Some Remarks on the Spectral Geometry of the Gribov Horizon

Abstract

We develop a local spectral framework for the Landau-gauge Gribov horizon that distinguishes gauge-orbit projection from degeneracy of the gauge-fixing Hessian. On a flat torus, spatially constant ghosts form a residual global-color kernel; after removal of this kernel, the reduced Faddeev-Popov operator is the normal Morse-Bott Hessian of the orbit-norm functional, whereas the covariant Laplacian defines the orthogonal connection. For the associated affine focal pencil, we prove a quadratic-form index theorem with a Morse-Bott endpoint. At a regular isolated crossing, the critical projector $P$ and invertible compressed derivative $Γ=P\dot{\mathcal{M}}P$ determine the spectral-flow jump and leading Laurent coefficient of the sourced ghost resolvent; for a simple zero, they also give the wall conormal. Crossings reached along affine rays from the positive region have negative-definite $Γ$, including symmetry-protected multiplets, while a second Schur reduction determines pole orders along tangential paths. A projected single-harmonic model checks the projected Feynman-Hellmann relation. For an $SU(2)$ hedgehog on $\mathbb R^3$, we construct a normalizable threshold state in every coupled spin-orbit channel and minimize over the full tower to obtain the exact stability interval $-2<g<1$. Dirichlet-box spectra approach these thresholds and serve as finite-volume comparisons; no numerical fit enters the continuum result.