Surface field theories of point group symmetry protected topological phases

We identify field theories that describe the surfaces of three-dimensional bosonic point group symmetry protected topological (pgSPT) phases. The anomalous nature of the surface field theories is revealed via a dimensional reduction argument. Specifically, we study three different surface field theories. The first field theory is quantum electrodynamics in three space-time dimensions (QED3) with four flavors of fermions. We show this theory can describe the surfaces of a majority of bosonic pgSPT phases protected by a single mirror reflection, or by $C_{nv}$ point group symmetry for $n=2,3,4,6$. The second field theory is a variant of QED3 with charge-1 and charge-3 Dirac fermions. This field theory can describe the surface of a reflection symmetric pgSPT phase built by placing an $E_{8}$ state on the mirror plane. The third field theory is an ${\rm O}(4)$ non-linear sigma model with a topological theta-term at $\theta=\pi$, or, equivalently, a non-compact ${\rm CP}^1$ model. Using a coupled wire construction, we show this is a surface theory for bosonic pgSPT phases with ${\rm U}(1) \times \mathbb{Z}_{2}^{P}$ symmetry. For the latter two field theories, we discuss the connection to gapped surfaces with topological order. Moreover, we conjecture that the latter two field theories can describe surfaces of more general bosonic pgSPT phases with $C_{nv}$ point group symmetry.