Topological quantum field theory and hyperkähler geometry (Q2719826)

Rozansky and Witten proposed a 3-dimensional sigma-model whose target space is a hyperkähler manifold. In the compact case, they conjectured that this theory has an associated topological quantum field theory (TQFT) with Hilbert spaces given by certain cohomology groups.NEWLINENEWLINENEWLINEThere is a modified TQFT constructed by Murakami and Ohtsuki using the universal quantum invariant. The vector spaces in this theory are certain spaces of diagrams, which are graded modules over a certain commutative ring.NEWLINENEWLINENEWLINERozansky-Witten theory naturally leads to a weight system on graph cohomology built from a hyperkähler manifold. The author of this paper constructed in [``A new weight system on chord diagrams via hyperkähler geometry'', Proceedings of the 2nd meeting on quaternionic structures in mathematics and physics, Roma, 1999, 349-363 (2001; Zbl 1035.53064), see also math. DG/0002218] a generalization of the weight system to chord diagrams on circles by adding vector bundles over the hyperkähler manifolds. In the present article he extends these ideas in order to apply a ``hyperkähler weight-system'' to the Murakami-Ohtsuki TQFT. The main result is given by proposition 4.1; all the observables can be obtained by pairing vectors from the TQFT with cohomology classes.