Stochastic quantisation of Yang-Mills-Higgs in 3D (Q6579143)

The purpose of this paper is to construct and study the Langevin dynamic associated to the Euclidean Yang-Mills-Higgs (YMH) measure on the torus \(\mathbf{T}^d\), for \(d=3\). The authors establish both a state space and a Markov process associated to the stochastic quantisation equation of Yang-Mills-Higgs (YMH) theories. The state space \(S\) is a nonlinear metric space of distributions that can serve as initial conditions for the (deterministic and stochastic) YMH flow with good continuity properties. By taking into account the gauge covariance of the deterministic YMH flow, they extend the gauge equivalence relation \(\sim\) to \(S\) in a canonic way, creating a quotient space of ``gauge orbits'' \(\mathcal{O}\). Using the theory of regularity structures, they demonstrate local in time solutions to the renormalised stochastic YMH flow. Additionally, symmetry arguments in the small noise limit result in a unique selection of renormalization counterterms, ensuring that these solutions are gauge covariant in law. This enables the definition of a canonical Markov process on the space of gauge orbits \(\mathcal{O}\) up to a potential finite time blow-up, associated with the stochastic YMH flow. \N\NThe work aloso includes four appendices. Appendix A discusses modelled distributions with singular behaviour at \(t=0\) (allowing for the construction of solutions starting from suitable singular initial conditions). Appendix B contains results on the deterministic YMH flow (helpful for defining the gauge equivalence). Appendix C extends the well-posedness result from \(\S\) 5 to the coupled system discussed in \(\S\) 6 and \(\S\) 7. Appendix D proves that the solution maps are injective in law as functions of renormalisation constants (which is useful for showing gauge covariance in \(\S\) 6).