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

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*Justin Sawon*

**E-mail:** sawon@maths.ox.ac.uk
**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