We study a class of meromorphic connections nabla(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 nabla(Z) as we rescale the central charge Z to 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.

Stability data, irregular connections and tropical curves / Filippini, S. A.; Garcia Fernandez, M.; Stoppa, Jacopo. - In: SELECTA MATHEMATICA. NEW SERIES. - ISSN 1420-9020. - 23:2(2017), pp. 1355-1418. [10.1007/s00029-016-0299-x]

Stability data, irregular connections and tropical curves

Stoppa, Jacopo
2017-01-01

Abstract

We study a class of meromorphic connections nabla(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 nabla(Z) as we rescale the central charge Z to 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.
2017
23
2
1355
1418
https://arxiv.org/abs/1403.7404
http://cdsads.u-strasbg.fr/abs/2014arXiv1403.7404F
Filippini, S. A.; Garcia Fernandez, M.; Stoppa, Jacopo
File in questo prodotto:
File Dimensione Formato  
2016 Stoppa.pdf

non disponibili

Tipologia: Versione Editoriale (PDF)
Licenza: Non specificato
Dimensione 1.08 MB
Formato Adobe PDF
1.08 MB Adobe PDF   Visualizza/Apri   Richiedi una copia
Stoppa.pdf

Open Access dal 20/11/2017

Tipologia: Documento in Post-print
Licenza: Non specificato
Dimensione 476.59 kB
Formato Adobe PDF
476.59 kB Adobe PDF Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/15966
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 14
  • ???jsp.display-item.citation.isi??? 13
social impact