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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
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