Anomaly cascade in (2+1)D fermionic topological phases

We develop a theory of anomalies of fermionic topological phases of matter in (2+1)D with a general fermionic symmetry group G_f. In general, G_f can be a non-trivial central extension of the bosonic symmetry group G_b by fermion parity. We encounter four layers of obstructions to lifting a G_f symmetry action on a super-modular category C to a minimal modular extension Č, which we dub the anomaly cascade: (i) An H¹(G_b,Z_T) obstruction to extending autoequivalences of C to Č, (ii) An H²(G_b, ker r) obstruction to extending the G_b group structure of the symmetry action to Č, where r is a map that restricts autoequivalences of Č to C, (iii) An H³(G_b, Z₂) obstruction to extending the symmetry fractionalization class to Č, and (iv) the well-known H⁴(G_b,U(1)) obstruction to developing a consistent G_b-crossed theory of symmetry defects for Č. A number of conjectures regarding symmetry actions on super-modular categories, guided by general expectations of anomalies in physics, are also presented.