Quantization of K\"ahler manifolds via differential operators

Abstract

In this paper, we study the quantization of classical observables (i.e., functions) on a K\"ahler manifold $X$ as differential operators acting on holomorphic sections of tensor powers $L^{\otimes k}$ of the pre-quantum line bundle $L$. We prove two global results as follows. (1). For a general smooth function $f \in C^\infty(X)$, we construct higher order generalizations of Kostant-Souriau's pre-quantum differential operators using our Fedosov-type constructions of Bargmann-Fock sheaves in previous works. We prove that these differential operators are asymptotic to the Berezin-Toeplitz operators $T_{f,k}$ acting on the Hilbert space $H^0(X, L^{\otimes k})$ as $k \to \infty$. (2). If a smooth function $f \in C^\infty(X)$ is furthermore the symbol of a level $k$ quantizable function , then we prove that the associated Berezin-Toeplitz operator $T_{f,k}$ is a holomorphic differential operator. Conversely, Berezin-Toeplitz operators that are holomorphic differential operators all arise in this way. This gives a complete characterization of when Berezin-Toeplitz operators are holomorphic differential operators. To prove these results, we establish new orthogonality relations which generalize the classical Tuynman's Lemma, and employ various differential-geometric and analytic technqiues such as H\"ormander's estimates.