A new weight system on chord diagrams via hyperkähler geometry (Q2715753)

The author presents some of his results contained in the author's PhD dissertation. The author studies the new invariants of hyper-Kähler manifolds which were first introduced by L. Rozansky and E. Witten. They occur as the weights \(b_\Gamma(X)\) in certain Feynman diagrams and depend only on the graph cohomology class which \(\Gamma\) represents. The weights \(b_\Gamma(X)\) are invariant under deformations of the hyper-Kähler metric and for certain choices of \(\Gamma\) give the characteristic numbers. On the other hand every characteristic number of \(X\) can be expressed as a Rozansky-Witten invariant for some choice of linear combinations of trivalent graphs. As an application of the theory the author gives the relation between the weights \(b_\Gamma(X)\) and the norm of the curvature of an irreducible compact hyper-Kähler manifold \(X\) and its volume for certain trivalent graphs \(\Gamma\). The author constructs also explicitely a weight system \(B_D(X, E_a)\) on chord diagrams \(D\) from a collection of holomorphic bundles \(E_a\) over a compact hyper-Kähler manifold \(X\).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00032].