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.
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Filippini, S.A., Garcia-Fernandez, M. & Stoppa, J. Stability data, irregular connections and tropical curves. Sel. Math. New Ser. 23, 1355–1418 (2017). https://doi.org/10.1007/s00029-016-0299-x
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DOI: https://doi.org/10.1007/s00029-016-0299-x