A Brief Introduction to Berezin Toeplitz Operators on Compact Kähler Manifolds

This text provides a comprehensive introduction to BerezinToeplitz operators on compact Kähler manifolds. The heart of the book is devoted to a proof of the main properties of these operators which have been playing a significant role in various areas of mathematics such as complex geometry, topological quantum field theory, integrable systems, and the study of links between symplectic topology and quantum mechanics. The book is carefully designed to supply graduate students with a unique accessibility to the subject. The first part contains a review of relevant material from complex geometry. Examples are presented with explicit detail and computation; prerequisites have been kept to a minimum. Readers are encouraged to enhance their understanding of the material by working through the many straightforward exercises.

Presents the first introduction to the subject for non-specialists Requires only a basic knowledge of differential geometry and analysis First chapters review the necessary geometric notions to study the field Includes several exercises and examples serving as tutorials or exam problems

Auteur

Yohann Le Floch is maître de conférences at Université de Strasbourg, in the Analysis team of the Institut de Recherche Mathématique Avancée. His areas of interest are semiclassical analysis, with a focus on Berezin-Toeplitz operators, and classical and quantum integrable systems.

Texte du rabat

This text provides a comprehensive introduction to Berezin Toeplitz operators on compact Kähler manifolds. The heart of the book is devoted to a proof of the main properties of these operators which have been playing a significant role in various areas of mathematics such as complex geometry, topological quantum field theory, integrable systems, and the study of links between symplectic topology and quantum mechanics. The book is carefully designed to supply graduate students with a unique accessibility to the subject. The first part contains a review of relevant material from complex geometry. Examples are presented with explicit detail and computation; prerequisites have been kept to a minimum. Readers are encouraged to enhance their understanding of the material by working through the many straightforward exercises.

Résumé

"The book ... represents an essential prerequisite for anyone who wants to work in the field. The author have managed to make it readable by non-specialists." (Béchir Dali, zbMATH 1452.32002, 2021)

Contenu
Preface.- 1. Introduction.- 2. A short introduction to Kähler manifolds.- 3. Complex line bundles with connections.- 4. Quantization of compact Kähler manifolds.- 5. BerezinToeplitz operators.- 6. Schwartz kernels.- 7. Asymptotics of the projector Pi_k.- 8. Proof of product and commutator estimates.- 9. Coherent states and norm correspondence.- A. The circle bundle point of view.- Bibliography.