Reflection positivity and a finite-a strong-coupling gap in lattice SU(N) Yang-Mills: Part (1)

Abstract

For G=SU(N) with N >= 2, we develop a reflection-positive transfer-matrix framework for four-dimensional lattice Yang-Mills which, on a nontrivial strong-coupling window 0<beta<beta(star)(N), yields a strictly positive spectral gap at fixed lattice spacing a, with bounds uniform in the spatial volume. The construction is compatible with OS reflection: on each Euclidean time slice we select a gauge-invariant transverse representative A(h) by Landau functional minimization within the fundamental modular region, and we insert a smooth "horizon" spectral projector as a slice-local positive weight that preserves reflection positivity. In the same regime 0<beta<beta(star)(N), a Kotecky-Preiss cluster expansion reorganizes the partition function and gauge-invariant correlators; it converges uniformly in the volume and implies exponential clustering for connected gauge-invariant observables with a decay rate bounded away from zero uniformly in the volume. OS reconstruction then promotes clustering to a nonzero lower bound for the spectral gap of the positive, self-adjoint transfer operator T (equivalently, of the transfer Hamiltonian H=-logT) at fixed a. We also establish a Wilson-loop area law throughout this window. The conclusions are stable under admissible variations of the slice-wise selector and of the smooth projector profile, and they quantify the existence of a finite-a mass gap for SU(N) Yang-Mills at strong coupling.