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

+ 1 like - 0 dislike
165 views

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 $srs=r^{-1}$. Suppose this SPT is put on a triangulated manifold $X$. Denote the gauge connection corresponding to the $Z_4$ by $a\in C^1(X, Z_4)$, and the gauge connection corresponding to $Z_2$ by $b\in C^1(X, Z_2)$. Here $a$ and $b$ are thought of as 1-cochains that represent the flat $Z_4$ and $Z_2$ gauge connections, respectively.

I have a couple of related questions:

1. Is the partition function of this SPT something like $\exp(i\pi\int_X a\cup b)$? I think it somehow makes sense, but I do not fully understand it. In fact, now that $Z_4$ and $Z_2$ do not commute, what does such a cup product mean?
2. If the above (or something like it) is indeed the partition function, 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 the parent Haldane chain 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?
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ys$\varnothing$csOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.