A note on Q-algebra and quantization

TL;DR

This paper investigates the application of Q-algebras to strict quantization, confirming Schwarz's formalism aligns with Rieffel's star product on tori and constructing examples on Kähler manifolds, including the fuzzy sphere.

Contribution

It proves Schwarz's conjecture for tori with constant Poisson structures and constructs twisted Fock modules on Kähler manifolds, linking to fuzzy sphere representations.

Findings

Schwarz's formalism matches Rieffel's star product on tori.

Constructed twisted Fock modules on compact Kähler manifolds.

Connected quantization modules to fuzzy sphere representations.

Abstract

In this note, we study Schwarz's conjecture on application of Q-algebras to strict quantization. We prove that in the case of a torus with a constant Poisson structure, Schwarz's formalism gives the same star product as Rieffel \cite{rif:quantization}. We construct twisted Fock modules as examples of quantization dg-modules in the case of a compact K\"ahler manifold. In particular, we relate this construction on $\complex\mathbb{P}^1$ to representations of a fuzzy sphere.