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Computations of Gromov-Witten invariants of toric varieties

Giosuè Muratore

TL;DR

This work provides a Julia-based tool to compute genus $0$ Gromov–Witten invariants for smooth projective toric varieties using Atiyah–Bott localization. By encoding toric data with Oscar.jl and enumerating fixed-point decorated graphs, the package delivers a flexible algorithm that handles standard GW invariants, small quantum products, and twisted invariants via explicit equivariant class formulas. The paper demonstrates explicit computations, including conic counts, quantum products on threefolds and fourfolds, and twisted invariants, showcasing both correctness against known results and practical computational performance. This contributes a reproducible, extensible framework for enumerative geometry and mirror-symmetry investigations in toric settings.

Abstract

We present the Julia package ToricAtiyahBott.jl, providing an easy way to perform the Atiyah-Bott formula on the moduli space of genus $0$ stable maps $\overline{M}_{0,m}(X,β)$ where $X$ is any smooth projective toric variety, and $β$ is any effective $1$-cycle. The list of the supported cohomological cycles contains the most common ones, and it is extensible. We provide a detailed explanation of the algorithm together with many examples and applications. The toric variety $X$, as well as the cohomology class $β$, must be defined using the package Oscar.jl.

Computations of Gromov-Witten invariants of toric varieties

TL;DR

This work provides a Julia-based tool to compute genus Gromov–Witten invariants for smooth projective toric varieties using Atiyah–Bott localization. By encoding toric data with Oscar.jl and enumerating fixed-point decorated graphs, the package delivers a flexible algorithm that handles standard GW invariants, small quantum products, and twisted invariants via explicit equivariant class formulas. The paper demonstrates explicit computations, including conic counts, quantum products on threefolds and fourfolds, and twisted invariants, showcasing both correctness against known results and practical computational performance. This contributes a reproducible, extensible framework for enumerative geometry and mirror-symmetry investigations in toric settings.

Abstract

We present the Julia package ToricAtiyahBott.jl, providing an easy way to perform the Atiyah-Bott formula on the moduli space of genus stable maps where is any smooth projective toric variety, and is any effective -cycle. The list of the supported cohomological cycles contains the most common ones, and it is extensible. We provide a detailed explanation of the algorithm together with many examples and applications. The toric variety , as well as the cohomology class , must be defined using the package Oscar.jl.
Paper Structure (10 sections, 2 theorems, 33 equations)

This paper contains 10 sections, 2 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Gromov--Witten invariants on toric varieties
  3. Toric varieties
  4. Moduli space of maps
  5. The algorithm
  6. Explicit computations
  7. Enumeration of curves
  8. Quantum cohomology ring of a threefold
  9. Quantum cohomology ring of a fourfold
  10. Twisted Gromov--Witten invariants

Key Result

Theorem 2.5

Let $P$ be a symmetric polynomial in Chern classes of equivariant vector bundle of $\overline{M}_{0,m}(X,\beta)$. Then where $N_\Gamma$ is the normal tangent bundle of $F_\Gamma$.

Theorems & Definitions (8)

  • Example 2.1
  • Example 2.2
  • Definition 2.4
  • Theorem 2.5
  • Lemma 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4