Gapped domain walls between 2+1D topologically ordered states

Abstract

The (2+1)-dimensional (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 (noninvertible) gravitational anomaly at the domain wall, and our theory can be viewed as finding all types of the gapped domain wall given a (noninvertible) gravitational anomaly. We also developed a systematic theory of gapped domain walls (boundaries) based on the structure coefficients of condensable algebras.