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Kontsevich Deformation Quantization and Flat Connections 

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Published in  Communications in Mathematical Physics. 2010, vol. 300, no. 1, p. 47  64  
Abstract  In arXiv:math/0105152, the second author used the Kontsevich deformation quantization technique to define a natural connection omega_n on the compactified configuration spaces of n points on the upper halfplane. This connection takes values in the Lie algebra of derivations of the free Lie algebra with n generators. In this paper, we show that omega_n is flat. The configuration space contains a boundary stratum at infinity which coincides with the (compactified) configuration space of n points on the complex plane. When restricted to this stratum, omega_n gives rise to a flat connection omega_n^infty. We show that the parallel transport Phi defined by omega_3^infty between configuration 1(23) and (12)3 verifies axioms of an associator. We conjecture that omega_n^infty takes values in the Lie algebra of infinitesimal braids. This conjecture implies that Phi is an even Drinfeld associator defining a new explicit solution of associator axioms. A proof of this conjecture has recently appeared in arXiv:0905.1789  
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Citation (ISO format)  ALEKSEEV, Anton, TOROSSIAN, Charles. Kontsevich Deformation Quantization and Flat Connections. In: Communications in Mathematical Physics, 2010, vol. 300, n° 1, p. 47  64. https://archiveouverte.unige.ch/unige:11956 