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Title: Chern-Simons theory
Author: Hinchliffe, R.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 1998
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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 HΣ , and to a 3-manifold M the theory assigns a numerical invariant (if M is closed), a vector in HδM (if M has closed boundary δM ) or a natural transformation of functors (if the boundary δM of M 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 G and focus on the particular case of G = SU(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 G 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 G 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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available