Stable maps to Looijenga pairs: orbifold examples

• Published in: Letters in mathematical physics 2021-08, Vol.111 (4), Article 109
• Main Authors: Bousseau, Pierrick, Brini, Andrea, Garrel, Michel van
• Format: Article
• Language: English
• Subjects: Boris Dubrovin Memorial Issue; Complex Systems; Geometry; Group Theory and Generalizations; Invariants; Mathematical and Computational Physics; Physics; Physics and Astronomy; Theoretical
• ISSN: 0377-9017; 1573-0530
• DOI: 10.1007/s11005-021-01451-9

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 = D 1 + ⋯ + D l an anticanonical divisor on Y with each D i 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 D i 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 ⨁ i O Y ( - D i ) , 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.