Rozansky-Witten invariants of hyperk\"ahler manifolds

TL;DR

This paper explores Rozansky-Witten invariants of compact hyperk"ahler manifolds, showing they encompass all characteristic numbers, relate to curvature and volume, and can be extended to include holomorphic vector bundles.

Contribution

It demonstrates that Rozansky-Witten invariants include all characteristic numbers and introduces a generalization involving holomorphic vector bundles.

Findings

Rozansky-Witten invariants include all characteristic numbers.

The norm of the Riemann curvature relates to volume and characteristic numbers.

A new generalization incorporating holomorphic vector bundles is proposed.

Abstract

We investigate invariants of compact hyperk{\"a}hler manifolds introduced by Rozansky and Witten: they associate an invariant to each graph homology class. It is obtained by using the graph to perform contractions on a power of the curvature tensor and then integrating the resulting scalar-valued function over the manifold, arriving at a number. For certain graph homology classes, the invariants we get are Chern numbers, and in fact all characteristic numbers arise in this way. We use relations in graph homology to study and compare these hyperk{\"a}hler manifold invariants. For example, we show that the norm of the Riemann curvature can be expressed in terms of the volume and characteristic numbers of the hyperk{\"a}hler manifold. We also investigate the question of whether the Rozansky-Witten invariants give us something more general than characteristic numbers. Finally, we introduce a generalization of these invariants which incorporates holomorphic vector bundles into the construction.