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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 ℍR2.

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Cite

BibTeXRIS
BibTeX
@misc{li2026heat,
  title = {{Heat Kernel and Resurgence}},
  author = {Si Li and Yong Li and Xinxing Tang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.21909},
  eprint = {2606.21909},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.21909},
}
RIS
TY  - UNPB
TI  - Heat Kernel and Resurgence
AU  - Si Li
AU  - Yong Li
AU  - Xinxing Tang
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2606.21909
UR  - https://arxiv.org/abs/2606.21909
AB  - 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_R^2.
ER  -