Stable maps to Looijenga pairs: orbifold examples

In [15], we established a series of correspondences relating five enumerative theories of log Calabi–Yau surfaces, i.e. pairs (Y, D) with Y a smooth projective complex surface and D=D1+⋯+Dl an anticanonical divisor on Y with each Di smooth and nef. In this paper, we explore the generalisation to Y being a smooth Deligne–Mumford stack with projective coarse moduli space of dimension 2 and Di nef Q-Cartier divisors. We consider in particular three infinite families of orbifold log Calabi–Yau surfaces, and for each of them, we provide closed-form solutions of the maximal contact log Gromov–Witten theory of the pair (Y, D), the local Gromov–Witten theory of the total space of ⨁iOY(−Di), and the open Gromov–Witten of toric orbi-branes in a Calabi–Yau 3-orbifold associated with (Y, D). We also consider new examples of BPS integral structures underlying these invariants and relate them to the Donaldson–Thomas theory of a symmetric quiver specified by (Y, D) and to a class of open/closed BPS invariants.