Stability data, irregular connections and tropical curves

Sara Angela Filippini (1) , Mario A Garcia-Fernandez (2) , Jacopo Stoppa (3)

1 I2M - Institut de Mathématiques de Marseille
2 XLIM-SRF - Systèmes RF
3 SISSA MathLab [Trieste]

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.

Domaines: Géométrie algébrique [math.AG]

Sara Angela Filippini, Mario A Garcia-Fernandez, Jacopo Stoppa. Stability data, irregular connections and tropical curves. Selecta Mathematica (New Series), 2017, 54, pp.103 - 1418. ⟨10.1007/s00029-016-0299-x⟩. ⟨hal-01502316⟩