Description:

This course provides an introduction to the deformation quantization of symplectic manifolds, centered around Fedosov’s landmark theorem and its interplay with other quantization schemes in Kähler geometry. Designed for graduate students, the course equips participants with both intuitive foundations and core techniques required for advanced work in this field.

This course consists of two parts:

Part I: Foundations and Fedosov's Approach

• Quantum mechanical background motivating deformation quantization.

• Geometric and algebraic framework of deformation quantization on symplectic manifolds, concluding with a full proof of Fedosov’s theorem.

Part II: Kähler Geometry and Advanced Topics

• Conceptual introduction to geometric quantization and Berezin–Toeplitz quantization.

Star products with separation of variables, Kapranov’s L∞ structures, and a Fedosov-type reconstruction of Berezin–Toeplitz quantization.

Prerequisite:

A solid foundation in differential geometry is required. Familiarity with symplectic geometry is strongly recommended. Prior exposure to basic homological algebra and formal deformation theory will also be beneficial.

References:

A series of suggested papers, including, but not limited to:

1) Fedosov, Boris V. A simple geometrical construction of deformation quantization. J. Differential Geom. 40 (1994), no. 2, 213–238.

2) Karabegov, Alexander V. Deformation quantizations with separation of variables on a Kähler manifold. Comm. Math. Phys. 180 (1996), no. 3, 745–755.

3) Schlichenmaier, Martin. Deformation quantization of compact Kähler manifolds by Berezin-Toeplitz quantization. Conférence Moshé Flato 1999, Vol. II (Dijon), 289--306, Math. Phys. Stud., 22, Kluwer Acad. Publ., Dordrecht, 2000.

4) Kapranov, M. Rozansky-Witten invariants via Atiyah classes. Compositio Math. 115 (1999), no. 1, 71–113.

5) Chan, Kwokwai; Leung, Naichung Conan; Li, Qin. Kapranov's $L_\infty$ structures, Fedosov's star products, and one-loop exact BV quantizations on Kähler manifolds. Commun. Number Theory Phys. 16 (2022), no. 2, 299–351.

Target Audience: Graduate students

Teaching Language: English

Registration: https://www.wjx.top/vm/e9RSsT7.aspx#