Topological structures, dark matter and gravitational waves in $E_6$

We discuss the appearance of topological structures from the spontaneous breaking of $E_6$ to the Standard Model via its maximal subgroup $SO(10)\times U(1{)}_{\psi }$. They include dumbbells, metastable strings, as well as domain walls bounded by necklaces. We provide a novel scenario for producing metastable strings based on the symmetry breaking $U(1{)}_{\psi }⟶Z_8⟶Z_4$. The metastable string arises from the merger of $Z_8$ strings that bound a domain wall. An unbroken gauge $Z_2$ symmetry from $SO(10)$ breaking yields viable stable dark matter candidates as well as topologically stable strings. We discuss the gravitational wave emission from two varieties of cosmic strings, namely the superheavy metastable ones and the intermediate scale topologically stable cosmic strings.