Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum cohomology of [C^N/μ_r]

Arend Bayer, Charles Cadman

TL;DR

The paper develops a comprehensive framework to compute genus-0 Gromov-Witten invariants of local orbifolds [\mathbb{C}^N/\mu_r] by constructing the moduli space of stable maps to B\mu_r via r-th root operations on \overline{M}_{0,n} and by introducing weighted stable maps. A central achievement is a closed formula for the equivariant Euler class of the obstruction bundle, assembled from μ_r-eigenspace data of the Hodge bundle and boundary divisors, which yields linear recursions for all genus-0 GW invariants, including explicit recursions for the local threefold [\mathbb{C}^3/\mu_r], notably [\mathbb{C}^3/\mu_3]. The methodology unifies twisted and weighted stable maps, leverages wall-crossing for weight changes, and connects to the twisted I-function and mirror-map in Givental's formalism, providing a practical computational toolkit for local orbifold GW theory and potential implications for crepant-resolution phenomena.

Abstract

We give a construction of the moduli space of stable maps to the classifying stack Bμ_r of a cyclic group by a sequence of r-th root constructions on M_{0, n}. We prove a closed formula for the total Chern class of μ_r-eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus zero Gromov-Witten theory of stacks of the form [C^N/μ_r]. We deduce linear recursions for all genus-zero Gromov-Witten invariants.

Quantum cohomology of [C^N/μ_r]

TL;DR

The paper develops a comprehensive framework to compute genus-0 Gromov-Witten invariants of local orbifolds [\mathbb{C}^N/\mu_r] by constructing the moduli space of stable maps to B\mu_r via r-th root operations on \overline{M}_{0,n} and by introducing weighted stable maps. A central achievement is a closed formula for the equivariant Euler class of the obstruction bundle, assembled from μ_r-eigenspace data of the Hodge bundle and boundary divisors, which yields linear recursions for all genus-0 GW invariants, including explicit recursions for the local threefold [\mathbb{C}^3/\mu_r], notably [\mathbb{C}^3/\mu_3]. The methodology unifies twisted and weighted stable maps, leverages wall-crossing for weight changes, and connects to the twisted I-function and mirror-map in Givental's formalism, providing a practical computational toolkit for local orbifold GW theory and potential implications for crepant-resolution phenomena.

Abstract

We give a construction of the moduli space of stable maps to the classifying stack Bμ_r of a cyclic group by a sequence of r-th root constructions on M_{0, n}. We prove a closed formula for the total Chern class of μ_r-eigenspaces of the Hodge bundle, and thus of the obstruction bundle of the genus zero Gromov-Witten theory of stacks of the form [C^N/μ_r]. We deduce linear recursions for all genus-zero Gromov-Witten invariants.

Paper Structure

This paper contains 35 sections, 25 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Introduction to $[\mathbb{C}^N/\mu_r]$
  3. Our methods and results
  4. Relation to other work
  5. Acknowledgements
  6. Notations and conventions
  7. Moduli space of stable maps to $B\mu_r$ via $r$-th roots
  8. User's guide to the $r$-th root construction
  9. Stable maps to $B\mu_r$
  10. Construction of the moduli space via root constructions
  11. The universal line bundle
  12. Base change to the gerbe
  13. Comments on the construction
  14. Weighted stable maps to $B\mu_r$
  15. Weighted stable maps
  16. ...and 20 more sections

Key Result

Theorem 1

$\overline{M}_{0, n}(e_1, \dots, e_n; B\mu_r)$ is a $\mu_r$-gerbe over the stack constructed from $\overline{M}_{0, n}$ by successively doing the $r_T$-th root construction at the boundary divisor $D^T \subset \overline{M}_{0, n}$ for all proper subsets $T \subset [n-1]$ having at least 2 elements.

Figures (1)

  • Figure 1: Comb for $n = 30$, $p= 7$ and $\underline{m} = (1, 1, 2, 3)$

Theorems & Definitions (28)

  • Theorem
  • Theorem
  • Definition 2.3.1
  • Lemma 2.3.2
  • Proposition 2.3.3
  • Lemma 2.4.1
  • Lemma 2.4.2
  • Definition 2.5.1
  • Theorem 2.5.2
  • Lemma 2.5.3
  • ...and 18 more