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 tau-functions of integrable equations is the fact that the (dual) partition functions of 4d N = 2 gauge theories solve Painleve 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-algebraicintegrable 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.

Automated summary

This paper discusses an important illustration in string theory, which is the tau-functions of integrable equations in the principle partition functions. It has been found that the (dual) partition functions of 4d N = 2 gauge theories can solve Painleve equations. Recent research has also suggested that the partition functions of topological string on local Calabi-Yau manifolds can solve q-difference equations of non-autonomous dynamics. The paper provides a detailed explanation of this proposal's solutions, presents a specific example, and suggests further research directions.