Metamorphosis of fractional instantons on a twisted $T^4$ with a double-trace deformation: a numerical study

Abstract

We use numerical minimization of the lattice action of trace-deformed Yang-Mills theory on $T^4$ with twisted boundary conditions to find the classical minimum action configurations of fractional topological charge. We vary the twists and ratios of torus periods to interpolate between different $R^{4-k}\times T^k$ geometries. This allows us to see how the corresponding minimum action saddle point configurations -- monopole-instantons ($k=1$), center vortices ($k=2$), and fractional instantons ($k=3,4$) -- morph into each other. We also study how the transition between them depends on the presence of a deformation potential. In particular, we argue that the recent analytic picture of chains of monopole-instantons collimating their flux into center-vortex sheets, while technically relying on the deformation potential, also holds in pure Yang-Mills theory, for tori whose shape causes the abelianization due to the deformation to align with the one due to the twists. Our results also indicate that with nonzero deformation potential, some transitions between different minimal-action fractional charge configurations may be discontinuous and involve level crossing.

Comments: 42 pages, 14 figures; Fig. 1 redrawn, Fig. 5 expanded, one figure added; references added; version to appear in PRD