Revision 2

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Changes against revision 1:

Title · changed: Revision 1: Does small-circle semiclassical confinement connect smoothly to four dimensions; Revision 2: Does small-circle semiclassical confinement connect smoothly to four dimensions?
Plain: 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: 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.
Settled by: 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.
Kind: well-posed
Answer type: yes-no
References: (empty)
Literature status: contested
Status note: Lattice studies of $N=1$ super-Yang-Mills with periodic fermions found no transition; for deformed pure Yang-Mills continuity remains unproven (2025).
Posed since: 2008

Decisions