Stability data, irregular connections and tropical curves

We study a class of meromorphic connections ∇ ( Z ) on 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...

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Bibliographic Details
Published inSelecta mathematica (Basel, Switzerland) Vol. 23; no. 2; pp. 1355 - 1418
Main Authors Filippini, Sara A., Garcia-Fernandez, Mario, Stoppa, Jacopo
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.04.2017
Springer Nature B.V
Springer Verlag
Subjects
Algebraic Geometry
Fields (mathematics)
Invariants
Lie groups
Markov analysis
Mathematical analysis
Mathematics
Mathematics and Statistics
Stability
Toruses
14T05
14N35
34M40
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Summary:We study a class of meromorphic connections ∇ ( Z ) on 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 ∇ ( Z ) as we rescale the central charge Z ↦ R Z . 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.
ISSN:1022-1824
1420-9020
DOI:10.1007/s00029-016-0299-x