Lattice construction of exotic invertible topological phases

In this paper, we provide state sum path integral definitions of exotic invertible topological phases proposed in the recent paper by Hsin, Ji, and Jian [arXiv:2105.09454 [cond-mat.str-el]. The phase has time-reversal $(T)$ symmetry, depends on a choice space-time structure called Wu structure. cannot be captured classification any bosonic or fermionic thus gives novel class phases. When $T$ symmetry defect admits spin structure, our construction reduces to sort decorated domain wall construction, terms theory with defects that defect. By utilizing integral, propose lattice for generates ${\mathbb{Z}}_{8}$ $(3+1)$d based This generalizes $T$-symmetric $(1+1)$d superconductor Fidkowski Kitaev. On oriented space-time, specific Crane-Yetter TQFT which ordered semion its boundary. Moreover, subclass $G$-SPT labeled pair cohomological data generic dimensions. Gu-Wen SPT