Kontsevich deformation quantization and flat connections

TL;DR

This paper proves the flatness of a natural connection derived from Kontsevich deformation quantization on configuration spaces and explores its relation to associators and braid groups.

Contribution

It demonstrates the flatness of the Kontsevich-derived connection and links it to associator axioms, proposing a new explicit solution related to infinitesimal braids.

Findings

Proves the flatness of the connection bla_n.

Shows the connection bla_n^ is flat and related to associators.

Conjectures bla_n^ takes values in the Lie algebra of infinitesimal braids.

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