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  Partition function of a 1+1 D bosonic SPT with D4 symmetry

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There should be a nontrivial 1+1 D bosonic SPT with D4 symmetry, which can be obtained by breaking the SO(3) symmetry of a Haldane chain to D4.

Concretely, write D4=Z4Z2, denote the generator of Z4 by r, and denote the generator of Z2 by s. Then srs=r1. Suppose this SPT is put on a triangulated manifold X. Denote the gauge connection corresponding to the Z4 by aC1(X,Z4), and the gauge connection corresponding to Z2 by bC1(X,Z2). Here a and b are thought of as 1-cochains that represent the flat Z4 and Z2 gauge connections, respectively.

I have a couple of related questions:

  1. Is the partition function of this SPT something like exp(iπXab)? I think it somehow makes sense, but I do not fully understand it. In fact, now that Z4 and Z2 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(iπXw2(SO(3))), where w2(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 D4 group into SO(3), and then pullback w2(SO(3)). But how should this be done properly?
  3. Now that the Z4 and Z2 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 ab have any special property?
asked Nov 13, 2020 in Theoretical Physics by Mr. Gentleman (270 points) [ revision history ]
recategorized Feb 18, 2021 by Mr. Gentleman

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