The consequences of the Fundamental Modular Region on the dynamics of an SU(N) Yang-Mills theory

In this article, we propose a Euclidean SU(N) Yang-Mills (YM) theory restricted to the Fundamental Modular Region (FMR) and show that, on a directed family of ``nice'' graphs, the FMR constraint leads to a quartic, gauge-invariant lattice action admitting a Gaussian representation via a justified Hubbard-Stratonovich transform. We prove the existence of a projective continuum limit equipped with the cylindrical topology, and supporting a probability measure consistent with all lattice projections. In this framework, we define fattened Wilson loop functionals and show that their Euclidean correlation functions satisfy Osterwalder-Schrader (OS)-type axioms. This gives us a rigorous reconstruction of a YM theory from global gauge-invariant observables. From them, we deduce the theory for local gauge-invariant observables and show that this is a non-trivial theory.