Nonperturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1) dimensions

In (1+1)-dimensional topological phases, unpaired Majorana zero modes (MZMs) can arise only if the internal symmetry group Gf of the ground state splits as Gf=Gb×Z₂ᶠ, where Z₂ᶠ is generated by fermion parity (-1)F. In contrast, (2+1)-dimensional [(2+1)D] topological superconductors (TSC) can host unpaired MZMs at defects even when Gf is not of the form Gb×Z₂ᶠ. In this paper we study how Gf together with the chiral central charge c_- strongly constrain the existence of unpaired MZMs and the quantum numbers of symmetry defects. Our results utilize a recent algebraic characterization of (2+1)D invertible fermionic topological states, which provides a nonperturbative approach based on topological quantum field theory, beyond free fermions. We study physically relevant groups such as U(1)ᶠH,0.28em0exSU(2)ᶠ×H,0.28em0exU(2)ᶠH, generic Abelian groups, as well as more general compact Lie groups, antiunitary symmetries, and crystalline symmetries. We present an algebraic formula for the fermionic crystalline equivalence principle, which gives an equivalence between states with crystalline and internal symmetries. In light of our theory, we discuss several previously proposed realizations of unpaired MZMs in TSC materials such as Sr₂RuO₄, transition metal dichalcogenides, and iron superconductors, in which crystalline symmetries are often important; in some cases we present additional predictions for the properties of these models.