2026/07/15 by Daniel G. Tedesco
#hep-th
After removing the constant adjoint modes associated with global gauge rotations, we formulate the Landau-gauge Faddeev-Popov zero-mode problem in Birman-Schwinger form. The resulting normalized operator, is dimensionless and self-adjoint for transverse backgrounds, and the first Gribov horizon is identified with the appearance of the spectral value -1. Because this reduction is a congruence rather than a similarity transformation, it preserves the inertia and nullity relevant to horizon crossings without identifying the numerical spectra of the two operators. We study these analytic properties on a periodic domain, using matrix-valued Cwikel estimates to control its singular-value behavior at the regularity scale selected by the first-order Faddeev-Popov interaction, with separate attention to the two-dimensional endpoint. The same formulation expresses the fixed-background ghost Green function through the diagonal resolvent, providing a common framework in which the exact spectral condition can be compared with the Born expansion and Gribov's no-pole construction while remaining distinct from statements involving the functional average over gauge fields. As an explicit application, we consider a periodic transverse SU(2) background for which the zero-mode equation reduces to a Mathieu-type recurrence and admits a systematic analysis through finite-channel and Feshbach reductions. This solvable sector also permits an examination of the volume dependence of horizon-touching configurations, without assigning them a statistical weight in the Yang-Mills measure.