Quantization of the Algebra of Chord Diagrams

Abstract

In this paper we define an algebra structure on the vector space $L(\Sigma)$ generated by links in the manifold $\Sigma \times [0,1]$ where $\Sigma $ is an oriented surface. This algebra has a filtration and the associated graded algebra $L_{Gr}(\Sigma)$ is naturally a Poisson algebra. There is a Poisson algebra homomorphism from the algebra of chord diagrams $ch(\Sigma)$ on $\Sigma $ to $L_{Gr}(\Sigma)$. We show that multiplication in $L(\Sigma)$ provides a geometric way to define a deformation quantization of the algebra of chord diagrams, provided there is a universal Vassiliev invariant for links in $\Sigma\times [0,1]$. The quantization descends to a quantization of the moduli space of flat connections on $\Sigma $ and it is universal with respect to group homomorphisms. If $\Sigma $ is compact with free fundamental group we construct a universal Vassiliev invariant.