Berezin-Toeplitz quantization revisited
Kwokwai Chan, Naichung Conan Leung, Qin Li, Yutung Yau
TL;DR
The paper resolves global aspects of Berezin--Toeplitz quantization on $H^0(X,L^{\otimes k})$ for a compact Kähler manifold by constructing higher Kostant--Souriau operators $P_{f,k,m}$ via Fedosov-type methods. It proves that $T_{f,k}$ admits an asymptotic expansion $T_{f,k}\sim\sum_{m\ge0}P_{f,k,m}$ with error $O(k^{-(m+1)/2})$, yielding asymptotic locality and a module-version of the BT star-product expansion. A complete characterization is given for when $T_{f,k}$ acts as a holomorphic differential operator: precisely when $f$ is the symbol of a level-$k$ quantizable function, with $T_{f,k}=P_{\alpha,k}$ for a flat $D_{BT,k}$-section $\alpha$. The work unifies Fedosov quantization, Bargmann--Fock modules, and Hörmander-type estimates to illuminate how deformation quantization on Kähler manifolds projects to finite-level quantum Hilbert spaces, improving understanding of locality, operator algebras, and the quantization of functions beyond Tuynman’s classical framework.
Abstract
This is a sequel to a series of works, where we studied the local aspects of the asymptotic action of deformation quantization on the Hilbert spaces $H^0(X, L^{\otimes k})$ of geometric quantization for a Kähler manifold $X$; here $L$ is a pre-quantum line bundle on $X$. In this paper, we consider the Berezin-Toeplitz deformation quantization and obtain the following global results concerning the asymptotic action of Berezin-Toeplitz operators on $H^0(X, L^{\otimes k})$: (1). For a general smooth function $f \in C^\infty(X)$, we prove that the Berezin-Toeplitz operators $T_{f,k}$ are asymptotic to differential operators acting on $H^0(X, L^{\otimes k})$ as $k \to \infty$. An immediate consequence is their asymptotic locality. (2). If $f \in C^\infty(X)$ is furthermore the symbol of a level $k$ quantizable function, then we prove that the associated Berezin-Toeplitz operators $T_{f,k}$ are all holomorphic differential operators. 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 construct higher order analgoues of Kostant-Souriau's pre-quantum differential operators using our Fedosov-type constructions in previous works, establish new orthogonality relations which generalize the classical Tuynman's Lemma, and employ various differential-geometric and analytic technqiues such as Hörmander's estimates.