Center-twisted Gribov spectra and the finite-volume Gaussian response in the refined Gribov--Zwanziger framework

Abstract

We develop a continuum framework for comparing the Gribov--Zwanziger and center-vortex descriptions of confinement through the gauge-invariant twisted partition function of the electric $\mathbb{Z}_N^{[1]}$ 1-form symmetry. A background 2-form field $B$, equivalently an 't~Hooft twist on a torus, labels a global sector and is not itself a dynamical center vortex. For a minimal irreducible twist on $T^4$, we derive the complete adjoint momentum lattice of $SU(N)$. The twisted spectrum is exactly the scalar spectrum on an enlarged torus with periods $(L_1,L_2,NL_3,NL_4)$ with the ordinary-torus sublattice removed. This yields a finite Faddeev--Popov gap at the flat representative and reduces twisted-minus-untwisted spectral traces to ordinary torus traces. For the refined Gribov--Zwanziger (RGZ) kernel, Poisson resummation gives an exact finite-volume Bessel-function winding sum with a universal center-twist projector. We evaluate the Gaussian one-loop integral at fixed RGZ parameters up to the finite-dimensional global zero-mode/stabilizer normalization. The Zwanziger determinants cancel, while the gauge-fixing/ghost sector leaves a universal massless primed determinant on $T^4$; the untwisted sector also contains constant gluon modes. The only normalization not fixed by the local quadratic Hessian is the relative zero-mode/stabilizer measure of the reducible untwisted and irreducibly twisted flat connections. We also derive closed finite-volume sources for the RGZ stationary equations. The massive response is exponentially suppressed at large volume, whereas the massless factor depends on the global zero-mode normalization. Thus the Gaussian calculation does not by itself establish the strong center-vortex-condensation criterion for $Z[B]/Z[0]$, but it isolates the remaining global normalization problem and separates a global twist from an actual dynamical vortex.