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$\newcommand{\D}{\mathrm{D}} \newcommand{\d}{\mathrm{d}} \newcommand{\Tr}{\mathrm{Tr}} \newcommand{\Ds}{D\kern-.6em/\kern.1em} \newcommand{\B}{\mathrm{B}} \newcommand{\H}{\mathrm{H}} \newcommand{\SU}{\mathrm{SU}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\U}{\mathrm{U}}$ As @4xion pointed out in a comment, the only anomalies you have to worry about are gauge ...


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$\newcommand{\Z}{\mathbb{Z}}$ I'm not entirely sure that this is what you're looking for, but it seems like the total symmetry is a $G=\Z_2\rtimes \Z_2^T$, with antiunitary $\Z_2^T$, so the number of SPT phases is inside [1] $$ \mathrm{H}^{d+1}(G,U(1)) = \begin{cases} \Z_2^{(d+2)/2} & d = 0 \mod 2 \\ \Z^{(d+1)/2}_{2} & d = 1 \mod 2 \end{cases}$$ So ...


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