Numerical fractional instantons in SU(2): center vortices, monopoles, and a sharp transition between them

Abstract

We use a numerical cooling algorithm to study fractional instantons in $SU(2)$ pure Yang-Mills on $\mathbb{R}^2\times\mathbb{T}^2_*$, $\mathbb{R}^3\times S^1$, and $\mathbb{R}\times \mathbb{T}^2_* \times S^1$. We confirm that the fractional instantons are center vortices on $\mathbb{R}^2\times\mathbb{T}^2_*$ and monopoles on $\mathbb{R}^3\times S^1$, and we calculate several properties relevant to using these solutions for semiclassical calculations. On $\mathbb{R}\times \mathbb{T}^2_* \times S^1$, we interpolate between the large $\mathbb{T}^2_*$ limit and the large $S^1$ limit to study how the solutions interpolate between center vortices and monopoles. We find that they are separated by a sharp transition, with 't Hooft's constant field strength solutions living at the transition point. These results contrast but do not contradict recent results suggesting continuity between vortices and monopoles.