Use this URL to cite or link to this record in EThOS:
Title: Chern-Simons theory
Author: Hinchliffe, R.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 1998
Availability of Full Text:
Full text unavailable from EThOS.
Please contact the current institution’s library for further details.
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