On the Rozansky–Witten weight systems

Abstract

Ideas of Rozansky and Witten, as developed by Kapranov, show that a complex symplectic manifold $X$ gives rise to Vassiliev weight systems. In this paper we study these weight systems by using $D\left(X\right)$, the derived category of coherent sheaves on $X$. The main idea (stated here a little imprecisely) is that $D\left(X\right)$ is the category of modules over the shifted tangent sheaf, which is a Lie algebra object in $D\left(X\right)$; 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 $D\left(X\right)$ 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 $\left(1+1+1\right)$–dimensional Rozansky–Witten TQFT, and to hyperkähler geometry.