Identification of Berezin-Toeplitz deformation quantization. (English) Zbl 0997.53067

Karabegov, A. V.; Schlichenmaier, M.

J. Reine Angew. Math. 540, 49-76 (2001).

A. V. Karabegov has shown in [Commun. Math. Phys. 180, 745-755 (1996; Zbl 0866.58037)] that there is a one to one correspondence between deformation quantizations with separation of variables on a Kähler manifold \((M,\omega)\) (i.e. differential star products on \(M\) such that on any chart of \(M\) left star multiplication by holomorphic functions and right star multiplication by antiholomorphic functions are point-wise multiplication of functions) and closed formal (1,1) forms \((1/\nu)\omega +\omega_0+\nu \omega_1+\). Moreover [A. V. Karabegov, Lett. Math. Phys. 43, 347-357 (1998; Zbl 0938.53049)] these star products are classified (up to equivalence) by formal cohomology classes \((1/i\nu)\omega +H^2(M, {\mathbb C}[[\nu]])\).
When \((M,\omega)\) is a (prequantizable) compact Kaehler manifold, it was shown by M. Schlichenmeier [Conférence Moshé Flato (Dijon, France 1999), G. Dito and D. Sternhaimer (eds.), Kluwer Vol. 2, 289-306 (2000; Zbl 1028.53085)] that the Berezin-Toeplitz quantization map leads to a star product on \(M\) called Berezin-Toeplitz deformation quantization. In the present work, the Berezin-Toeplitz deformation quantization is proved to be a star product with separation of variables and the corresponding formal form is explicitly determined especially by use of results on Szegő kernels from L. Boutet de Monvel and S. Sjöstrand [Astérisque 34/35, 123-164 (1976; Zbl 0344.32010)]. The formal cohomology class of Berezin-Toeplitz deformation quantization is also obtained.

Reviewer: Benjamin Cahen (Metz)

MSC:

53D55	Deformation quantization, star products
53D50	Geometric quantization
81S10	Geometry and quantization, symplectic methods

Keywords:

Kähler manifold; Berezin-Toeplitz deformation quantization; star product; Szegő kernels

Citations:

Zbl 0866.58037; Zbl 0938.53049; Zbl 1028.53085; Zbl 0344.32010

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