# Berezin-Toeplitz Quantization for Compact Kähler Manifolds. A Review of Results

@article{Schlichenmaier2010BerezinToeplitzQF, title={Berezin-Toeplitz Quantization for Compact K{\"a}hler Manifolds. A Review of Results}, author={Martin Schlichenmaier}, journal={Advances in Mathematical Physics}, year={2010}, volume={2010}, pages={643-680} }

This article is a review on Berezin-Toeplitz operator and Berezin-Toeplitz deformation quantization for compact quantizable Kahler manifolds. The basic objects, concepts, and results are given. This concerns the correct semiclassical limit behaviour of the operator quantization, the unique Berezin-Toeplitz deformation quantization (star product), covariant and contravariant Berezin symbols, and Berezin transform. Other related objects and constructions are also discussed.

## 76 Citations

### Berezin-Toeplitz quantization for compact Kähler manifolds. An introduction (3 Lectures)

- Mathematics
- 2011

The Berezin-Toeplitz operator and Berezin-Toeplitz deformation quan- tization schemes give quantization methods adapted to a Kahler structure on a manifold to be quantized. Here we present an…

### Berezin-Toeplitz quantization and naturally defined star products for Kähler manifolds

- Mathematics
- 2018

For compact quantizable Kähler manifolds the Berezin-Toeplitz quantization schemes, both operator and deformation quantization (star product) are reviewed. The treatment includes Berezin’s covariant…

### Berezin’s Coherent States, Symbols and Transform for Compact Kähler Manifolds

- Mathematics
- 2013

We review coherent state techniques for general quantizable compact Kahler manifolds. Discussed are co- and contravariant Berezin symbols, Berezin-Toeplitz quantization, the Berezin transform, and…

### BEREZIN–TOEPLITZ QUANTIZATION, HYPERKÄHLER MANIFOLDS, AND MULTISYMPLECTIC MANIFOLDS

- MathematicsGlasgow Mathematical Journal
- 2016

Abstract We suggest a way to quantize, using Berezin–Toeplitz quantization, a compact hyperkähler manifold (equipped with a natural 3-plectic form), or a compact integral Kähler manifold of complex…

### Berezin–Toeplitz quantization for lower energy forms

- Mathematics
- 2014

ABSTRACT Let M be an arbitrary complex manifold and let L be a Hermitian holomorphic line bundle over M. We introduce the Berezin–Toeplitz quantization of the open set of M where the curvature on L…

### Berezin–Toeplitz Quantization for Eigenstates of the Bochner Laplacian on Symplectic Manifolds

- Mathematics
- 2017

We study the Berezin–Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian corresponding to eigenvalues localized near the origin on a symplectic…

### Berezin-Toeplitz Quantization and Star Products for Compact Kaehler Manifolds

- Mathematics
- 2012

For compact quantizable K\"ahler manifolds certain naturally defined star products and their constructions are reviewed. The presentation centers around the Berezin-Toeplitz quantization scheme which…

### Some naturally defined star products for Kähler manifolds by

- Mathematics
- 2012

We give for the Kähler manifold case an overview of the constructions of some naturally defined star products. In particular, the Berezin-Toeplitz, Berezin, Geometric Quantization,…

### Some naturally defined star products for Kaehler manifolds

- Mathematics
- 2016

We give for the Kahler manifold case an overview of the constructions of some naturally defined star products. In particular, the Berezin-Toeplitz, Berezin, Geometric Quantization,…

### Semiclassical spectral analysis of Toeplitz operators on symplectic manifolds: the case of discrete wells

- MathematicsMathematische Zeitschrift
- 2020

We consider Toeplitz operators associated with the renormalized Bochner Laplacian on high tensor powers of a positive line bundle on a compact symplectic manifold. We study the asymptotic behavior,…

## References

SHOWING 1-10 OF 98 REFERENCES

### Berezin-Toeplitz quantization of compact Kähler manifolds

- Mathematics
- 1996

In this lecture results are reviewed obtained by the author together with Martin Bordemann and Eckhard Meinrenken on the Berezin-Toeplitz quantization of compact Kaehler manifolds. Using global…

### Berezin-Toeplitz quantization and Berezin symbols for arbitrary compact Kähler manifolds

- Mathematics
- 1999

For phase-space manifolds which are compact Kahler manifolds relations between the Berezin-Toeplitz quantization and the quantization with the help of Berezin's coherent states and symbols are…

### Deformation quantization of compact Kahler manifolds by Berezin-Toeplitz quantization

- Mathematics
- 1999

For arbitrary compact quantizable Kahler manifolds it is shown how a natural formal deformation quantization (star product) can be obtained via Berezin-Toeplitz operators. Results on their…

### Deformation quantization of compact Kähler manifolds via Berezin-Toeplitz operators

- Mathematics
- 1996

This talk reports on results on the deformation quantization (star products) and on approximative operator representations for quantizable compact Kahler manifolds obtained via Berezin-Toeplitz…

### Toeplitz quantization of Kähler manifolds anggl(N), N→∞ limits

- Mathematics
- 1993

For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes…

### Identification of Berezin-Toeplitz Deformation Quantization

- Mathematics
- 2000

We give a complete identification of the deformation quantization which was obtained from the Berezin- Toeplitz quantization on an arbitrary compact Kahler manifold. The deformation quantization with…

### Toeplitz Operators on Symplectic Manifolds

- Mathematics
- 2008

We study the Berezin-Toeplitz quantization on symplectic manifolds making use of the full off-diagonal asymptotic expansion of the Bergman kernel. We give also a characterization of Toeplitz…

### BEREZIN–TOEPLITZ QUANTIZATION OF THE MODULI SPACE OF FLAT SU(N) CONNECTIONS

- Mathematics
- 2007

Communicated by Theodore Voronov Abstract. The moduli space of flat SU(n) connections on Riemann surfaces is of fundamental importance in TQFT. There is an associated representation of the mapping…

### Deformation estimates for the Berezin-Toeplitz quantization

- Mathematics
- 1992

Deformation estimates for the Berezin-Toeplitz quantization ofCn are obtained. These estimates justify the description of CCR+K as a first-order quantum deformation of AP+C0, where CCR is the…

### Berezin-Toeplitz Quantization and Berezin Transform

- Mathematics
- 2001

In this lecture results on the Berezin-Toeplitz quantization of arbitrary compact quantizable Kahler manifolds are presented. These results are obtained in joint work with M. Bordemann and E.…