On a Graph Theoretic Formula of Gammelgaard for Berezin-Toeplitz Quantization
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.
- Publication:
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Letters in Mathematical Physics
- Pub Date:
- February 2013
- DOI:
- 10.1007/s11005-012-0585-2
- arXiv:
- arXiv:1204.2259
- Bibcode:
- 2013LMaPh.103..145X
- Keywords:
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- Mathematics - Quantum Algebra;
- Mathematical Physics
- E-Print:
- 18 pages