The authors study a topological sigma model which is like Chern-Simons theory with reversed statistics. The paper makes heavy use of concepts from field theory; however, some of the main points may be made without reference to field theory. In particular, weight systems may be constructed from hyper-Kähler manifolds. A weight system is a map from the collection of trivalent graphs into the real numbers which satisfies the IHX relation. Given a weight system we can produce a finite type 3-manifold invariant [see T. Ohtsuki, J. Knot Theory Ramifications 5, 101-115 (1996)]. The invariant will be of the form \[ \sum a_\Gamma I_\Gamma(M) \] where the sum is taken over all trivalent graphs \(\Gamma\), the weight system is \(a_\Gamma\) and \(I_\Gamma(M)\) are certain integrals. Up to now, the standard method to define weight systems has used the structure constraints of a Lie algebra.
In the present paper, the authors show that the Bianchi idenity on a hyper-Kähler manifold may be written in a way reminiscent of the Jacobi identity. The invariants arise in perturbative expansions of a certain path integral. This is yet another example of interesting mathematics that was discovered through physical intuitions.