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On a Graph Theoretic Formula of Gammelgaard for Berezin–Toeplitz Quantization

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Abstract

We give a proof of (a slightly refined version of) a graph theoretic formula due to Gammelgaard, Karabegov and Schlichenmaier for Berezin–Toeplitz quantization on Kähler manifolds. We obtain the formula by inverting the Berezin transform using a composition formula for the ring of differential operators encoded by linear combinations of strongly connected graphs. The same method is also used to identify the dual Karabegov–Bordemann–Waldmann star product. Our proof has the merit of giving more insight into Karabegov–Schlichenmaier’s identification theorem (Karabegov in J Reine Angew Math 540:49–76, 2001) that the Karabegov classifying form of the Berezin and Berezin–Toeplitz star products are, respectively, obtained by deforming the Kähler metric along the Ricci curvature and the logarithm of the Bergman kernel.

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Authors and Affiliations

  1. Center of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, Zhejiang, China

    Hao Xu

  2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Hao Xu

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  1. Hao Xu
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Xu, H. On a Graph Theoretic Formula of Gammelgaard for Berezin–Toeplitz Quantization. Lett Math Phys 103, 145–169 (2013). https://doi.org/10.1007/s11005-012-0585-2

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