# Boundary conformal field theory and symmetry protected topological phases in $2+1$ dimensions

## TL;DR

This paper introduces a diagnostic approach for identifying non-trivial 2+1D SPT phases by analyzing their 1+1D edge conformal field theories and the obstructions to boundary states that preserve symmetry.

## Contribution

It establishes a direct link between edge CFT properties and the presence of SPT phases, providing a new method to detect topological order via boundary state analysis.

## Key findings

- Obstruction to boundary states indicates non-trivial SPT phases.
- Applicable to time-reversal symmetric topological insulators and bosonic SPTs.
- Demonstrates relation between edgeability, gappability, and SPT detection.

## Abstract

We propose a diagnostic tool for detecting non-trivial symmetry protected topological (SPT) phases protected by a symmetry group $G$ in 2+1 dimensions. Our method is based on directly studying the 1+1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests in the in-existence of boundary states that preserves both the conformal symmetry and the global symmetry $G$. We discuss the relation between edgeability, the ability to find a consistent boundary state, and gappability, the ability to gap out a CFT, in the presence of $G$. We study several cases including time-reversal symmetric topological insulators, $\mathbb{Z}_N$ symmetric bosonic SPTs, and $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetric topological superconductors.