  We study perturbative and global anomalies at the boundaries of bosonic
analogues of integer quantum Hall (BIQH) and topological insulator (BTI) phases
using a description of the boundaries of these phases in terms of a nonlinear
sigma model (NLSM) with Wess-Zumino term. One of the main results of the paper
is that these anomalies are robust against arbitrary smooth deformations of the
target space of the NLSM which describes the phase, provided that the
deformations also respect the symmetry of the phase. In the first part of the
paper we discuss the perturbative $U(1)$ anomaly at the boundary of BIQH states
in all odd (spacetime) dimensions. In the second part we study global anomalies
at the boundary of BTI states in even dimensions. In a previous work [Phys.
Rev. B 95, 035149 (2017)] we argued that the boundary of the BTI phase exhibits
a global anomaly which is an analogue of the parity anomaly of Dirac fermions
in three dimensions. Here we elevate this argument to a proof for the boundary
of the two-dimensional BTI state by explicitly computing the partition function
of the gauged NLSM describing the boundary. We then use the powerful
equivariant localization technique to show that this global anomaly is robust
against all smooth deformations of the target space of the NLSM which preserve
the $U(1)\rtimes\mathbb{Z}_2$ symmetry of the BTI state. We also comment on the
difficulties of generalizing this latter proof to higher dimensions. Finally,
we discuss the expected low energy behavior of the boundary theories studied in
this paper when the coupling constants are allowed to flow under the
renormalization group.
