# Anomaly indicators for topological orders with $U(1)$ and time-reversal symmetry

## TL;DR

This paper develops anomaly indicators for topological orders with $U(1)$ and time-reversal symmetry, enabling detection of anomalies that relate to surface states of 3D SPT phases, covering both bosonic and fermionic cases.

## Contribution

It introduces new anomaly indicators for symmetric topological orders with $U(1)$ and time-reversal symmetry, applicable to both bosonic and fermionic systems, and conjectures their completeness.

## Key findings

- Constructed anomaly indicators for $U(1)\rtimes\mathbb{Z}_2^T$ and $U(1)\times\mathbb{Z}_2^T$ symmetries.
- Indicators are conjectured to be complete, matching known 3D SPT classifications.
- One indicator has a mathematical interpretation as a bulk partition function.

## Abstract

We study anomalies in time-reversal ($\mathbb{Z}_2^T$) and $U(1)$ symmetric topological orders. In this context, an anomalous topological order is one that cannot be realized in a strictly $(2+1)$-D system but can be realized on the surface of a $(3+1)$-D symmetry-protected topological (SPT) phase. To detect these anomalies we propose several anomaly indicators -- functions that take as input the algebraic data of a symmetric topological order and that output a number indicating the presence or absence of an anomaly. We construct such indicators for both structures of the full symmetry group, i.e. $U(1)\rtimes\mathbb{Z}_2^T$ and $U(1)\times\mathbb{Z}_2^T$, and for both bosonic and fermionic topological orders. In all cases we conjecture that our indicators are complete in the sense that the anomalies they detect are in one-to-one correspondence with the known classification of $(3+1)$-D SPT phases with the same symmetry. We also show that one of our indicators for bosonic topological orders has a mathematical interpretation as a partition function for the bulk $(3+1)$-D SPT phase on a particular manifold and in the presence of a particular background gauge field for the $U(1)$ symmetry.