Semiclassical analysis for Yang--Mills random connections on compact surfaces

We introduce anisotropic Banach spaces of distributional 1-forms on compact surfaces designed to capture the fine regularity properties of Morse gauge-fixed Yang–Mills random connections. Using singular connections supported on unstable curves of a Morse gradient flow, we provide a new description of the moduli space of flat connections and of its canonical Atiyah–Bott–Goldman symplectic form. When the group is the special unitary group, we prove that in the zero-area limit, the Yang–Mills random connection converges, within these anisotropic spaces, to a random distributional 1-form whose law coincides with our Morse-theoretic representative of the Atiyah–Bott–Goldman measure. Our approach extends the works of Witten, Forman, Liu, and Sengupta by establishing the zero-area limit of the Yang–Mills measure at the level of random distributional connections, rather than only at the level of holonomies. This answers a question of Thierry Lévy on the semiclassical limits of Yang–Mills random connections.