GV and GW invariants via the enhanced movable cone

TL;DR

This paper provides a combinatorial framework linking GV and GW invariants via the movable cone, describing their transformations under flops and proving the Crepant Transformation Conjecture for certain singularities.

Contribution

It introduces a novel combinatorial characterization of non-zero GV invariants, describes the critical locus of the quantum potential, and establishes explicit transformations of invariants under flops.

Findings

Characterization of non-zero GV invariants via wall multiplicities

Description of the critical locus as an infinite hyperplane arrangement

Explicit isomorphism between quantum cohomologies of crepant resolutions

Abstract

Given any smooth germ of a threefold flopping contraction, we first give a combinatorial characterisation of which Gopakumar-Vafa (GV) invariants are non-zero, 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 Iyama and the second author, 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.