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

TL;DR

This paper develops a comprehensive mathematical framework for understanding gapless edges of 2D topological orders, including boundary-bulk relations and non-chiral edges, linking topological orders with rational conformal field theories.

Contribution

It introduces the boundary-bulk relation as a monoidal equivalence, generalizes the Drinfeld center to an enriched setting, and connects topological orders with rational CFTs through dimensional reduction.

Findings

Established a rigorous boundary-bulk relation as a monoidal equivalence.

Developed the mathematical theory of non-chiral gapless edges and 0+1D walls.

Showed all anomaly-free 1+1D boundary-bulk rational CFTs can be derived from 2D topological orders.

Abstract

This is the second part of a two-part work on the unified mathematical theory of gapped and gapless edges of 2+1D topological orders. In Part I, we have developed the mathematical theory of chiral gapless edges. In Part II, we study boundary-bulk relation and non-chiral gapless edges. In particular, we explain how the notion of the center of an enriched monoidal category naturally emerges from the boundary-bulk relation. After the study of 0+1D gapless walls, we give the complete boundary-bulk relation for 2+1D topological orders with chiral gapless edges (including gapped edges) and 0d walls between edges. This relation is stated precisely and proved rigorously as a monoidal equivalence, which generalizes the functoriality of the usual Drinfeld center to an enriched setting. We also develop the mathematical theory of non-chiral gapless edges and 0+1D walls, and explain how to gap out certain non-chiral 1+1D gapless edges and 0+1D gapless walls categorically. In the end, we show that all anomaly-free 1+1D boundary-bulk rational CFT's can be recovered from 2d topological orders with chiral gapless edges via a dimensional reduction process. This provides physical meanings to some mysterious connections between mathematical results in fusion categories and those in rational CFT's.