function of one of such $2+1D$ SPTs, it should exhibit the same flux responses.

We therefore introduce two additional static gauge fields ($B^{\sigma},B^{\tau}$) which are coupled to matter in the usual manner: we trade each $\tau_{k} \tau_{l}$ with $\tau_{k} B^{\tau}_{kl} \tau_{l}$ and each $\sigma_{i} \sigma_{j}$ with $\sigma_{i} B^{\sigma}_{i j} \sigma_{j}$. The adjective static refers to the fact that they are not summed over in the partition function, which is then: $$\begin{aligned}
  Z(\{B^\tau\},\{B^{\sigma} \})
  &=
    \frac{1}{Z}
  \sum_{\tau,\sigma} 
  \prod_{p}
  e^{\tilde{J_2} \tau_{k}B^\tau_{kl} \tau_{l} }
  (\tau_{k}B_{kl}^\tau \tau_{l})^{\omega_p(B^\sigma)}
  \, ,\end{aligned}$$ where $Z\equiv Z(\{1\},\{1\})$ as above. If we require that both fluxes are zero everywhere, namely $\prod_{(ij)\in\partial p} B_{ij}^\sigma=\prod_{(kl)\in\partial p^*} B_{ij}^\tau=1$, where $p^*$ is a dual plaquette, we can rewrite $B_{ij}^\sigma=\tilde{\sigma_i}\tilde{\sigma}_j$, $B_{kl}^\tau=\tilde{\tau_k}\tilde{\tau}_l$, and reabsorb the $B$’s in the definition of $\sigma,\tau$. Thus introducing gauge fields with zero flux is equivalent to set them to $1$. When coupling to gauge fields, from formula (5) of the main paper the vorticity becomes $$\begin{aligned}
  \label{eq:omegap_B}
  (-)^{\omega_p(B^\sigma)} 
  = 
  \prod_{(ij)\in\partial p}
  \exp\left(i\pi\frac{1-\sigma_iB_{ij}^\sigma \sigma_j}{4} \epsilon^p_{ij}\right)
  \, .\end{aligned}$$ If we now violate the zero flux constraint, then $(-)^{\omega_{p}(B^\sigma)}$ can assume the additional values $\pm i$ on top of $\pm 1$ which it had before. A related issue to be discussed is the definition 