Determine whether a center-vortex profile closes the twist-induced Faddeev–Popov gap

Determine whether a center-vortex configuration in a fixed ’t Hooft-flux sector closes the twist-induced Faddeev–Popov spectral gap and drives one or more eigenvalues toward the boundary of the sectorwise first Gribov region.

Background

The paper distinguishes the irreducibly twisted flat representative, whose free Faddeev–Popov operator has a strictly positive gap, from an actual dynamical center-vortex configuration in the same flux sector. The unresolved issue is whether the vortex profile modifies the Faddeev–Popov spectrum sufficiently to reach or approach the Gribov horizon.

The proposed investigation compares the spectrum of the sectorwise Faddeev–Popov operator evaluated on a vortex configuration with its spectrum at the flat representative. Resolving this question would clarify whether the connection between center vortices and the Gribov horizon is dynamical rather than merely a consequence of global twisting.

References

The sharp question is whether the vortex profile closes the twist-induced gap and drives one or more eigenvalues toward the boundary of $\Omega_B$.

— Center-twisted Gribov spectra and the finite-volume Gaussian response in the refined Gribov--Zwanziger framework  (2609.03525 - Morikawa, 3 Sep 2026) in Section 7.1, “The spectral question” (Eq. 7.1–7.2)

A closed analytic expression for the required self-dual $\mathbb{R}2\times T2$ vortex profile is not presently available.

— Center-twisted Gribov spectra and the finite-volume Gaussian response in the refined Gribov--Zwanziger framework  (2609.03525 - Morikawa, 3 Sep 2026) in Section 7.2, “$\mathbb{R}^2\times T^2$ as a controlled bridge”