Quantization of the Algebra of Chord Diagrams

Abstract

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