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... Show moreImportant 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-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. Show less
My PhD research is devoted to studies of the conjectural cluster-algebraic symmetry in the theory of topological string, which is the simplest, however already non-trivial sector in the theory of... Show moreMy PhD research is devoted to studies of the conjectural cluster-algebraic symmetry in the theory of topological string, which is the simplest, however already non-trivial sector in the theory of string. The language of cluster algebras is the modern tool which was initially developed for solving problems in linear algebra, and has recently been applied to the theory of integrable systems. In this dissertation we identify the cluster-algebraic nature of new classes of integrable systems of string-theoretic origin. We then show how the partition functions of topological string on the corresponding geometries can be naturally fit into a cluster-algebraic context. Show less
