Kontsevich Deformation Quantization and Flat Connections

Abstract

In Torossian (J Lie Theory 12(2):597–616, 2002), the second author used the Kontsevich deformation quantization technique to define a natural connection ω _n on the compactified configuration spaces \({\overline{C}_{n,0}}\) of n points on the upper half-plane. Connections ω _n take values in the Lie algebra of derivations of the free Lie algebra with n generators. In this paper, we show that ω _n is flat.

The configuration space \({\overline{C}_{n,0}}\) 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, ω _n gives rise to a flat connection \({\omega_n^\infty}\). We show that the parallel transport \({\Phi}\) defined by the connection \({\omega_3^\infty}\) between configuration 1(23) and (12)3 verifies axioms of an associator.

We conjecture that \({\omega_n^\infty}\) takes values in the Lie algebra \({\mathfrak{t}_n}\) of infinitesimal braids. If correct, this conjecture implies that \({\Phi \in \exp(\mathfrak{t}_3)}\) is a Drinfeld’s associator. Furthermore, we prove \({\Phi \neq \Phi_{KZ}}\) showing that \({\Phi}\) is a new explicit solution of associator axioms.