Block and Göttsche have defined a q -number refinement of counts of tropical curves in . Under the change of variables , we show that the result is a generating series of higher genus log Gromov–Witten invariants with insertion of a lambda class. This gives a geometric interpretation of the Block-Göttsche invariants and makes their deformation invariance manifest.

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Gross, Hacking and Keel have constructed mirrors of log Calabi-Yau surfaces in terms of counts of rational curves. Using q-deformed scattering diagrams defined in terms of higher genus log Gromov-Witten invariants, we construct deformation quantizations of these mirrors and we produce canonical bases of the corresponding noncommutative algebras of functions.

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 + • • • + D l 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 i OY (−Di), and the open Gromov-Witten of toric orbi-branes in a Calabi-Yau 3-orbifold associated to (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.