Thermodynamic limit of random partitions and dispersionless Toda hierarchy

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.

💡 Research Summary

The paper investigates the thermodynamic limit of random partition models that represent the instanton sums of four‑dimensional (4D) and five‑dimensional (5D) N=2 supersymmetric U(1) gauge theories, incorporating an infinite set of external potentials. By employing a two‑dimensional free fermion formalism, the authors rewrite the Nekrasov partition functions as sums over ordinary Young diagrams (partitions) with weights expressed through hook‑length formulas, Barnes G‑functions (in 4D), and q‑deformed Schur functions (in 5D). The external potentials Φ_k are identified with moments of the Maya diagram density ρ(x), which encodes the positions of particles at x = μ_i – i.

In the large‑size (thermodynamic) limit, the dominant contributions come from a “limit shape” ρ_*(x) that minimizes an energy functional E