Bonthonneau, G. Rivière and T. D. Tô, as a genuinely random distributional connection rather than a limit of discrete models. I will explain how randomising the curvature and solving for the connection allows one to recover the classical formulas of Migdal, Witten and Lévy from a new probabilistic construction.
No prior exposure to gauge theory or Yang–Mills is assumed; the talk should be accessible to anyone with a general background in probability or geometry.
Based on the preprint arXiv:2607.24640.
