Topologically Massive Yang-Mills Theory and Link Invariants (Thesis)

TL;DR

This thesis explores the relationship between topologically massive Yang-Mills theory and Chern-Simons theory, revealing a splitting into two Chern-Simons components at large distances and connecting observables like Wilson and 't Hooft loops.

Contribution

It demonstrates the near Chern-Simons limit of topologically massive Yang-Mills theory splits into two Chern-Simons theories and extends this concept to pure Yang-Mills theory at large scales.

Findings

Near Chern-Simons limit yields two split Chern-Simons theories.

Observable relations connect Wilson and 't Hooft loops to skein relations.

At large scales, pure Yang-Mills acts as two cancelling Chern-Simons theories.

Abstract

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 Chern-Simons 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.