Inflation, Proton Decay and Gravitational Waves from Metastable Strings in SU (4) C SU (2) L U (1) R Model

We present a realistic supersymmetric -hybrid inflation model within the framework of SU (4) C SU (2) L U (1) R gauge symmetry, wherein the symmetry breaking SU (4) C SU (2) L U (1) R SU (3) C SU (2) L U (1) ₁-₋ U (1) R occurs before observable inflation, effectively eliminating topologically stable primordial monopoles. Subsequent breaking of U (1) ₁-₋ U (1) R U (1) Y after inflation leads to the formation of superheavy metastable cosmic strings (CSs), capable of producing a stochastic gravitational wave background (SGWB) consistent with the recent PTA data. Moreover, the scalar spectral index nₛ and the tensor-to-scalar ratio r align with Planck 2018 observations. A consistent scenario for reheating and non-thermal leptogenesis is employed to explain the observed matter content of the universe. Finally, the embedding of G₄₂₁ into the Pati-Salam gauge symmetry G₄₂₂ is briefly discussed, predicting potentially observable proton decay rates detectable at facilities such as Hyper Kamiokande and DUNE.