Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tau functions of the UC hierarchy as partition functions of matrix models

Chuanzhong Li, Andrei Mironov, Alexander Yu. Orlov

TL;DR

This work establishes a bridge between the universal character (UC) hierarchy and exactly solvable matrix models by showing that carefully constructed matrix integrals yield UC tau functions. It develops two complementary constructions: matrix models tied to the product of two spheres with embedded graphs glued by a gluing matrix (Method IV) and multi-matrix ensembles on graphs that realize the multi-component UC hierarchy. The authors provide explicit unitary-matrix examples and outline precise conditions under which the partition functions become UC tau functions through UC matings of Schur functions. The framework generalizes to complex and unitary multi-matrix ensembles on embedded graphs, enabling a broad family of UC tau function realizations and offering potential connections to two-dimensional Yang–Mills theory and topological string theories.

Abstract

We present a family of matrix models such that their partition functions are tau functions of the universal character (UC) hierarchy. This develops one of the topics of our previous paper arXiv:2410.14823. We found new matrix models associated with the product of two spheres with embedded graphs via a gluing matrix. We also generalize these studies to multi-matrix models case, which corresponds to the multi-component UC hierarchy.

Tau functions of the UC hierarchy as partition functions of matrix models

TL;DR

This work establishes a bridge between the universal character (UC) hierarchy and exactly solvable matrix models by showing that carefully constructed matrix integrals yield UC tau functions. It develops two complementary constructions: matrix models tied to the product of two spheres with embedded graphs glued by a gluing matrix (Method IV) and multi-matrix ensembles on graphs that realize the multi-component UC hierarchy. The authors provide explicit unitary-matrix examples and outline precise conditions under which the partition functions become UC tau functions through UC matings of Schur functions. The framework generalizes to complex and unitary multi-matrix ensembles on embedded graphs, enabling a broad family of UC tau function realizations and offering potential connections to two-dimensional Yang–Mills theory and topological string theories.

Abstract

We present a family of matrix models such that their partition functions are tau functions of the universal character (UC) hierarchy. This develops one of the topics of our previous paper arXiv:2410.14823. We found new matrix models associated with the product of two spheres with embedded graphs via a gluing matrix. We also generalize these studies to multi-matrix models case, which corresponds to the multi-component UC hierarchy.

Paper Structure

This paper contains 26 sections, 9 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Schur functions and universal characters.
  3. Matings of pairs of the Schur functions
  4. KP and UC tau functions.
  5. Relations of the UC hierarchy to matrix models.
  6. Method I.
  7. Method II.
  8. Method III.
  9. Method IV.
  10. Matrix models associated with the product of two spheres with embedded graphs
  11. General description
  12. Examples. Integrals over unitary matrices
  13. Example 1
  14. Example 2
  15. Multi-matrix models of complex and unitary matrices associated with embedded graphs
  16. ...and 11 more sections

Key Result

Proposition 1

where $s_\lambda(I_N):=s_\lambda(\mathbf{p}(I_N))$ is known to be dimension of the irreducible representation $\rho_\lambda$ of the linear group $\mathbb{GL}_N$ lebelled by $\lambda$Mac, which can be written down in an explicit way in terms of the partition $\lambda$ :

Theorems & Definitions (11)

  • Remark 1
  • Proposition 1
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Proposition 2
  • Proposition 3
  • Lemma 9
  • Proposition 4
  • ...and 1 more