ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: not given Sawon, Justin, New College, University of Oxford, Great Britain The Rozansky–Witten invariants of hyperkähler manifolds We investigate invariants of hyperkähler manifolds introduced by Rozansky and Witten. For each trivalent graph with 2n vertices we get an invariant of a hyperkähler manifold of dimension 4n. The invariants are the same for cohomologous graphs (graphs equivalent under the IHX relations). This allows us to use hyperkähler 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 \underline{virtual} hyperkähler manifolds. Certain combinations of graphs give rise to invariants which can be identified with the Chern numbers, and we use the virtual hyperkähler manifolds to obtain further unexpected relations between the graph invariants and Chern numbers.