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=D₁+⋯+D_l∈|−K_Y| 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):<br/><br/>1. the log Gromov–Witten theory of the pair (Y,D),<br/>2. the Gromov–Witten theory of the total space of ⊕_iO_Y(−D_i),<br/>3. the open Gromov–Witten theory of special Lagrangians in a Calabi–Yau 3–fold determined by (Y,D),<br/>4. the Donaldson–Thomas theory of a symmetric quiver specified by (Y,D), and<br/>5. a class of BPS invariants considered in different contexts by Klemm and Pandharipande, Ionel and Parker, and Labastida, Mariño, Ooguri and Vafa.<br/><br/>We furthermore provide a complete closed-form solution to the calculation of all these invariants.<br/>