On the Generalized Casson Invariant

The path integral generalization of the Casson invariant as developed by Rozansky and Witten is investigated. The path integral for various three manifolds is explicitly evaluated. A new class of topological observables is introduced that may allow for more effective invariants. Finally it is shown how the dimensional reduction of these theories corresponds to a generalization of the topological B sigma model

A Geometric Interpretation of the \chi-{y} Genus on Hyper-Kahler Manifolds

The group SL(2) acts on the space of cohomology groups of any hyper-Kahler manifold X. The \chi_{y} genus of a hyper-Kahler X is shown to have a geometric interpretation as the super trace of an element of SL(2). As a by product one learns that the generalized Casson invariant for a mapping torus is essentially the \chi_{y} genus

Chern-Simons Theory on Seifert 3-Manifolds

We study Chern-Simons theory on 3-manifolds M that are circle-bundles over 2-dimensional orbifolds S by the method of Abelianisation. This method, which completely sidesteps the issue of having to integrate over the moduli space of non-Abelian flat connections, reduces the complete partition function of the non-Abelian theory on M to a 2-dimensional Abelian theory on the orbifold S which is easily evaluated.Comment: 27 page