Chern-Simons Splitting of 2+1D Pure Yang-Mills Theory at Large Distances

Abstract

Geometric quantization of 2+1 dimensional pure Yang-Mills theory is studied with focusing on finite large scales. It is previously shown that (Yildirim, 2015, Int. J. Mod. Phys A, 30(7), 1550034), topologically massive Yang-Mills theory exhibits a Chern-Simons splitting behavior at large scales, similar to the topologically massive AdS gravity model in its near Chern-Simons limit. This splitting occurs as two Chern-Simons parts with levels $+k/2$ each, where $k$ is the original Chern-Simons level in the Lagrangian. In the pure Chern-Simons limit, split parts combine to give the original Chern-Simons level. The opposite limit of the gravitational analogue gives an Einstein-Hilbert term with a negative cosmological constant, which can be written as a two half Chern-Simons terms with opposite signs. With this motivation, the gauge theory analogue of this limit is investigated, showing that at finite large distances pure Yang-Mills theory acts like a topological theory that consists of split Chern-Simons parts with levels $+k/2$ and $-k/2$. At very large distances these split terms cancel, making the Yang-Mills theory trivial, as expected, due to the existence of a mass gap. Gauge invariance of the split parts is discussed. It is also shown that this splitting behavior can be exploited to incorporate link invariants of knot theory, just like in the topologically massive case.