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.
Categories: hep-lat hep-th