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 D 1 C C D l 2 j K Y j an anticanonical singular nodal curve. Under some natural 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 L i 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 and Pandharipande, Ionel and Parker, and Labastida, Mariño, Ooguri and Vafa.We furthermore provide a complete closed-form solution to the calculation of all these invariants.14J33, 14J81, 14N35, 16G20, 53D45 1. Introduction 394 2. Nef Looijenga pairs 410 3. Local Gromov-Witten theory 416 4. Log Gromov-Witten theory 423 5. Log-local correspondence 438 6. Open Gromov-Witten theory 445 7. KP and quiver DT invariants 463 8. Higher-genus BPS invariants 469 9. Orbifolds 474 Appendix A. Proof of Theorem 3.3 476 Appendix B. Infinite scattering 480 Appendix C. Proof of Theorem 5.4 482 Appendix D. Symmetric functions 488 References 491