A wall-crossing formula for Gromov–Witten invariants under variation of git quotient

Eduardo González, Chris T. Woodward

Research output: Contribution to journalArticlepeer-review

Abstract

We prove a quantum version of a wall-crossing formula of Kalkman (Kalkman in J Reine Angew Math 485:37–52, 1995; Lerman in Math Res Lett 2:247–258, 1995) that compares intersection pairings on geometric invariant theory (git) quotients related by a change in polarization. Each expression in the classical formula is quantized in the sense that it is replaced by an integral over moduli spaces of certain stable maps; in particular, the wall-crossing terms are gauged Gromov–Witten invariants with smaller structure group. As an application, we show that the genus zero graph Gromov–Witten potentials of quotients related by wall-crossings of crepant type are equivalent up to a distribution in one of the quantum parameters that is almost everywhere zero. This is a version of the crepant transformation conjecture of Li–Ruan (Invent Math 145(1):151–218, 2001), Bryan–Graber (Algebraic Geometry—Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol 80. American Mathematical Society, Providence, pp 23–42, 2009), Coates–Ruan (Quantum cohomology and crepant resolutions: a conjecture, arXiv:0710.5901, 2007) etc. in cases where the crepant transformation is obtained by variation of git.

Original languageEnglish (US)
Pages (from-to)4135-4199
Number of pages65
JournalMathematische Annalen
Volume388
Issue number4
DOIs
StatePublished - Apr 2024

All Science Journal Classification (ASJC) codes

  • General Mathematics

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