TY - JOUR

T1 - A wall-crossing formula for Gromov–Witten invariants under variation of git quotient

AU - González, Eduardo

AU - Woodward, Chris T.

N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2023.

PY - 2024/4

Y1 - 2024/4

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85159299748&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85159299748&partnerID=8YFLogxK

U2 - 10.1007/s00208-023-02622-w

DO - 10.1007/s00208-023-02622-w

M3 - Article

AN - SCOPUS:85159299748

SN - 0025-5831

VL - 388

SP - 4135

EP - 4199

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 4

ER -