$\mathbf{\mu}$-Hybrid Inflation and Metastable Cosmic Strings in $\mathbf{SU(3)_c\times SU(2)_L\times SU(2)_R \times U(1)_{B-L}}$

We develop a scenario based on $\mu$-hybrid inflation within the framework of a left-right symmetric model defined by the gauge symmetry group $G_{3221} = SU(3)_c \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}$. This model features a two-stage symmetry breaking to the Standard Model, with inflation occurring between the two stages, resulting in the dilution of primordial monopoles produced during the symmetry breaking of $SU(2)_R$ to $U(1)_R$. The second symmetry breaking gives rise to a network of metastable cosmic strings, which produces gravitational waves that can be observed by current and future experiments. A crucial aspect of this model is the inclusion of $R$-symmetry breaking terms in the superpotential, which helps explain the origin of the first symmetry breaking scale and ensures experimentally viable inflation with a minimal (canonical) K\"ahler potential. We show that the model successfully implements consistent inflation, leptogenesis, metastable cosmic strings, and gravitino dark matter, all in agreement with the observational data and constraints on reheating temperature from leptogenesis and gravitino overproduction bounds. We explore the potential embedding of $G_{3221}$ within the gauge symmetry $G_{422} = SU(4)_c \times SU(2)_L \times SU(2)_R$, and provide a brief discussion on the proton decay predictions, which can be tested by the upcoming experiments.