Stability data, irregular connections and tropical curves

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 $R \rightarrow 0$ “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 $R \rightarrow \infty$ “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.