Speaker
Description
We develop a model-independent framework to construct quark--lepton families in flipped trinification, $SU(3)_C\times SU(3)_L\times SU(3)_R\times U(1)_X$, for arbitrary charge embedding. The approach identifies and classifies Irreducible Anomaly-Free Sets (IAFS)---minimal fermion multiplet combinations that cancel, by themselves, all gauge and mixed anomalies: $[SU(3)_{L,R}]^3$, $[SU(3)_{L,R}]^2U(1)_X$, $[SU(3)_C]^2U(1)_X$, $U(1)_X^3$, and $\text{grav}^2U(1)_X$. We show how Standard-Model families arise as unions of a small number of IAFS and derive general constraints relating family replication to color, recovering well-known 331-like non-universality as a limiting case. For each IAFS we provide electroweak-viable charge assignments, discuss the scalar content needed for the sequential breaking $SU(3)_L\times SU(3)_R\times U(1)_X\!\to\! SU(2)_L\times U(1)_Y$, and outline renormalizable Yukawa structures that generate realistic quark and lepton masses (including right-handed neutrinos) while controlling exotic states. The classification maps directly onto phenomenology, predicting patterns for extra neutral currents ($Z'/Z''$), exotic quark/lepton charges, residual discrete symmetries, and typical flavor textures. Our results provide a modular ``building-block'' toolkit to engineer flipped-trinification models from quark--lepton families and their irreducible anomaly-free combinations, clarifying which embeddings remain compatible with current collider and flavor constraints.
