Stability data, irregular connections and tropical curves

We study a class of meromorphic connections ∇ (Z) on P1, 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 ∇ (Z) as we rescale the central charge Z↦ RZ. In the R→ 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→ ∞ “large complex structure” limit the connections ∇ (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.