Scattering diagrams, theta functions, and refined tropical curve counts
Travis Mandel · Journal of the London Mathematical Society · 2021
In the Gross–Siebert mirror symmetry program, certain enumerations of tropical disks are encoded in combinatorial objects called scattering diagrams and broken lines. These, in turn, are used to construct a mirror scheme equipped with a canonical basis of regular functions called theta functions. This paper serves to develop the relationships between tropical disks, scattering diagrams, and broken lines in the quantum setting; for example, we express quantum theta functions in terms of refined enumerations of tropical disks.We apply these tropical descriptions to prove a refined version of the Carl–Pumperla–Siebert (Preprint) result on consistency of theta functions, and also to prove the quantum Frobenius Conjecture of Fock and Goncharov (Ann. Sci. Éc. Norm. Supér. (4) 42 (2009) 865–930).