Framed Wilson Operators, Fermionic Strings, and Gravitational Anomaly in 4d
Ryan Thorngren
TL;DR
This work analyzes gapped phases with time-reversal and gravitational anomalies in 3+1 and 4D by leveraging the cobordism-based classification of bosonic SPT phases, with Stiefel-Whitney classes $w_j$ encoding the underlying topology. It introduces framed Wilson operators, electric and magnetic operators, and higher-structure generalizations that reveal fermionic quasiparticles and fermionic strings on the boundaries of such phases. The authors demonstrate boundary theories for actions like $w_2$, $w_2^2$, and $w_2 w_3$, including a construction of a 4D all-fermion topological order and a QED system with a fermionic monopole that exhibits a global gravitational anomaly described by $w_2 w_3$. These results illuminate how cobordism invariants dictate boundary excitations, braiding, and anomaly cancellation, linking topological field theory, framing, and gravitational anomalies in a coherent framework.
Abstract
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 $π$-flux.