Stability data, irregular connections and tropical curves

TL;DR

This paper investigates a class of meromorphic connections parametrized by stability conditions, revealing their connections to wall-crossing phenomena, tropical geometry, and enumerative invariants in two key limits.

Contribution

It introduces a new family of connections related to stability conditions, providing explicit formulas and linking to tropical geometry and enumerative invariants in different limits.

Findings

In the conformal limit, connections recover Bridgeland and Toledano Laredo's structures.

In the large complex structure limit, connections relate to tropical geometry and Gromov-Witten invariants.

Flat sections exhibit tropical behaviour and encode enumerative invariants.

Abstract

We study a class of meromorphic connections $\nabla(Z)$ on $\mathbb{P}^1$, parametrised by the central charge $Z$ of a stability condition, with values in a Lie algebra of formal vector fields on a torus. Their definition is motivated by the work of Gaiotto, Moore and Neitzke on wall-crossing and three-dimensional field theories. Our main results concern two limits of the families $\nabla(Z)$ as we rescale the central charge $Z \mapsto RZ$. In the $R \to 0$ "conformal limit" we recover a version of the connections introduced by Bridgeland and Toledano Laredo (and so the Joyce holomorphic generating functions for enumerative invariants), although with a different construction yielding new explicit formulae. In the $R \to \infty$ "large complex structure" limit the connections $\nabla(Z)$ make contact with the Gross-Pandharipande-Siebert approach to wall-crossing based on tropical geometry. Their flat sections display tropical behaviour, and also encode certain tropical/relative Gromov-Witten invariants.