The Z3 Center Symmetry: Arithmetic Lattices, Automorphic Spectral Triples and QCD

This work introduces a framework for a Z3 center symmetry theory, linking arithmetic number theory, geometric representation theory, and noncommutative geometry to describe confining SU(3) gauge theories, such as QCD. It proposes that a Z3 symmetry necessitates two distinct, conjugate "twists", which are realized arithmetically through the factorization of Dedekind zeta functions into Dirichlet L-functions, and geometrically through the decomposition of the automorphic category for SU(3). This structure is then translated into an operator algebraic formalism, defining an algebra of observables generated by ’t Hooft-Wilson loop operators, and shown to decompose into superselection sectors corresponding to different N-alities. The physical implications are explored, detailing the internal structure and fusion rules of chromoelectric flux tubes, and demonstrating the spectral equivalence of vortex and anti-vortex states due to a charge conjugation symmetry. It further establishes a connection between Casimir scaling and the analytic invariants of L-functions, and constructs a spectral triple, where the Dirac operator's spectrum is defined by the zeros of these L-functions, to provide a potential non-perturbative description of Z3 theories and a generalization to SU(N) gauge theories.