Year: 2026
Type: preprint
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 $\mathbb{H}_{\R}^2$.
Source: arXiv (Cornell University)
Authors: Li, Si, Li, Yong, Tang, Xinxing
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7165687298