All-fermion electrodynamics and fermion number anomaly inflow

We demonstrate that ($3+1$)-dimensional quantum electrodynamics with fermionic charges, fermionic monopoles, and fermionic dyons arises at the edge of a ($4+1$)-dimensional gapped phase with short-range entanglement. The ground state of this phase cannot be adiabatically connected to a product state, even in the absence of any symmetry. Our construction provides independent evidence for the obstruction found by Wang et al. [Science 343, 629 (2014)] to a ($3+1$)-dimensional short-distance completion of all-fermion electrodynamics. The nontriviality of the bulk is demonstrated by a fermion number anomaly.