linear combination of $e^{\pm i\phi}$ and $\tau$ of $e^{\pm i\theta}$, $\theta$ being the dual field. Therefore, the symmetry is realized as anticipated in the main text: $\phi \rightarrow \phi + \pi$ and $\theta \rightarrow \theta + \pi$, as it does in quantum SPTs [@YuanMing2012; @Nayak2014]. We also note that even though the local weight has no such symmetry, the global weight still has it, due to the global constraint $\prod_p (-1)^{\omega_p}=1$ for a closed manifold. From this analysis it follows that the lattice ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry is realized in the field theory in an anomalous chiral way: $\phi \rightarrow \phi + \pi$ and $\theta \rightarrow \theta + \pi$, where $\theta$ is the dual field.

Let us consider perturbations to this surface model. Adding a $\sigma\sigma$ term to the boundary action corresponds to the six vertex model in an external field. Denoted by $H/2$ and $V/2$ the horizontal and vertical couplings, the theory remains critical within the region $(e^{2|H|}-1)(e^{2|V|}-1)\le 1$ [@Reshetikhin], the only effect of $H,V\neq 0$ being renormalizing the stiffness of $\phi$ [@Kim]. A ferromagnetic coupling between the $\tau$’s would generically induce the RG-irrelevant term $\cos(2\theta)$. Interestingly, the relevant $\cos(\theta)$ term is forbidden without requiring any fine tuning of the 