Equations on knot polynomials and 3d/5d duality

TL;DR

This paper explores the deep connections between knot polynomials, integrable systems, and dualities in quantum field theories, highlighting the role of A-polynomials and Baxter equations in a 3d/5d AGT framework.

Contribution

It reveals the identity between A-polynomial equations for knots and Baxter equations for integrable systems, advancing the understanding of 3d-5d dualities in quantum theories.

Findings

A-polynomial equations correspond to Baxter equations in integrable systems.

Knot parameters relate to points in the moduli space of integrable systems.

The work links knot invariants to quantum field theory dualities.

Abstract

We briefly review the current situation with various relations between knot/braid polynomials (Chern-Simons correlation functions), ordinary and extended, considered as functions of the representation and of the knot topology. These include linear skein relations, quadratic Plucker relations, as well as "differential" and (quantum) A-polynomial structures. We pay a special attention to identity between the A-polynomial equations for knots and Baxter equations for quantum relativistic integrable systems, related through Seiberg-Witten theory to 5d super-Yang-Mills models and through the AGT relation to the q-Virasoro algebra. This identity is an important ingredient of emerging a 3d-5d generalization of the AGT relation. The shape of the Baxter equation (including the values of coefficients) depend on the choice of the knot/braid. Thus, like the case of KP integrability, where (some, so far torus) knots parameterize particular points of the Universal Grassmannian, in this relation they parameterize particular points in the moduli space of many-body integrable systems of relativistic type.