The Rozansky-Witten invariants of hyperkähler manifolds

Abstract

We investigate invariants of hyperkähler manifolds introduced by Rozansky and Witten. For each tri-valent 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 virtual hyperkähler 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.

Appears in the Proceedings of the 7th International Conference on Differential Geometry and Applications (available electronically).

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