Resurgence and perverse sheaves
Mikhail Kapranov, Yan Soibelman
TL;DR
This work reframes resurgence theory in the language of perverse sheaves on $\mathbb{C}$ with a convolution-algebra structure, lifting alien derivatives to a sheaf-theoretic context. It develops a toy resurgent formalism using the localized category $\overline{\operatorname{Perv}}(\mathbb{C})$ and interprets Stokes- and tunneling-type data via vanishing/nearby cycles and transport maps, linking Borel-plane topology to Picard–Lefschetz data. The paper also outlines avenues to extend beyond finitely many singularities and sketches concrete potential realizations in COHA, wall-crossing structures, and Chern–Simons theory. Overall, it provides a conceptual bridge between resurgence and perverse-sheaf theory and sketches a roadmap toward a full generalized framework for infinite singularities.
Abstract
We propose a point of view on resurgence theory based on the study of perverse sheaves on the complex line carrying an algebraic structure with respect to additive convolution. In particular, we lift the concept of alien derivatives introduced originally by J. Écalle, to the framework of perverse sheaves and study its behavior under sheaf-theoretic convolution. The full fledged resurgence theory needs a (yet undeveloped) generalization of the concept of perverse sheaves allowing infinite, possibly dense, sets of singularities. We discuss possible approaches to defining such objects and some potential examples of them coming from Cohomological Hall Algebras, wall-crossing structures and Chern-Simons theory.