Vershik-Kerov in higher times

TL;DR

This paper explores generalizations of the Vershik-Kerov limit shape problem motivated by advanced theories like topological string theory and gauge theory, analyzing specific quiver theories and their geometric limits.

Contribution

It introduces new limit shape results for quiver theories and connects them to elliptic curves and dualities in higher-dimensional gauge theories.

Findings

Limit shape governed by a genus two algebraic curve.

Connections between enumerative parameters and dualities.

Extension to double-elliptic and six-dimensional theories.

Abstract

Several generalizations of Vershik-Kerov limit shape problem are motivated by topological string theory and supersymmetric gauge theory instanton count. In this paper specifically we study the circular and linear quiver theories. We also briefly discuss the double-elliptic generalization of the Vershik-Kerov problem, related to six dimensional gauge theory compactified on a torus, and to elliptic cohomology of the Hilbert scheme of points on a plane. We prove that the limit shape in that setting is governed by a genus two algebraic curve, suggesting unexpected dualities between the enumerative and equivariant parameters.