CMM formula as superintegrability property of unitary model

TL;DR

This paper explores the superintegrability property of the CMM formula within the context of unitary models, linking it to knot invariants and Macdonald polynomials, and discusses its potential generalizations.

Contribution

It reveals that the CMM identity for Macdonald residues encapsulates superintegrability in unitary models and connects to knot polynomial relations.

Findings

CMM identity reformulates superintegrability properties.

Connections established between Macdonald polynomials and knot invariants.

Potential for extending formulas to arbitrary knots and links.

Abstract

A typical example of superintegrability is provided by expression of the Hopf link hyperpolynomial in an arbitrary representation through a pair of the Macdonald polynomials at special points. In the simpler case of the Hopf link HOMFLY-PT polynomial and a pair of the Schur functions, it is a relation in the unitary matrix model. We explain that the Cherednik-Mehta-Macdonald (CMM) identity for bilinear Macdonald residues with an elliptic weight function is nothing but a reformulation of these same formulas. Their lifting to arbitrary knots and links, even torus ones remains obscure.