Gapped domain walls between 2+1D topologically ordered states

TL;DR

This paper develops a systematic theory to classify gapped domain walls and boundaries in 2+1D topologically ordered states using mapping-class-group representations and fixed-point partition functions.

Contribution

It introduces a comprehensive framework linking topological order, mapping-class-group representations, and gravitational anomalies to classify gapped domain walls and boundaries.

Findings

Determines conditions for gapped domain walls using genus-1 and genus-2 surface representations.

Characterizes gapped boundaries via fixed-point partition functions.

Connects bulk topological order with gravitational anomalies at domain walls.

Abstract

The 2+1D topological order can be characterized by the mapping-class-group representations for Riemann surfaces of genus-1, genus-2, etc. In this paper, we use those representations to determine the possible gapped boundaries of a 2+1D topological order, as well as the domain walls between two topological orders. We find that mapping-class-group representations for both genus-1 and genus-2 surfaces are needed to determine the gapped domain walls and boundaries. Our systematic theory is based on the fixed-point partition functions for the walls (or the boundaries), which completely characterize the gapped domain walls (or the boundaries). The mapping-class-group representations give rise to conditions that must be satisfied by the fixed-point partition functions, which leads to a systematic theory. Such conditions can be viewed as bulk topological order determining the (non-invertible) gravitational anomaly at the domain wall, and our theory can be viewed as finding all types of the gapped domain wall given a (non-invertible) gravitational anomaly. We also developed a systematic theory of gapped domain walls (boundaries) based on the structure coefficients of condensable algebras.