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Kontsevich Deformation Quantization and Flat Connections

Torossian, Charles
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 half-plane. 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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ALEKSEEV, Anton, TOROSSIAN, Charles. Kontsevich Deformation Quantization and Flat Connections. In: Communications in Mathematical Physics, 2010, vol. 300, n° 1, p. 47 - 64. https://archive-ouverte.unige.ch/unige:11956

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Deposited on : 2010-09-30

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