Thermodynamic limit of random partitions and dispersionless Toda hierarchy

TL;DR

This paper investigates the thermodynamic limit of random partition models related to supersymmetric gauge theories, connecting the limit shape problem to the dispersionless Toda hierarchy and Seiberg-Witten theory.

Contribution

It establishes a link between the limit shape of Young diagrams in gauge theory models and solutions to the dispersionless Toda hierarchy via a Riemann-Hilbert problem.

Findings

Limit shape characterized by a variational problem.

Solution of Riemann-Hilbert problem using complex curves.

Relation of string equations to hidden symmetries.

Abstract

We study the thermodynamic limit of random partition models for the instanton sum of 4D and 5D supersymmetric U(1) gauge theories deformed by some physical observables. The physical observables correspond to external potentials in the statistical model. The partition function is reformulated in terms of the density function of Maya diagrams. The thermodynamic limit is governed by a limit shape of Young diagrams associated with dominant terms in the partition function. The limit shape is characterized by a variational problem, which is further converted to a scalar-valued Riemann-Hilbert problem. This Riemann-Hilbert problem is solved with the aid of a complex curve, which may be thought of as the Seiberg-Witten curve of the deformed U(1) gauge theory. This solution of the Riemann-Hilbert problem is identified with a special solution of the dispersionless Toda hierarchy that satisfies a pair of generalized string equations. The generalized string equations for the 5D gauge theory are shown to be related to hidden symmetries of the statistical model. The prepotential and the Seiberg-Witten differential are also considered.