Resurgence and perverse sheaves

TL;DR

This paper introduces a novel perspective on resurgence theory using perverse sheaves with algebraic convolution, extending the concept of alien derivatives within a sheaf-theoretic framework to better understand complex singularities.

Contribution

It proposes lifting alien derivatives to perverse sheaves, explores their behavior under convolution, and discusses potential generalizations for infinite singularity sets in resurgence theory.

Findings

Alien derivatives are extended to perverse sheaves.

Behavior of sheaf-theoretic convolution is analyzed.

Potential applications in Cohomological Hall Algebras and Chern-Simons theory.

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. \'Ecalle, 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.