Stability data, irregular connections and tropical curves

• Sara A. Filippini ^[1] ; Mario Garcia-Fernandez ^[3] ; Jacopo Stoppa ^[2]
  1. [1] Institut de Mathématiques de Marseille

     Institut de Mathématiques de Marseille

     Arrondissement de Marseille, Francia

     
  2. [2] International School for Advanced Studies

     International School for Advanced Studies

     Trieste, Italia

     
  3. [3] CSIC

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 23, Nº. 2, 2017, págs. 1355-1418
• Idioma: inglés
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• Resumen

  • 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.