Kontsevich Deformation Quantization and Flat Connections

In Torossian (J Lie Theory 12(2): 597-616, 2002), the second author used the Kontsevich deformation quantization technique to define a natural connection omega(n) on the compactified configuration spaces (C) over bar (n), 0 of n points on the upper half-plane. Connections omega(n) take 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 (C) over bar (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, omega(n) gives rise to a flat connection omega(infinity)(n). We show that the parallel transport Phi defined by the connection. omega(infinity)(3) between configuration 1(23) and (12)3 verifies axioms of an associator. We conjecture that omega(infinity)(n) takes values in the Lie algebra t(n) of infinitesimal braids. If correct, this conjecture implies that Phi is an element of exp(t(3)) is a Drinfeld's associator. Furthermore, we prove Phi not equal Phi(KZ) showing that Phi is a new explicit solution of associator axioms.