Rozansky-Witten invariants via Atiyah classes

Abstract

Recently, L.Rozansky and E.Witten (hep-th/9612216) associated to any hyperKaehler manifold X a system of "weights" (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a very simple cohomological definition of these weights in terms of the Atiyah class of X (the obstruction to existence of a holomorphic connection). We show that the analogy between the tensor of curvature of a hyperKaehler metric and the tensor of structure constants of a Lie algebra observed by Rozansky and Witten, holds in fact for any complex manifold, if we work on the level of cohomology and for any Kaehler manifold, if we work on the level of Dolbeault forms. In particular, for any sheaf A of commutative algebras on any complex manifold the shifted cohomology of the tangent sheaf tensored with A is a graded Lie algebra. We also show that a certain system of Gilkey-type complexes of "natural tensors" on Kaehler manifolds is, up to suspension (accounting for various changes of signs), identified with the PROP (in the sense of Adams and MacLane) describing Lie algebras "up to higher homotopies". As an outcome of our considerations, we give a formula for the Rozansky-Witten classes using any Kaehler metric on a holomorphic symplectic manifold.