  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.
