Heat Kernel and Resurgence

Abstract

We study the resurgent structure of short-time heat kernel asymptotics from the viewpoint of Picard-Lefschetz theory. For a real analytic Riemannian manifold, we show the heat kernel admits a 1-Gevrey small-time expansion whose Borel transform detects complex-geometric data beyond the real geodesic sector. We formulate an infinite-dimensional Picard-Lefschetz problem of Morse-Floer type for the holomorphic energy functional on the complexified path space, and propose a heat-kernel analogue of the Picard-Lefschetz/Alien correspondence. In this framework, pointed alien operators acting on the asymptotic expansion associated with the real geodesic are predicted to produce the formal heat-kernel sectors associated with other holomorphic geodesics, with coefficients given by signed counts of connecting trajectories of the Morse flow. We perform a confirming test of this proposal on the hyperbolic plane $H^2$.