Berezin-Toeplitz Quantization in Real Polarizations with Toric Singularities

Abstract

On a compact Kähler manifold $X$, Toeplitz operators determine a deformation quantization $(C^{\infty }(X,\mathbb{C})[[\hslash ]],\star )$ with separation of variables [10] with respect to transversal complex polarizations $T^{1,0}X,T^{0,1}X$ as $\hslash \to 0^{+}$ [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 $\hslash \to 0^{+}$.

16 pages