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arXiv · 2606.21909

Heat Kernel and Resurgence

Si Li·Yong Li·Xinxing Tang
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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 $H^2$.

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Si Li, Yong Li, Xinxing Tang. 2026-06-20. Heat Kernel and Resurgence. https://arxiv.org/abs/2606.21909

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