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

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Abstract

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 symbols and the Berezin transform. The general compact quantizable case was done by Bordemann–Meinrenken–Schlichenmaier, Schlichenmaier, and Karabegov–Schlichenmaier. For star products on Kähler manifolds, separation of variables, or equivalently star product of (anti-) Wick type, is a crucial property. As canonically defined star products the Berezin-Toeplitz, Berezin, and the geometric quantization are treated. It turns out that all three are equivalent, but different.

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Notes

  1. For E not a line bundle the Berezin-Toeplitz star product is a star product in \(C^\infty (X,End(E))[[\nu ]]\). This might be considered as a quantization with additional internal degrees of freedom, see [32, Remark 2.27].

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  1. Mathematics Research Unit, FSTC, University of Luxembourg, Maison du Nombre, 6, avenue de la Fonte, 4364, Esch-sur-Alzette, Luxembourg

    Martin Schlichenmaier

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  1. Martin Schlichenmaier
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Correspondence to Martin Schlichenmaier.

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To the memory of our dear friend Sasha Vasiliev who passed away so unexpectedly in the middle of his life. We all miss him.

Partial support by the Internal Research Project GEOMQ15, University of Luxembourg, and by the OPEN programme of the Fonds National de la Recherche (FNR), Luxembourg, project QUANTMOD O13/570706 is gratefully acknowledged.

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Schlichenmaier, M. Berezin-Toeplitz quantization and naturally defined star products for Kähler manifolds. Anal.Math.Phys. 8, 691–710 (2018). https://doi.org/10.1007/s13324-018-0225-9

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