Berezin-Toeplitz quantization asssociated with higher Landau levels of the Bochner Laplacian

TL;DR

This paper develops a Berezin-Toeplitz quantization framework associated with higher Landau levels of the Bochner Laplacian on symplectic manifolds, generalizing almost Kähler quantization and analyzing spectral clustering.

Contribution

It introduces a new quantization method linked to higher Landau levels, extending the scope of Berezin-Toeplitz quantization beyond the lowest Landau level.

Findings

Spectral clusters of the Bochner Laplacian are characterized around pΛ.

A Toeplitz operator calculus is constructed for these clusters.

The lowest Landau level recovers the known almost Kähler quantization.

Abstract

In this paper, we construct a family of Berezin-Toeplitz type quantizations of a compact symplectic manifold. For this, we choose a Riemannian metric on the manifold such that the associated Bochner Laplacian has the same local model at each point (this is slightly more general than in almost-K\"ahler quantization). Then the spectrum of the Bochner Laplacian on high tensor powers $L^p$ of the prequantum line bundle $L$ asymptotically splits into clusters of size ${\mathcal O}(p^{3/4})$ around the points $p\Lambda$, where $\Lambda$ is an eigenvalue of the model operator (which can be naturally called a Landau level). We develop the Toeplitz operator calculus with the quantum space, which is the eigenspace of the Bochner Laplacian corresponding to the eigebvalues frrom the cluster. We show that it provides a Berezin-Toeplitz quantization. If the cluster corresponds to a Landau level of multiplicity one, we obtain an algebra of Toeplitz operators and a formal star-product. For the lowest Landau level, it recovers the almost K\"ahler quantization.