Date of Degree
PhD (Doctor of Philosophy)
Vincent G. J. Rodgers
In this thesis, topologically massive Yang-Mills theory is studied in the framework of geometric quantization. This theory has a mass gap that is proportional to the topological mass m. Thus, Yang-Mills contribution decays exponentially at very large distances compared to 1/m, leaving a pure Chern-Simons theory with level number k. The focus of this research is the near Chern-Simons limit of the theory, where the distance is large enough to give an almost topological theory, with a small contribution from the Yang-Mills term. It is shown that this almost topological theory consists of two copies of Chern-Simons with level number k/2, very similar to the Chern-Simons splitting of topologically massive AdS gravity model. As m approaches to infinity, the split parts add up to give the original Chern-Simons term with level k. Also, gauge invariance of the split CS theories is discussed for odd values of k. Furthermore, a relation between the observables of topologically massive Yang-Mills theory and Chern-Simons theory is obtained. It is shown that one of the two split Chern-Simons pieces is associated with Wilson loops while the other with 't Hooft loops. This allows one to use skein relations to calculate topologically massive Yang-Mills theory observables in the near Chern-Simons limit. Finally, motivated with the topologically massive AdS gravity model, Chern-Simons splitting concept is extended to pure Yang-Mills theory at large distances. It is shown that pure Yang-Mills theory acts like two Chern-Simons theories with level numbers k/2 and -k/2 at large scales. At very large scales, these two terms cancel to make the theory trivial, as required by the existence of a mass gap.
Yang-Mills theory is a very important part of the Standard Model of particle physics. Despite its success in explaining the high-energy physics, its mathematical behavior is very difficult to understand at low energies. This difficulty gave birth to a simplified program, pioneered by Feynman(1981). Feynman suggested that, in order to understand the qualitative behavior of the theory, one can study it in two space and one time dimensions instead of the actual 3+1 dimensional case.
There is another important 2+1 dimensional theory in mathematical physics, called the Chern-Simons theory. This theory is related to knot theory, which was shown by Witten(1989). Chern-Simons theory is very useful in condensed matter physics, especially in the fractional quantum hall effect phenomena.
This thesis studies a mixture of these two 2+1 dimensional theories, called the "Topologically Massive Yang-Mills theory". Unlike Chern-Simons theory, Yang-Mills theory is not directly related to knot theory. But Topologically Massive Yang-Mills theory has an interesting low energy behavior that allows one to use knot theory. This relation is the main result of this thesis and it provides a nice extension to the relations between field theories and knot theory. Furthermore, a similar result is obtained for pure Yang-Mills theory. The results of this thesis can be used in some condensed matter theory applications and also, it may provide valuable insight in studying 2+1 dimensional particle physics.
publicabstract, Chern Simons, Geometric Quantization, Knot Theory, Link Invariants, Topological Field Theory, Yang Mills
ix, 80 pages
Includes bibliographical references (pages 77-80).
Copyright 2014 Tuna Yildirim