Identify defect-anomaly interpretations of the four-dimensional class-D boundary invariant

Establish whether the difference of the three-dimensional \(\zeta\) invariants associated with the boundary restrictions of a four-dimensional class-D particle-hole symmetry can be identified with real-space defect anomalies.

Background

In four-dimensional class D, the paper relates the parity of the second Chern character to the difference $\zeta[C_0;\mathfrak s]-\zeta[C_\pi;\mathfrak s]$ of three-dimensional invariants of the symmetry matrices on the two boundary momentum slices. The paper also gives quasilocal odd-parity constructions and finite-range no-go results.

The authors compare these results with a real-space picture involving monopole-line-bound states and anomaly cancellation, but explicitly refrain from identifying the momentum-space invariant difference with a defect anomaly. Establishing that identification is left unresolved.