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Stable maps to Looijenga pairs

Pierrick Bousseau, Andrea Brini, Michel van Garrel

Abstract

A log Calabi-Yau surface with maximal boundary, or Looijenga pair, is a pair $(Y,D)$ with $Y$ a smooth rational projective complex surface and $D=D_1+\dots + D_l \in |-K_Y|$ an anticanonical singular nodal curve. Under some positivity conditions on the pair, we propose a series of correspondences relating five different classes of enumerative invariants attached to $(Y,D)$: 1) the log Gromov-Witten theory of the pair $(Y,D)$, 2) the Gromov-Witten theory of the total space of $\bigoplus_i \mathcal{O}_Y(-D_i)$, 3) the open Gromov-Witten theory of special Lagrangians in a Calabi-Yau 3-fold determined by $(Y,D)$, 4) the Donaldson-Thomas theory of a symmetric quiver specified by $(Y,D)$, and 5) a class of BPS invariants considered in different contexts by Klemm-Pandharipande, Ionel-Parker, and Labastida-Marino-Ooguri-Vafa. We furthermore provide a complete closed-form solution to the calculation of all these invariants.

Stable maps to Looijenga pairs

Abstract

A log Calabi-Yau surface with maximal boundary, or Looijenga pair, is a pair with a smooth rational projective complex surface and an anticanonical singular nodal curve. Under some positivity conditions on the pair, we propose a series of correspondences relating five different classes of enumerative invariants attached to : 1) the log Gromov-Witten theory of the pair , 2) the Gromov-Witten theory of the total space of , 3) the open Gromov-Witten theory of special Lagrangians in a Calabi-Yau 3-fold determined by , 4) the Donaldson-Thomas theory of a symmetric quiver specified by , and 5) a class of BPS invariants considered in different contexts by Klemm-Pandharipande, Ionel-Parker, and Labastida-Marino-Ooguri-Vafa. We furthermore provide a complete closed-form solution to the calculation of all these invariants.

Paper Structure

This paper contains 78 sections, 61 theorems, 225 equations, 25 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Looijenga pairs
  3. Summary of the main results
  4. Geometries
  5. Enumerative theories
  6. Correspondences
  7. Integrality
  8. Solutions
  9. Enumerative problems
  10. Higher genus log Gromov--Witten invariants
  11. Local Gromov--Witten invariants
  12. Higher genus open Gromov--Witten invariants
  13. Quiver DT invariants
  14. Open/closed BPS invariants
  15. The web of correspondences: geometric motivation
  16. ...and 63 more sections

Key Result

Theorem 1.4

For every nef Looijenga pair $Y(D)$, the genus $0$ log invariants $N_{0,d}^{\rm log}(Y(D))$ and the genus $0$ local invariants $N^{\rm loc}_{0,d}(Y(D))$ are related by Moreover, we provide a closed-form solution to the calculation of both sets of invariants in eq:loglocthm.

Figures (25)

  • Figure 1.1: Enumerative invariants of $Y(D)$ and their mutual relations.
  • Figure 1.2: Exchanging log and open conditions.
  • Figure 4.1: The toric model of $\mathbb{P}^2(1,4)$.
  • Figure 4.2: The toric model of $\mathop{\mathrm{dP}}\nolimits_3(1,1)$
  • Figure 4.3: $\mathop{\mathrm{Scatt}}\nolimits\mathop{\mathrm{dP}}\nolimits_3(1,1)$
  • ...and 20 more figures

Theorems & Definitions (137)

  • Conjecture 1.1: The stationary log/local correspondence for maximal log CY pairs
  • Conjecture 1.2: Log-open correspondence for surfaces
  • Conjecture 1.3: The all-genus log-open correspondence for surfaces
  • Theorem 1.4: =\ref{['thm_log_local', 'lem:localgw', 'thm:dP5', 'thm:dP33comp', 'thm:F04comp']}
  • Theorem 1.5: =\ref{['prop:dp311', 'thm:log_dp3_0_0_0', 'thm:logf0_0000', 'thm:logopen']}
  • Theorem 1.6: =\ref{['thm:kpdt']}
  • Theorem 1.7: =\ref{['thm:openbps']}
  • Definition 2.1
  • Lemma 2.1: Rocco
  • Proposition 2.2
  • ...and 127 more