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.

Latex2e, 19 pages (US letter format), 8 eps-Figures

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