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Superintegrability of $q,t$-matrix models and quantum toroidal algebra recursions

Luca Cassia, Victor Mishnyakov

TL;DR

The paper develops a universal algebraic framework for $q,t$-deformed matrix models that realize representations of the deformed Virasoro and quantum toroidal $U_{q,t}(\hat{\hat{\mathfrak{gl}}}_1)$ algebras. By exploiting $q,t$-Virasoro constraints and a resummed recursion operator, it proves Macdonald (and in special cases Skew-Macdonald) superintegrability across ten distinct models, including refined Chern–Simons and various matter configurations, with explicit formulas for $\langle P_\lambda\rangle^w$ in terms of eigenvalues and evaluation loci. The authors connect these recursions to the W-representation and gauge-transform constructions, unifying proofs and offering new closed formulas, while highlighting links to skein recursions in the unrefined limit and to symmetric-orthogonal polynomials (e.g., interpolation Macdonald, Al-Salam–Carlitz). The work thus clarifies hidden $U_{q,t}(\hat{\hat{\mathfrak{gl}}}_1)$ symmetry in these models and illuminates the role of superintegrability in both multivariate orthogonal polynomials and knot-theoretic refinements. Its framework provides a versatile toolkit for deriving exact correlator formulas and for exploring connections to refined topological invariants and categorification.

Abstract

$q,t$-deformed matrix models give rise to representations of the deformed Virasoro algebra and more generally of the quantum toroidal $\mathfrak{gl}_1$ algebra. These representations are described in terms of finite difference equations that induce recursion relations for correlation functions. Under suitable assumptions, these recursions admit unique solutions expressible through "superintegrability" formulas, i.e. explicit closed formulas for averages of Macdonald polynomials. In this paper, we discuss examples arising from localization of 3d $\mathcal{N}=2$ theories, which include $q,t$-deformation of well known classical ensembles: Gaussian, Laguerre and Jacobi. We explain how relations in the quantum toroidal algebra can be used to give a new and universal proof of the known superintegrability formulas, as well as to derive new formulas for models that have not been previously studied in the literature. Finally, we make some remarks regarding the relation between superintegrability and orthogonal polynomials.

Superintegrability of $q,t$-matrix models and quantum toroidal algebra recursions

TL;DR

The paper develops a universal algebraic framework for -deformed matrix models that realize representations of the deformed Virasoro and quantum toroidal algebras. By exploiting -Virasoro constraints and a resummed recursion operator, it proves Macdonald (and in special cases Skew-Macdonald) superintegrability across ten distinct models, including refined Chern–Simons and various matter configurations, with explicit formulas for in terms of eigenvalues and evaluation loci. The authors connect these recursions to the W-representation and gauge-transform constructions, unifying proofs and offering new closed formulas, while highlighting links to skein recursions in the unrefined limit and to symmetric-orthogonal polynomials (e.g., interpolation Macdonald, Al-Salam–Carlitz). The work thus clarifies hidden symmetry in these models and illuminates the role of superintegrability in both multivariate orthogonal polynomials and knot-theoretic refinements. Its framework provides a versatile toolkit for deriving exact correlator formulas and for exploring connections to refined topological invariants and categorification.

Abstract

-deformed matrix models give rise to representations of the deformed Virasoro algebra and more generally of the quantum toroidal algebra. These representations are described in terms of finite difference equations that induce recursion relations for correlation functions. Under suitable assumptions, these recursions admit unique solutions expressible through "superintegrability" formulas, i.e. explicit closed formulas for averages of Macdonald polynomials. In this paper, we discuss examples arising from localization of 3d theories, which include -deformation of well known classical ensembles: Gaussian, Laguerre and Jacobi. We explain how relations in the quantum toroidal algebra can be used to give a new and universal proof of the known superintegrability formulas, as well as to derive new formulas for models that have not been previously studied in the literature. Finally, we make some remarks regarding the relation between superintegrability and orthogonal polynomials.
Paper Structure (37 sections, 11 theorems, 280 equations, 1 table)

This paper contains 37 sections, 11 theorems, 280 equations, 1 table.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Acknowledgments
  4. Preliminaries on symmetric functions and Young tableaux
  5. Classical Matrix Models
  6. Hermitian matrix models
  7. Superintegrability
  8. Virasoro constraints
  9. Cut-and-join or W-representation
  10. q,t-deformation
  11. Definition of the models
  12. Supersymmetric gauge theories on D^2xS^1
  13. Refined Chern--Simons theory on S^3
  14. q,t-Virasoro constraints
  15. Superintegrability from recursions
  16. ...and 22 more sections

Key Result

Lemma 4.1

Suppose there exist two non-zero polynomials $M^+(z)$ and $M^-(z)$, such that $\max(d_{+},d_{-})\leq2$ and Then the generating function of the $q,t$-deformed matrix model with weight function $w(x)$ satisfies the equations where $\hat{\mathsf{U}}^w_m$ are the modes of the generating current $\hat{\mathsf{U}}^w(z)=\sum\limits_{m\in\mathbb{Z}}\hat{\mathsf{U}}^w_m z^{-m}$, defined as and we intro

Theorems & Definitions (31)

  • Example 3.1
  • Example 3.2
  • Definition 3.3: Schur superintegrability of Hermitian eigenvalue models
  • Example 3.4
  • Example 3.5
  • Remark 4.1
  • Remark 4.2
  • Lemma 4.1
  • proof
  • Corollary 4.1
  • ...and 21 more