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Superintegrability for some $(q,t)$-deformed matrix models

Fan Liu, Rui Wang, Jie Yang, Wei-Zhong Zhao

TL;DR

This paper addresses the problem of establishing superintegrability for a class of $q,t$-deformed matrix models by analyzing Macdonald's $q,t$-deformed hypergeometric functions. The authors develop a concise method based on $W$- and Lassalle-type operator representations to convert hypergeometric constraints into a uniqueness statement for the solution space, then apply it to concrete models. They prove superintegrability for the refined Chern-Simons model, the $q$-Selberg integral, and the $(q,t)$-deformed Hermite and Laguerre ensembles, and they further formulate a general degraded $(q,t)$-deformed integral that encompasses these cases. This framework provides a robust, adaptable tool for verifying superintegrability and connects refined knot invariants with $(q,t)$-deformed matrix-model observables, with potential extensions to broader parameter regimes.

Abstract

We analyze the Macdonald's $(q,t)$-deformed hypergeometric functions with one and two set variables and present their constraints. We prove the uniqueness to the solution of these constraints. We propose a concise method to prove the superintegrability relations for some well-known $(q,t)$-deformed matrix models, where the constraints of hypergeometric functions play a crucial role.

Superintegrability for some $(q,t)$-deformed matrix models

TL;DR

This paper addresses the problem of establishing superintegrability for a class of -deformed matrix models by analyzing Macdonald's -deformed hypergeometric functions. The authors develop a concise method based on - and Lassalle-type operator representations to convert hypergeometric constraints into a uniqueness statement for the solution space, then apply it to concrete models. They prove superintegrability for the refined Chern-Simons model, the -Selberg integral, and the -deformed Hermite and Laguerre ensembles, and they further formulate a general degraded -deformed integral that encompasses these cases. This framework provides a robust, adaptable tool for verifying superintegrability and connects refined knot invariants with -deformed matrix-model observables, with potential extensions to broader parameter regimes.

Abstract

We analyze the Macdonald's -deformed hypergeometric functions with one and two set variables and present their constraints. We prove the uniqueness to the solution of these constraints. We propose a concise method to prove the superintegrability relations for some well-known -deformed matrix models, where the constraints of hypergeometric functions play a crucial role.
Paper Structure (12 sections, 13 theorems, 152 equations)

This paper contains 12 sections, 13 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. ($q,t$)-deformed hypergeometric functions and their constraints
  3. Preliminaries associated with Macdonald polynomials
  4. $(q,t)$-deformed hypergeometric functions
  5. Constraints of $(q,t)$-deformed hypergeometric functions
  6. Some $(q,t)$-analogues of matrix models
  7. The refined Chern-Simons matrix model
  8. $q$-Selberg integral
  9. $(q,t)$-deformed Hermite and Laguerre ensembles
  10. A general $(q,t)$-deformed matrix model
  11. Conclusions
  12. $q$-derivative and $q$-integral

Key Result

Lemma 2.1

The Lassalle's operators $\mathcal{E}_k(\mathbf{x})$ with $k\in\mathbb{N}$ (Edef) can be rewritten as the operators with the collective variables $\mathbf{p}=(p_1,p_2,\cdots)$ where we take $p_n=\sum_{i=1}^Nx_i^n$.

Theorems & Definitions (26)

  • Lemma 2.1
  • proof
  • Definition 2.1
  • Proposition 2.1
  • Definition 2.2
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 16 more