Theta Function Basis of the Cox ring of Postive 2d Looijenga pairs

We give a canonical basis of theta functions for the Cox ring of two dimensional Looijenga pairs with affine interior, with structure constants naive counts of k-analytic disks in the total space of the universal deformation of the mirror (which, as this is dimension two, is isomorphic to the log Calabi-Yau surface itself)