Trivializing group cohomology SPT phases
Published:
Phases of invertible topological quantum field theories with a global symmetry $G$ are known to be classified by (the Anderson dual of) cobordism. More precisely, for bosonic theories in $d$ spacetime dimensions, the group of invertible phases with unitary $G$ symmetry is $h^d(G)=(D\Omega^{\rm SO})_d(BG)$ ($D$ stands for the Anderson dual). An earlier attempt to the classification for $G$ is unitary is the group cohomology $\mathcal{H}^d(G, \mathrm{U}(1))$. It was noted in arXiv:1403.1467 that in general the homomorphism from $\mathcal{H}^d(G, \mathrm{U}(1))$ to $h^d(G)$ is neither surjective nor injective. That is, on one hand there are “beyond group-cohomology” SPT phases, which exist in $d=4$ for $G=\mathbb{Z}_2^{\mathsf{T}}$ (anti-unitary $\mathbb{Z}_2$), and $d=5$ for $G=\mathbb{Z}_2$. On the other hand, the map can have nontrivial kernel: a nontrivial element of $\mathcal{H}^d(G, \mathrm{U}(1))$ may actually correspond to a trivial SPT phase. The latter phenomenon only occurs for $d\geq 7$, and the simplest group is $G=\mathbb{Z}_3\times \mathbb{Z}_3$.