Inflation, Proton Decay and Gravitational Waves from Metastable Strings in $SU(4{)}_C\times SU(2{)}_L\times U(1{)}_R$ Model

We present a realistic supersymmetric $\mu$-hybrid inflation model within the framework of $SU(4{)}_C\times SU(2{)}_L\times U(1{)}_R$ gauge symmetry, wherein the symmetry breaking $SU(4{)}_C\times SU(2{)}_L\times U(1{)}_R\to SU(3{)}_C\times SU(2{)}_L\times U(1{)}_{B-L}\times U(1{)}_R$ occurs before observable inflation, effectively eliminating topologically stable primordial monopoles. Subsequent breaking of $U(1{)}_{B-L}\times U(1{)}_R\to U(1{)}_Y$ 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 $n_s$ 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 $G_{421}$ into the Pati-Salam gauge symmetry $G_{422}$ is briefly discussed, predicting potentially observable proton decay rates detectable at facilities such as Hyper Kamiokande and DUNE.