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We study perturbative and global anomalies at the boundaries of bosonicanalogues of integer quantum Hall (BIQH) and topological insulator (BTI) phasesusing a description of the boundaries of these phases in terms of a nonlinearsigma model (NLSM) with Wess-Zumino term. One of the main results of the paperis that these anomalies are robust against arbitrary smooth deformations of thetarget space of the NLSM which describes the phase, provided that thedeformations also respect the symmetry of the phase. In the first part of thepaper we discuss the perturbative $U(1)$ anomaly at the boundary of BIQH statesin all odd (spacetime) dimensions. In the second part we study global anomaliesat 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 exhibitsa global anomaly which is an analogue of the parity anomaly of Dirac fermionsin three dimensions. Here we elevate this argument to a proof for the boundaryof the two-dimensional BTI state by explicitly computing the partition functionof the gauged NLSM describing the boundary. We then use the powerfulequivariant localization technique to show that this global anomaly is robustagainst all smooth deformations of the target space of the NLSM which preservethe $U(1)\rtimes\mathbb{Z}_2$ symmetry of the BTI state. We also comment on thedifficulties of generalizing this latter proof to higher dimensions. Finally,we discuss the expected low energy behavior of the boundary theories studied inthis paper when the coupling constants are allowed to flow under therenormalization group.

Perturbative and global anomalies in bosonic analogues of integer quantum Hall and topological insulator phases