On the holomorphic point of view in the theory of quantum knot invariants

TL;DR

This paper explores a geometric approach to quantum knot invariants, connecting quantum groups and Weyl quantizations, and introduces a basis of quantum observables relevant to quantum computing.

Contribution

It advances the geometric understanding of the Witten-Reshetikhin-Turaev theory and relates quantum groups to moduli space quantizations, providing new insights into quantum invariants.

Findings

Reconstruction of modular functor from Weyl quantization.

Description of a basis of quantum observables on the torus.

Answers to questions related to quantum computing.

Abstract

In this paper we describe progress made toward the construction of the Witten-Reshetikhin-Turaev theory of knot invariants from the geometric point of view. This is done in the perspective of a joint result of the author with A. Uribe which relates the quantum group and the Weyl quantizations of the moduli space of flat SU(2)-connections on the tours. Two results are emphasized: the reconstruction from Weyl quantization of the restriction to the torus of the modular functor, and a description of a basis of the space of quantum observables on the torus in terms of colored curves, which answers a question related to quantum computing.