Symmetry-preserving boundary of (2+1)D fractional quantum Hall states

TL;DR

This paper develops a general algebraic framework for understanding symmetry-preserving gapped boundaries in (2+1)D topological phases, including fermionic cases, and identifies new obstructions beyond known invariants.

Contribution

It extends the algebraic description of gapped boundaries to symmetry-enriched and fermionic phases, deriving new obstructions using a Gauss-Milgram type formula.

Findings

Derived new obstructions to gapped boundaries for fermionic phases.

Extended algebraic framework to include symmetry and fermionic cases.

Identified higher invariants beyond central charge and Hall conductivity.

Abstract

We investigate symmetry-preserving gapped boundary of (2+1)D topological phases with global symmetry, which can be either bosonic or fermionic. We develop a general algebraic description for gapped boundary condition for symmetry-enriched or fermionic topological phases, extending the framework of Lagrangian algebra anyon for bosonic phases without symmetry. We then focus on application to the case with U(1) symmetry. We derive new obstructions to symmetry-preserving gapped boundary for U(1)$^f$-symmetric (2+1)D fermionic topological phases, which are beyond chiral central charge $c_-$ and electric Hall conductivity $\sigma_H$. These obstructions are given by a simple Gauss-Milgram type formula valid for super-modular category, and regarded as a higher version of $c_-$ and $\sigma_H$.