Matrix models for the nested hypergeometric tau-functions

TL;DR

This work extends hypergeometric tau-functions by introducing nested hypergeometric tau-functions within the free-fermion framework, yielding a broad class of 2D Toda tau-functions with a diagonal-weight structure. It establishes a skew Schur function expansion and multiple cut-and-join descriptions, then constructs universal multi-matrix models—complex, unitary, and normal—organized as chains that reduce to eigenvalue integrals. The paper also develops W-operator formalisms and superintegrability properties, and it provides detailed examples for hypergeometric, skew hypergeometric, and higher $m$ cases, including connections to fully simple maps and hypermaps. Finally, it discusses the onset of broken integrability in more general settings, outlining potential extensions to other hierarchies and deformations and highlighting the enumerative geometry implications. The framework offers a flexible toolkit for encoding weighted Hurwitz-type invariants through matrix-model realizations and fermionic representations, with potential applications to topological string theory and related combinatorial geometries.

Abstract

We introduce and investigate a family of tau-functions of the 2D Toda hierarchy, which is a natural generalization of the hypergeometric family associated with Hurwitz numbers. For this family we prove a skew Schur function expansion formula. For arbitrary rational weight generating functions we construct the multi-matrix models. Two different types of cut-and-join descriptions are derived. Considered examples include generalized fully simple maps, which we identify with the recently introduced skew hypergeometric tau-functions.

Theorems & Definitions (23)

• Proposition 2.1
• Lemma 3.1
• Definition 1
• Theorem 1
• Remark 4.1
• Lemma 4.1
• Remark 4.2
• Remark 4.3
• Remark 4.4
• Proposition 4.2