Stable maps to Looijenga pairs

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 $D=D_1+\cdots +D_l\in |-K_Y|$ 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 ${⨁}_i{\mathcal{O}}_Y(-D_i)$, 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.

v1: 114 pages (80pp+appendices), 40 figures. v2: minor changes, references added. v3: 94 pages, exposition streamlined and shortened, typos fixed, references added. v4: introduction substantially revised, 98 pages

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Cited by in corpus (5)

• Stable maps to Looijenga pairs: orbifold examples
• Open/Closed BPS Correspondence and Integrality
• The local/logarithmic correspondence and the degeneration formula for quasimaps
• The log-open correspondence for two-component Looijenga pairs
• Open WDVV Equations and Frobenius Structures for Toric Calabi-Yau 3-Folds