Invariant measure and universality of the 2D Yang-Mills Langevin dynamic

TL;DR

This paper proves the invariance and universality of the 2D Yang-Mills measure under Langevin dynamics, using advanced mathematical techniques to connect discrete lattice models with continuum theories.

Contribution

It introduces a novel combination of regularity structures, gauge fixing, and Bourgain's method to establish measure invariance and universality for 2D Yang-Mills Langevin dynamics.

Findings

Yang-Mills measure is invariant under the Langevin dynamic.

Universality of the YM measure from discrete lattice gauge theories.

Proof of uniqueness for mass renormalisation in gauge-covariant Langevin dynamics.

Abstract

We prove that the Yang-Mills (YM) measure for the trivial principal bundle over the two-dimensional torus, with any connected, compact structure group, is invariant for the associated renormalised Langevin dynamic. Our argument relies on a combination of regularity structures, lattice gauge-fixing, and Bourgain's method for invariant measures. Several corollaries are presented including a gauge-fixed decomposition of the YM measure into a Gaussian free field and an almost Lipschitz remainder, and a proof of universality for the YM measure that we derive from a universality for the Langevin dynamic for a wide class of discrete approximations. The latter includes standard lattice gauge theories associated to Wilson, Villain, and Manton actions. An important step in the argument, which is of independent interest, is a proof of uniqueness for the mass renormalisation of the gauge-covariant continuum Langevin dynamic, which allows us to identify the limit of discrete approximations. This latter result relies on Euler estimates for singular SPDEs and for Young ODEs arising from Wilson loops.