Scattering diagrams, theta functions, and refined tropical curve counts

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. ec. Norm. Super. (4) 42 (2009) 865-930).

Automated summary

This paper focuses on the relationships between tropical disks, scattering diagrams, and broken lines in the quantum setting, and how they are used to prove significant mathematical results.