QFT In the literature: contested

Does small-circle semiclassical confinement connect smoothly to four dimensions

In plain words

When one space direction is curled into a small circle, confinement can be computed by hand. Whether that calculation describes the same physics as ordinary uncurled space is unknown.

Precise statement

Pure $\operatorname{SU}(N)$ Yang-Mills on $R^{3} x S^{1}_L$ with a center-stabilizing double-trace deformation, or $N=1$ super-Yang-Mills with periodic fermions, confines semiclassically at small L via monopole-instantons and bions (Unsal and collaborators, 2007-2008). Determine whether the small-L regime is connected to the $R^4$ theory as $L \to \infty$ without a phase transition, so that the semiclassical mechanism is the 4D mechanism.

What would settle it

Lattice simulation of the deformed theory over the full range of L times the strong scale, showing analytic L-dependence of the string tension and order parameters, or identifying a transition.

Status in the literature

Unverified note

Lattice studies of $N=1$ super-Yang-Mills with periodic fermions found no transition; for deformed pure Yang-Mills continuity remains unproven (2025).