Let us consider a Chern-Simons theory on a $3$-manifold $M$ (can be a spin manifold with a given spin structure if needed) with a discrete-symmetry gauge field e.g. $\mathbb{Z}_n$ symmetry. It can be embedded in a $U(1)$ gauge group with constraints as \begin{eqnarray} dA&=&0,\\ \oint_\gamma A&\in&2\pi\mathbb{Z}/n, \end{eqnarray} where $\gamma$ is any closed loop on $M$, and the action is \begin{eqnarray} S_\text{CS}[A]=\int_M\frac{k}{4\pi}A\wedge dA. \end{eqnarray} However, by one of the constraints, $dA=0$, I naively arrive at the conclusion that \begin{eqnarray} S_\text{CS}[A]=0, \end{eqnarray} for instance on $M=T^3$ a $3$-torus.
However, discrete Chern-Simons theories are generically nontrivial with $k\in\mathbb{Z}_{2n}$, namely, it gives a $\mathbb{Z}_{2n}$-classification of fermionic symmetry-protected topological (SPT) phase with an onsite $\mathbb{Z}_n$ symmetry. (It can be obtained by braiding of defects of Higgs fields in a Higgs approach.)
My question is, since the action vanishes $S_\text{CS}[A]=0$ for discrete gauge field $A$, how can we see its nontriviality directly from the action $S_\text{CS}[A]$? Or I cannot understand why a nontrivial SPT bulk has a vanishing effective response $S_\text{CS}[A]=0$.
