Analyticity and resurgence in wall-crossing formulas

TL;DR

This paper introduces analytic stability data on vector fields, proves their topological properties, and conjectures their role in generating resurgent series, supported by multiple examples.

Contribution

It defines analytic stability data, establishes their topological nature, and proposes a conjecture linking them to resurgent series, with validation in specific cases.

Findings

Analytic stability data form an open and closed subset.

Conjecture connecting analytic data to resurgent series.

Validation of the conjecture in several examples.

Abstract

We introduce the notion of analytic stability data on the Lie algebra of vector fields on a torus. We prove that the subspace of analytic stability data is open and closed in the topological space of all stability data. We formulate a general conjecture which explains how analytic stability data give rise to resurgent series. This conjecture is checked in several examples.