Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux

Important illustration to the principle "partition functions in string theory are τ-functions of integrable equations" is the fact that the (dual) partition functions of 4dN = 2 gauge theories solve Painlevé equations. In this paper we show a road to self-consistent proof of the recently suggested generalization of this correspondence: partition functions of topological string on local Calabi-Yau manifolds solve q-difference equations of non-autonomous dynamics of the "cluster-algebraic"integrable systems.We explain in details the "solutions" side of the proposal. In the simplest non-trivial example we show how 3d box-counting of topological string partition function appears from the counting of dimers on bipartite graph with the discrete gauge field of "flux" q. This is a new form of topological string/spectral theory type correspondence, since the partition function of dimers can be computed as determinant of the linear q-difference Kasteleyn operator. Using WKB method in the "melting" q → 1 limit we get a closed integral formula for Seiberg-Witten prepotential of the corresponding 5d gauge theory. The "equations" side of the correspondence remains the intriguing topic for the further studies.