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

## TL;DR

This paper investigates the robustness of perturbative and global anomalies at the boundaries of bosonic quantum Hall and topological insulator phases using nonlinear sigma models, providing proofs of anomaly invariance under symmetry-preserving deformations.

## Contribution

It offers a proof that boundary anomalies in bosonic topological phases are stable under smooth, symmetry-respecting deformations of the model's target space.

## Key findings

- Perturbative $U(1)$ anomaly at BIQH boundary in all odd dimensions.
- Global anomaly at BTI boundary in two dimensions.
- Anomaly robustness proven using equivariant localization technique.

## Abstract

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.