Recovering the full Z8 Dai–Freed anomaly in the generalized GW Majorana formulation

Determine how the generalized Ginsparg–Wilson/overlap lattice formulation for 0+1-dimensional Majorana fermions with time-reversal symmetry realizes the complete Z8 Dai–Freed anomaly, rather than only the manifest Z4 anomaly.

Background

The authors extend Ginsparg–Wilson constructions to Majorana fermions and discrete symmetries, deriving a lattice symmetry and anomaly that reproduce aspects of known continuum results. For 1D Majorana edge modes of the Fidkowski–Kitaev chain, their framework yields a Jacobian consistent with a Z2 anomaly for two flavors, implying a Z4 anomaly for a single 1D Majorana.

However, the complete Dai–Freed anomaly in the continuum for a single 1D Majorana with time-reversal symmetry is Z8. The paper explicitly notes that their current formulation does not yet capture the full Z8 structure, leaving a gap to be resolved.