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.
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BibTeX
@misc{bringmann2026construction,
title = {{Construction of the 2D non-Abelian Yang-Mills-Higgs measure via stochastic quantisation}},
author = {Bjoern Bringmann and Sky Cao and Martin Hairer and Wenhao Zhao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03640},
eprint = {2611.03640},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03640},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Construction of the 2D non-Abelian Yang-Mills-Higgs measure via stochastic quantisation
AU - Bjoern Bringmann
AU - Sky Cao
AU - Martin Hairer
AU - Wenhao Zhao
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03640
UR - https://arxiv.org/abs/2611.03640
AB - 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.
ER -