Vershik–Kerov in higher times

Abstract

Several generalizations of the Vershik–Kerov limit shape problem are motivated by topological string theory and supersymmetric gauge theory instanton counts. In this paper, we specifically study the ${\hat{A}}_r$- and $A_r$-quiver-type models (circular and linear quiver theories). We also briefly discuss the double-elliptic generalization of the Vershik–Kerov problem, related to six-dimensional $\mathcal{N}=2$ gauge theory compactified on a torus and to elliptic cohomology of the Hilbert scheme of points on ${\mathbb{C}}^2$. 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.