Stochastic quantization of Yang-Mills
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One of the central problems in mathematical physics is the rigorous construction of the quantum Yang-Mills theory on \mathbb{R}^4 and the proof of the existence of a positive mass gap. In this talk, I will present some recent progress on the lower dimensional cases T^2 and T^3. In particular, we construct gauge orbit spaces \mathcal{A}/~ where \mathcal{A} denotes a space of connections of negative regularity, equipped with a suitable notion of gauge equivalence ~. We then define stochastic dynamics on \mathcal{A} that respect gauge symmetry (in probability law) and induce Markov processes on \mathcal{A}/~. On lattices in any dimensions, the mixing properties of these dynamics yield the existence of a mass gap, in the strong coupling regime. Based on joint work with A. Chandra, I. Chevyrev, M. Hairer, R. Zhu and X. Zhu.