Landau levels on a compact manifold

TL;DR

This paper investigates higher Landau levels on compact manifolds with constant magnetic fields, computing their dimensions, analyzing associated Toeplitz algebras, and establishing isomorphisms with twisted quantizations.

Contribution

It extends the understanding of Landau levels beyond the first, providing explicit dimension formulas and algebraic structures for higher levels.

Findings

Dimensions of higher Landau levels are given by Riemann-Roch numbers.

Associated Toeplitz algebras are characterized and studied.

Higher levels are shown to be isomorphic to twisted quantizations.

Abstract

We consider a magnetic Laplacian on a compact manifold, with a constant non-degenerate magnetic field. In the large field limit, it is known that the eigenvalues are grouped in clusters, the corresponding sums of eigenspaces being called the Landau levels. The first level has been studied in-depth as a natural generalization of the Kaehler quantization. The current paper is devoted to the higher levels: we compute their dimensions as Riemann-Roch numbers, study the associated Toeplitz algebras and prove that each level is isomorphic with a quantization twisted by a convenient auxiliary bundle.