SPT indices emerging from translation invariance in two dimensional quantum spin systems

We consider SPT-phases with on-site $G$ (where $G$ is any finite group) symmetry for two-dimensional quantum spin systems. We then impose translation invariance in one direction and observe that on top of the $H^3(G,\mathbb{T})$-valued index constructed in \cite{ogata2021h3gmathbb}, an additional $H^2(G,\mathbb{T})$-valued index emerges. We also show that if we impose translation invariance in two directions, on top of the expected $H^3(G,\mathbb{T})\oplus H^2(G,\mathbb{T})\oplus H^2(G,\mathbb{T})$ valued index, an additional $H^1(G,\mathbb{T})$-valued index emerges.