Construction of the 2D non-Abelian Yang-Mills-Higgs measure via stochastic quantisation

Abstract

We provide the first mathematical construction of the non-Abelian Yang-Mills-Higgs measure on a two-dimensional torus by implementing Parisi and Wu's ``stochastic quantisation'' procedure. We use a specific gauge covariant (but non-Markovian) approximation to the previously constructed associated Langevin dynamics and develop a covariant paracontrolled calculus for it. This allows us to obtain short-time control over times that are much longer than what the cubic nonlinearity naïvely suggests. When combined with the decay properties of the Yang-Mills-Higgs action under the deterministic gradient flow, as well as a quantitative measure of how the noise pushes the dynamic away from Yang-Mills connections, this is good enough to yield uniform-in-time bounds. We finally show that there exists a renormalisation under which our gauge covariant approximation converges to the Langevin dynamics (modulo gauge transformation) and that the latter is topologically irreducible and strong Feller. It therefore admits a unique invariant measure, providing the sought-after construction. Our construction is non-perturbative and works in particular for any fixed values of the coupling constants.