# Partition functions of descendent SPTs of the Haldane chain

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}?