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 ${\mathbb{R}}^2\times {\mathbb{T}}_{\ast }^2$, ${\mathbb{R}}^3\times S^1$, and $\mathbb{R}\times {\mathbb{T}}_{\ast }^2\times S^1$. We confirm that the fractional instantons are center vortices on ${\mathbb{R}}^2\times {\mathbb{T}}_{\ast }^2$ and monopoles on ${\mathbb{R}}^3\times S^1$, and we calculate several properties relevant to using these solutions for semiclassical calculations. On $\mathbb{R}\times {\mathbb{T}}_{\ast }^2\times S^1$, we interpolate between the large ${\mathbb{T}}_{\ast }^2$ 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.

47 pages, 27 figures. Updated to add references and correct typos. This version was submitted to JHEP