The wall-crossing formula and Lagrangian mutations

James Pascaleff, Dmitry Tonkonog

Research output: Contribution to journalArticle

Abstract

We prove a general form of the wall-crossing formula which relates the disk potentials of monotone Lagrangian submanifolds with their Floer-theoretic behaviour away from a Donaldson divisor. We define geometric operations called mutations of Lagrangian tori in del Pezzo surfaces and in toric Fano varieties of higher dimension, and study the corresponding wall-crossing formulas that compute the disk potential of a mutated torus from that of the original one. In the case of del Pezzo surfaces, this justifies the connection between Vianna's tori and the theory of mutations of Landau-Ginzburg seeds. In higher dimension, this provides new Lagrangian tori in toric Fanos corresponding to different chambers of the mirror variety, including ones which are conjecturally separated by infinitely many walls from the chamber containing the standard toric fibre.

Original languageEnglish (US)
Article number106850
JournalAdvances in Mathematics
Volume361
DOIs
StatePublished - Feb 12 2020

Keywords

  • Cluster transformation
  • Holomorphic disk
  • Lagrangian torus
  • Landau-Ginzburg potential
  • Mutation
  • Wall-crossing

ASJC Scopus subject areas

  • Mathematics(all)

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