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Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds

D. Maulik, A. Oblomkov, A. Okounkov, R. Pandharipande

Abstract

We prove the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Marino, and Vafa of the Gromov-Witten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting.

Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds

Abstract

We prove the equivariant Gromov-Witten theory of a nonsingular toric 3-fold X with primary insertions is equivalent to the equivariant Donaldson-Thomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Marino, and Vafa of the Gromov-Witten theory of local Calabi-Yau toric 3-folds are proven to be correct in the full 3-leg setting.

Paper Structure

This paper contains 47 sections, 12 theorems, 195 equations, 5 figures.

Table of Contents

  1. The GW/DT correspondence
  2. Gromov-Witten theory
  3. Donaldson-Thomas theory
  4. Primary GW/DT correspondence
  5. Relative GW/DT correspondence
  6. Degeneration formulas
  7. $\mathcal{A}_n$ geometries
  8. Plan of the paper
  9. Acknowledgments
  10. Localization
  11. Toric geometry
  12. Standard localization
  13. Structure
  14. Gromov-Witten edges
  15. Donaldson-Thomas edges
  16. ...and 32 more sections

Key Result

Theorem 1

The primary GW/DT correspondence holds for all nonsingular toric 3-folds $X$ in $T$-equivariant cohomology.

Figures (5)

  • Figure 1: An edge in the toric polytope of $X$
  • Figure 2: Capped localization for the $\mathcal{A}_1$-cap
  • Figure 3: $\mathcal{A}_1$-cap minus the capped rubber
  • Figure 4: Capped localization for the $\mathcal{A}_2$-cap
  • Figure :

Theorems & Definitions (19)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • proof
  • Lemma 5
  • Theorem 2
  • Lemma 6
  • proof
  • ...and 9 more