We present a realistic supersymmetric
μ-hybrid inflation model within the
framework of
SU(4)C×SU(2)L×U(1)R gauge symmetry, wherein the
symmetry breaking $SU(4)_C \times SU(2)_L \times U(1)_R\rightarrow
SU(3)_C\times SU(2)_L \times U(1)_{B-L}\times U(1)_R$ occurs before observable
inflation, effectively eliminating topologically stable primordial monopoles.
Subsequent breaking of
U(1)B−L×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.