Letters in Mathematical Physics

, Volume 103, Issue 2, pp 145–169 | Cite as

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



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.

Mathematics Subject Classification (2010)

53D55 32J27 


Berezin–Toeplitz quantization Berezin transform Karabegov form 


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Copyright information

© Springer 2012

Authors and Affiliations

  1. 1.Center of Mathematical SciencesZhejiang UniversityHangzhouChina
  2. 2.Department of MathematicsHarvard UniversityCambridgeUSA

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