JournalsjstVol. 12, No. 1pp. 143–167

Berezin–Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian

  • Yuri A. Kordyukov

    Russian Academy of Sciences, Ufa; and Kazan Federal University, Kazan, Russia
Berezin–Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian cover
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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-Kähler quantization). Then the spectrum of the Bochner Laplacian on high tensor powers LpL^p of the prequantum line bundle LL asymptotically splits into clusters of size O(p3/4){\mathcal O}(p^{3/4}) around the points pΛp\Lambda, where Λ\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 eigenvalues from 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 Kähler quantization.

Cite this article

Yuri A. Kordyukov, Berezin–Toeplitz quantization associated with higher Landau levels of the Bochner Laplacian. J. Spectr. Theory 12 (2022), no. 1, pp. 143–167

DOI 10.4171/JST/397