# A mathematical theory of gapless edges of 2d topological orders. Part I

## TL;DR

This paper develops a unified mathematical framework for describing both gapped and gapless edges of 2D topological orders, revealing their observables form an enriched fusion category and establishing a connection to bulk properties.

## Contribution

It introduces a comprehensive mathematical theory for chiral gapless edges, unifying their description with gapped edges via enriched fusion categories and classifying both types.

## Key findings

- Observables form an enriched unitary fusion category.
- The Drinfeld center of this category matches the bulk's modular tensor category.
- All chiral gapless edges can be derived from topological Wick rotations.

## Abstract

This is the first part of a two-part work on a unified mathematical theory of gapped and gapless edges of 2d topological orders. We analyze all the possible observables on the 1+1D world sheet of a chiral gapless edge of a 2d topological order, and show that these observables form an enriched unitary fusion category, the Drinfeld center of which is precisely the unitary modular tensor category associated to the bulk. This mathematical description of a chiral gapless edge automatically includes that of a gapped edge (i.e. a unitary fusion category) as a special case. Therefore, we obtain a unified mathematical description and a classification of both gapped and chiral gapless edges of a given 2d topological order. In the process of our analysis, we encounter an interesting and reoccurring phenomenon: spatial fusion anomaly, which leads us to propose the Principle of Universality at RG fixed points for all quantum field theories. Our theory also implies that all chiral gapless edges can be obtained from a so-called topological Wick rotations. This fact leads us to propose, at the end of this work, a surprising correspondence between gapped and gapless phases in all dimensions.