Stability conditions and Stokes factors

TL;DR

This paper establishes a connection between stability conditions on module categories over finite-dimensional algebras and isomonodromic families of irregular connections, linking algebraic invariants to differential equations.

Contribution

It demonstrates that the space of stability conditions parametrizes an isomonodromic family of irregular connections with residues related to Joyce's counting invariants.

Findings

Space of stability conditions corresponds to an isomonodromic family of connections

Residues of connections are given by Joyce's generating function

Provides a geometric interpretation of algebraic invariants

Abstract

Let A be the category of modules over a complex, finite-dimensional algebra. We show that the space of stability conditions on A parametrises an isomonodromic family of irregular connections on P^1 with values in the Hall algebra of A. The residues of these connections are given by the holomorphic generating function for counting invariants in A constructed by D. Joyce.