Stable maps to Looijenga pairs

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+⋯+Dl∈|−KY| 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 ⊕iOY(−Di),

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.