| Summary: | This dissertation is concerned with various mathematical aspects of Topological Quantum Field Theories (TQFTs) known as Chern-Simons theories. Although this subject has its origins in theoretical physics, the treatment here is in terms of the axiomatic approach due to Segal and Atiyah. A key feature of the thesis is the notion of a 3-tier (axiomatic) TQFT. This involves assigning a category to a closed I-manifold and a functor to a 2-manifold with boundary which is viewed as a cobordism between I-manifolds. To a closed 2-manifold Σ the theory assigns a vector space <I>H<SUB>Σ</SUB></I> , and to a 3-manifold <I>M</I> the theory assigns a numerical invariant (if <I>M </I>is closed), a vector in <I>H<SUB>δM</SUB></I> (if <I>M</I> has closed boundary <I>δM</I> ) or a natural transformation of functors (if the boundary <I>δM</I> of <I>M</I> has a 1-dimensional boundary). After a brief introduction, we introduce in chapter 1 the definition of a TQFT and that of a 3-tier TQFT. We then describe the geometrical set-up for Chern-Simons Theory for a Lie group <I>G </I>and focus on the particular case of <I>G = SU</I>(2). Finally we describe quite concisely how it might fit into a 3-tier TQFT structure. Roughly the next half of the thesis treats the specific case of Chern-Simons theory for the circle group T. In chapter 2 we describe a number of interesting topological aspects of the theory. In chapter 3 we go on to show how the theory fits into 3-tier TQFT framework. In the next two chapters we begin to deal with Chern-Simons theories for <I>G </I>a non-compact group. In chapters 6 and 7 we deal with a rather more algebraic theory which is the abelian version of a theory which is meant to compute the Casson invariant for oriented homology 3-spheres. For this reason we call it the abelian Casson-type theory. From the physics viewpoint, it coincides with the Chern-Simons theory where <I>G</I> is a supergroup. This is rather difficult to motivate mathematically, so we adopt an algebraic-topological definition of the theory and show it satisfies the TQFT axioms. We then go on to show how it fits into the 3-tier TQFT structure. The novelty here is that the category assigned to a 1-manifold is not semisimple.
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