The Haldane chain can be viewed as a $1+1$ D SPT protected by an $SO(3)$ symmetry. If this SPT is put on a triangulated closed manifold $X$, its partition function can be written as $$ e^{i\pi\int_Xw_2(SO(3))} \tag{1}\label{SO(3)} $$ where $w_2(SO(3))$ is the second Stiefel-Whitney class of the probe $SO(3)$ gauge bundle that this SPT is coupled to.

If the $SO(3)$ symmetry is weakly broken to $Z_2\times Z_2$, it is known that the descendent state is still a nontrivial SPT, with a partition function $$ e^{i\pi\int_Xa\cup b} \tag{2}\label{Z2xZ2} $$ where $a, b\in C^1(X, Z_2)$ are 1-cochains that can be thought of as the gauge connections corresponding to the two remaining $Z_2$ symmetries, respectively. One can verify that this partition function is consistent with the result obtained from group cohomology as in this paper.

Besides directly writing down Eq. \eqref{Z2xZ2} for the $Z_2\times Z_2$ descendent of the Haldane chain and verifying that it is correct, we can also derive it from Eq. \eqref{SO(3)}. To do so, suppose we represent the $SO(3)$ group elements as $3\times 3$ orthogonal matrices with positive determinant, and represent the generators of the two $Z_2$ symmetries as diagonal matrices ${\rm diag}(-1, -1, 1)$ and ${\rm diag}(1, -1, -1)$, respectively. This specifies a group homomorphism that tells us how the $Z_2\times Z_2$ is embedded into the $SO(3)$. Now we can use Whitney's product formula to pull back Eq. \eqref{SO(3)} along this homomorphism to obtain the partition function of the $Z_2\times Z_2$ descendent: $$w_2(SO(3))=a\cup (a+b)+a\cup b+b\cup(a+b)=3a\cup b+a\cup a+b\cup b=3a\cup b+\frac{1}{2}(da+db)=a\cup b \tag{3}\label{pullback}$$ In the above we have lifted $a,b$ to integer 1-cochains and simplified the result using the fact that we only care about the integral over $X$ modulo 2. This derivation indeed yields Eq. \eqref{Z2xZ2}.

Although I am aware of the above procedure, I feel I do not fully understand all details. My first question is

  1. Usually the explicit definition of $w_2(SO(3))$ is written in terms of the transition functions of the $SO(3)$ gauge bundle, and usually we think of $a, b$ as gauge connections rather than transition functions. Then in the first step of the above derivation, how exactly is $w_2(SO(3))$ related to $a$ and $b$? Or should I really also think of $a$ and $b$ as transition functions, rather than gauge connections? But if I do not think of $a,b$ as gauge connections, why should $e^{i\pi\int_{C_1}a}$ and $e^{i\pi\int_{C_1}b}$ be viewed as the Wilson lines along a $1$-cycle $C_1$?

Next, consider breaking the $SO(3)$ symmetry of the Haldane chain to $D_{2n}$, and we can take $D_4$ ($n=2$) for simplicity. 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 $a\in C^1(X, Z_4)$, and the gauge connection corresponding to $Z_2$ by $b\in C^1(X, Z_2)$ (let me still call them gauge connections for now).

I have a couple of related questions:

  1. Is the partition function of this $D_4$ descendent something like $\exp(i\pi\int_X a\cup b)$? I think it somehow makes sense, but I do not fully understand it. In particular, now that $Z_4$ and $Z_2$ do not commute, what does such a cup product mean? The definition in this paper does not seem to apply.

  2. More generally, how should the cup product for gauge connections corresponding to a non-Abelian symmetry be defined, and what special properties does it have due to the non-Abelian nature of the symmetry?

  3. If the above (or something else) is the correct partition function for the $D_4$ descendent, how should it be derived from Eq. \eqref{SO(3)}, in a way similar to Eq. \eqref{pullback}?


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