GV and GW invariants via the enhanced movable cone

Nabijou, N. and Wemyss, M. (2023) GV and GW invariants via the enhanced movable cone. Moduli, (Accepted for Publication)

[img] Text
309638.pdf - Accepted Version
Restricted to Repository staff only



Given any smooth germ of a threefold flopping contraction, we first give a combinatorial characterisation of which Gopakumar–Vafa (GV) invariants are nonzero, by prescribing multiplicities to the walls in the movable cone. On the Gromov– Witten (GW) side, this allows us to describe, and even draw, the critical locus of the associated quantum potential. We prove that the critical locus is the infinite hyperplane arrangement of [IW], and moreover that the quantum potential can be reconstructed from a finite fundamental domain. We then iterate, obtaining a combinatorial description of the matrix which controls the transformation of the non-zero GV invariants under a flop. There are three main ingredients and applications: (1) a construction of flops from simultaneous resolution via cosets, which describes how the dual graph changes, (2) a closed formula which describes the change in dimension of the contraction algebra under flop, and (3) a direct and explicit isomorphism between quantum cohomologies of different crepant resolutions, giving a Coxeter-style, visual proof of the Crepant Transformation Conjecture for isolated cDV singularities.

Item Type:Articles
Status:Accepted for Publication
Glasgow Author(s) Enlighten ID:Wemyss, Professor Michael and Nabijou, Mr Navid
Authors: Nabijou, N., and Wemyss, M.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Moduli
Publisher:Cambridge University Press
ISSN (Online):2977-1382

University Staff: Request a correction | Enlighten Editors: Update this record

Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
301581The Homological Minimal Model ProgramMichael WemyssEngineering and Physical Sciences Research Council (EPSRC)EP/R009325/1M&S - Mathematics
310007MMiMMAMichael WemyssEuropean Commission (EC)101001227M&S - Mathematics