Chern-Simons theory, link invariants and the Askey-Wilson algebra

TL;DR

This paper investigates the appearance of the Askey-Wilson algebra in SU(2) Chern-Simons theory and link invariants, revealing algebraic relations governing Wilson loop expectation values and link invariants.

Contribution

It demonstrates the connection between the Askey-Wilson algebra and link invariants in Chern-Simons theory, providing new algebraic insights into Wilson loop relations.

Findings

Wilson loop expectation values satisfy Askey-Wilson algebra relations

Link invariants do not distinguish certain linear link combinations

The Askey-Wilson algebra appears naturally in the RT construction

Abstract

The occurrence of the Askey-Wilson (AW) algebra in the $SU(2)$ Chern-Simons (CS) theory and in the Reshetikhin-Turaev (RT) link invariant construction with quantum algebra $U_q(\mathfrak{su}_2)$ is explored. Tangle diagrams with three strands with some of them enclosed in a spin-$1/2$ closed loop are associated to the generators of the AW algebra. It is shown in both the CS theory and RT construction that the link invariant of these tangles obey the relations of the AW generators. It follows that the expectation values of certain Wilson loops in the CS theory satisfy relations dictated by the AW algebra and that the link invariants do not distinguish the corresponding linear combinations of links.