Resurgence for general (possibly infinite-dimensional) thimble integrals

Establish that for thimble integrals I_a = ∫_{𝒞_a^θ} e^{-z f} ν, associated to holomorphic maps f from complex manifolds X to translation surfaces B, without assuming that f is a holomorphic Morse function and allowing possibly infinite-dimensional analogues, the position-domain function ι_a obtained by the thimble projection formula is resurgent; equivalently, show that the Borel transform of the asymptotic expansion of I_a is endlessly analytically continuable. This would extend the known Morse-case result asserting resurgence of ι_a.

Background

The paper explains that for one-dimensional thimble integrals with a holomorphic Morse function in the exponent, geometric arguments imply that the corresponding position-domain function (the Borel transform obtained via the thimble projection formula) is resurgent. This provides a rigorous link between thimble integrals and the analytic structure captured by resurgence theory.

It is conjectured that this resurgence property persists beyond the Morse setting to more general thimble integrals, and possibly even to infinite-dimensional settings relevant to quantum field theory. Formalizing and proving this extension would generalize current results and clarify the scope of resurgence phenomena in geometric and physical applications.

References

Geometric arguments show that when a thimble integral has a holomorphic Morse function in the exponent, as we have in at least some of our examples, the corresponding function on the position domain is always resurgentSection 6.2. Conjecturally, this property extends to more general thimble integrals---perhaps even infinite-dimensional ones examples~5--6.

— The regularity of ODEs and thimble integrals with respect to Borel summation  (2407.01412 - Fantini et al., 2024) in Section 1.3, Other results — Links to resurgence

This is indicative of a change of regime at \alpha_1 \sim 1 (as expected on physical grounds) and it leaves open the interesting possibility that the integral representation may contain non-perturbative contributions of order \exp[-A/\alpha_1] for some constant A. This could suggest a phenomenon of resurgence in high-energy gravitational scattering in this specific high-energy regime, and it could potentially pave the way towards strong-coupling calculations for \alpha_1 \geq 1 that might bypass the series expansion entirely.

— A Note on a High-Energy Limit of Gravitational Perturbation Theory  (2608.19941 - Damgaard et al., 20 Aug 2026) in Conclusion