Ideas of Rozansky and Witten, as developed by Kapranov, show that a complex symplectic
manifold
gives rise to Vassiliev weight systems. In this paper we study these weight systems by using
, the derived category of
coherent sheaves on .
The main idea (stated here a little imprecisely) is that
is the
category of modules over the shifted tangent sheaf, which is a Lie algebra object in
; the
weight systems then arise from this Lie algebra in a standard way. The other main
results are a description of the symmetric algebra, universal enveloping algebra and
Duflo isomorphism in this context, and the fact that a slight modification of
has the structure of a braided ribbon category, which gives another way
to look at the associated invariants of links. Our original motivation for
this work was to try to gain insight into the Jacobi diagram algebras used
in Vassiliev theory by looking at them in a new light, but there are other
potential applications, in particular to the rigorous construction of the
–dimensional
Rozansky–Witten TQFT, and to hyperkähler geometry.
