Berezin-Toeplitz Quantization in Real Polarizations with Toric Singularities

TL;DR

This paper extends Berezin-Toeplitz quantization to singular real polarizations on compact toric symplectic manifolds, demonstrating the resulting star product through norm estimates despite singularities.

Contribution

It introduces a novel approach to quantization in the presence of toric singularities, incorporating half-form correction and localization techniques.

Findings

Toeplitz operators determine a star product as  +

Quantization extends to singular real polarizations on toric manifolds

Norm estimates establish the star product in the singular setting

Abstract

On a compact K\"ahler manifold $X$, Toeplitz operators determine a deformation quantization $(\operatorname{C}^\infty(X, \mathbb{C})[[\hbar]], \star)$ with separation of variables [10] with respect to transversal complex polarizations $T^{1, 0}X, T^{0, 1}X$ as $\hbar \to 0^+$ [15]. The analogous statement is proved for compact symplectic manifolds with transversal non-singular real polarizations [13]. In this paper, we establish the analogous result for transversal singular real polarizations on compact toric symplectic manifolds $X$. Due to toric singularities, half-form correction and localization of our Toeplitz operators are essential. Via norm estimations, we show that these Toeplitz operators determine a star product on $X$ as $\hbar \to 0^+$.