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  Partition function of a $1+1$ D bosonic SPT with $D_4$ symmetry

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There should be a nontrivial $1+1$ D bosonic SPT with $D_4$ symmetry, which can be obtained by breaking the $SO(3)$ symmetry of a Haldane chain to $D_4$.

Concretely, write $D_4=Z_4\rtimes Z_2$, denote the generator of $Z_4$ by $r$, and denote the generator of $Z_2$ by $s$. Then $srs=r^{-1}$. Denote the gauge connection corresponding to the $Z_4$ by $a$, and the gauge connection corresponding to $Z_2$ by $b$.

I have a couple of related questions:

  1. If this SPT is put on a triangulated manifold, $X$, is its partition function $\exp(i\pi\int_X a\cup b)$?
  2. If so, how should I obtain it from the known partition function of the parent $SO(3)$ SPT, $\exp(i\pi\int_X w_2(SO(3)))$, where $w_2(SO(3))$ is the second Stiefel-Whitney class of the $SO(3)$ gauge bundle that this SPT is coupled to? I understand perhaps I need to first embed this $D_4$ group into $SO(3)$, and then pullback $w_2(SO(3))$. But how should this be done properly?
  3. Now that the $Z_4$ and $Z_2$ do not commute, I think there should be a twist if we perform a coboundary operation on $a$. But besides this, is there any other constraint or relation between $a$ and $b$? For example, does $a\cup b$ have any special property?
asked Nov 13 in Q&A by Mr. Gentleman (260 points) [ no revision ]

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