The Weyl quantization and the quantum group quantization of the moduli space of flat SU(2)-connections on the torus are the same

TL;DR

This paper demonstrates the unitary equivalence between Weyl quantization and quantum group quantization for the moduli space of flat SU(2)-connections on a torus, linking two prominent quantization methods.

Contribution

It establishes the unitarily equivalence of Weyl and quantum group quantizations for the torus moduli space, comparing operator matrices and analyzing the *-product structure.

Findings

Weyl and quantum group quantizations are unitarily equivalent.

Operator matrices for cosine functions match in both quantizations.

The *-product satisfies the product-to-sum formula for noncommutative cosines.

Abstract

We prove that, for the moduli space of flat SU(2)-connections on the torus, the Weyl quantization and the quantization using the quantum group of SL(2,C) are unitarily equivalent. This is done by comparing the matrices of the operators associated by the two quantization to cosine functions. We also discuss the *-product of the Weyl quantization and show that it satisfies the product-to-sum formula for noncommutative cosines on the noncommutative torus.