# Gapless edges of 2d topological orders and enriched monoidal categories

## TL;DR

This paper provides a unified mathematical framework for describing both gapped and gapless edges of 2D topological orders using enriched monoidal categories, revealing boundary-bulk duality and classifying edge types.

## Contribution

It introduces a comprehensive mathematical model for gapless edges via enriched monoidal categories, extending the understanding of boundary-bulk duality in topological phases.

## Key findings

- Unified description of gapped and gapless edges
- Boundary-bulk duality holds for gapless edges
- Classification of all gapped and gapless edges for a given bulk

## Abstract

In this work, we give a precise mathematical description of a fully chiral gapless edge of a 2d topological order (without symmetry). We show that the observables on the 1+1D world sheet of such an edge consist of a family of topological edge excitations, boundary CFT's and walls between boundary CFT's. These observables can be described by a chiral algebra and an enriched monoidal category. This mathematical description automatically includes that of gapped edges as special cases. Therefore, it gives a unified framework to study both gapped and gapless edges. Moreover, the boundary-bulk duality also holds for gapless edges. More precisely, the unitary modular tensor category that describes the 2d bulk phase is exactly the Drinfeld center of the enriched monoidal category that describes the gapless/gapped edge. We propose a classification of all gapped and fully chiral gapless edges of a given bulk phase. In the end, we explain how modular-invariant bulk conformal field theories naturally emerge on certain gapless walls between two trivial phases.