Computing genus-zero twisted Gromov-Witten invariants
type: Item
instance of: scholarly article
title: Computing genus-zero twisted Gromov-Witten invariants
zbMATH Open document ID: 1176.14009
DOI: 10.1215/00127094-2009-015
author: Tom Coates
author: Alessio Corti
author: Hiroshi Iritani
author: Hsian-Hua Tseng
published in: Duke Mathematical Journal
publication date: 2009-05-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0702234
review text: The paper under review concerns twisted Gromov-Witten invariants of orbifolds. Fixing an orbifold \(\mathcal X\), we let \(\mathcal X_{g,n,d}\) be the moduli space of stable maps to \(\mathcal X\). Each vector bundle \(E\) over \(\mathcal X\) gives rise to a \(K\)-theory class \[ E_{g,n,d}=\pi_{!} f^{\star} E \] via pushforward from the universal family of the pullback of \(E\). The twisted Gromov-Witten invariants are integrals of the form \[ \int_{[\mathcal X_{g, n,d}]^{vir}} c(E_{g,n,d}) \prod_{i=1}^{n} {\text{ev}}_i^{\star} \alpha_i \psi_i^{k_i} \] for nonnegative integers \(k_i\) and classes \(\alpha_i\in H^{\star}_{orb}(\mathcal X).\) Here \(c\) denotes an invertible multiplicative characteristic class. The computation of twisted invariants is considered in this paper. In particular, the authors prove a ``quantum Lefschetz theorem'' expressing genus zero one point twisted invariants of complete intersections in terms of the invariants of the ambient orbifold. Furthermore, the genus zero potential of \(\left[\mathbb C^2/\mathbb Z_n\right]\) are computed. Calculations for \(\left[\mathbb C^3/\mathbb Z_3\right]\) also also presented. The appendix proves Ruan's Crepant Resolution Conjecture for type A surface singularity. Letting \(Y\) be the crepant resolution of the singularity \(\mathcal X=\left[\mathbb C^2/\mathbb Z_n\right]\), the Conjecture gives an isomorphism between the small quantum cohomology of \(Y\) and the quantum cohomology of \(\mathcal X\) after analytic continuation in the quantum parameters and specializiation of some quantum parameters to roots of unity. In fact, a stronger statement of Bryan-Graber is proved here, matching the big quantum cohomology after analytic continuation and specialization, via a linear isomorphism preserving the orbifold Poincaré pairing.
reviewed by: Dragos Oprea
Mathematics Subject Classification ID: 14N35
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 53D45
zbMATH DE Number: 5550765
MaRDI profile type: Publication
OpenAlex ID: W2042381550
arXiv ID: math/0702234
cites work: Q4409406
cites work: Gromov-Witten theory of Deligne-Mumford stacks
cites work: Topological strings and (almost) modular forms
cites work: Torus actions on symplectic manifolds
cites work: Quantum periods. I: Semi-infinite variations of Hodge structures
cites work: Quantum cohomology of [ℂN/μr]
cites work: Another way to enumeration of rational curves with torus actions
cites work: The orbifold Chow ring of toric Deligne-Mumford stacks
cites work: Q3622185
cites work: Root systems and the quantum cohomology of ADE resolutions
cites work: Q3622186
cites work: The orbifold quantum cohomology of $\mathbb{C}^{2}/\mathbb{Z}_3$ and Hurwitz-Hodge integrals
cites work: A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory.
cites work: Q4409407
cites work: A new cohomology theory of orbifold
cites work: Local mirror symmetry: Calculations and interpretations
cites work: Givental's Lagrangian cone and \(S^1\)--equivariant Gromov-Witten theory
cites work: The quantum orbifold cohomology of weighted projective spaces
cites work: Quantum Riemann-Roch, Lefschetz and Serre
cites work: Wall-crossings in toric Gromov-Witten theory. I: Crepant examples
cites work: Q4237088
cites work: Hodge integrals and Gromov-Witten theory
cites work: Hodge structures for orbifold cohomology
cites work: Q5465207
cites work: Equivariant Gromov - Witten Invariants
cites work: A mirror theorem for toric complete intersections
cites work: Q4534370
cites work: Q5465208
cites work: Localization of virtual classes
cites work: Q3438270
cites work: Convergence of quantum cohomology by quantum Lefschetz
cites work: Autoequivalences of derived categories on the minimal resolutions of \(A_n\)-singularities on surfaces
cites work: Q4409410
cites work: Quantum hyperplane section theorem for homogeneous spaces
cites work: Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee
cites work: Quantum Lefschetz hyperplane theorem
cites work: Mirror principle. I
cites work: Mirror principle. II
cites work: Q4820358
cites work: Über die Lösung algebraischer Gleichungssysteme durch hypergeometrische Funktionen
cites work: Q4788849
cites work: Q3216763
cites work: Q4248266
cites work: CHEN–RUAN COHOMOLOGY OF ADE SINGULARITIES
cites work: GKZ Hypergeometric Structures
cites work: Riemann-Roch theorems for Deligne-Mumford stacks
description: scientific article