Topological boundary

We investigate (3+1)d topological orders in fermionic systems with an anomalous ℤF2N symmetry, where its ℤF2 subgroup is the fermion parity. Such an anomalous symmetry arises as the discrete subgroup of the chiral U(1) symmetry of ν copies of Weyl fermions of the same chirality. Guided by the crystalline correspondence principle, we construct (3+1)d symmetry-preserving gapped states on the boundary of a closely related (4+1)d CN×ℤF2 symmetry-protected topological (SPT) state (with CN being the N-fold rotation), whenever it is possible. In particular, for ν=N, we show that the (3+1)d symmetric gapped state admits a topological ℤ4 gauge theory description at low energy, and propose that a similar theory saturates the corresponding ℤF2N anomaly. For N∤ν, our construction cannot produce any topological quantum field theory (TQFT) symmetric gapped state; but for ν=N/2, we find a non-TQFT symmetric gapped state via stacking lower-dimensional (2+1)d non-discrete-gauge-theory topological orders inhomogeneously. For other values of ν, no symmetric gapped state is possible within our construction, which is consistent with the theorem by Cordova-Ohmori.