The Rozansky-Witten invariants of hyperk{\"a}hler manifolds

Justin Sawon


Abstract. We investigate invariants of hyperk{\"a}hler manifolds introduced by Rozansky and Witten. For each tri-valent graph with $2n$ vertices we get an invariant of a hyperk{\"a}hler manifold of dimension $4n$. The invariants are the same for cohomologous graphs (graphs equivalent under the IHX relations). This allows us to use hyperk{\"a}hler manifolds to define elements in the dual of the graph cohomology. Conversely, we regard elements in the dual of the graph cohomology (such as those arising in Chern-Simons theory) as {\em virtual\/} hyperk{\"a}hler manifolds. Certain combinations of graphs give rise to invariants which can be identified with the Chern numbers, and we use the virtual manifolds to obtain further unexpected relations between the graph invariants and Chern numbers.

AMSclassification. 32C17, 53C15, 57R20

Keywords. Hyperk{\"a}hler manifolds, graph cohomology, invariants, Chern numbers