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 $R\to 0$ "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.

v1: 80 pages, 7 figures. Sections 4 and 7 supersede most of arXiv:1306.3852 [math.AG]. v2: 51 pages, 4 figures. Abridged revised version. Differential-geometric material will appear elsewhere. v3 update to published version

References in corpus (6)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Instanton Correction, Wall Crossing And Mirror Symmetry Of Hitchin's Moduli Spaces
• Stability conditions on K3 surfaces
• Harmonic Maps to Buildings and Singular Perturbation Theory
• TBA type equations and tropical curves
• Opers and TBA