Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d

We study gapped systems with anomalous time-reversal symmetry and global gravitational anomaly in three and four spacetime dimensions. These systems describe topological order on the boundary of bosonic Symmetry Protected Topological (SPT) Phases. Our description of these phases is via the recent cobordism proposal for their classification. In particular, the behavior of these systems is determined by the geometry of Stiefel-Whitney classes. We discuss electric and magnetic operators defined by these classes, and new types of Wilson lines and surfaces that sit on their boundary. The lines describe fermionic particles, while the surfaces describe a sort of fermionic string. We show that QED with a fermionic monopole exhibits the 4d global gravitational anomaly and has a fermionic $\pi$-flux.