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

TL;DR

This work extends a mathematical framework for gapless and gapped edges of 2d topological orders by formulating a boundary–bulk correspondence via centers of enriched monoidal categories. It introduces 0d walls and 0+1d walls, analyzes their observables, fusion, and anomalies, and proves that the bulk is captured by the center of the edge data, yielding a monoidal equivalence between edge-centered and bulk-centered categories. It then generalizes to non-chiral gapless edges through full field algebras, providing a classification and a mechanism for purely edge transitions, all while demonstrating that dimensional reduction recovers the full spectrum of boundary–bulk RCFTs and modular invariants. The results unify fusion-categorical, RCFT, and topological-order perspectives, offering a comprehensive, hierarchical description of gapped and gapless boundaries and their higher-codimension defects with potential extensions to higher dimensions and phase diagrams.

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.

Figures (20)

• Figure 1: This picture depicts a 2d topological order $(\EuScript{C},0)$ with three different gapped edges given by UFC's $\EuScript{L},\EuScript{M},\EuScript{N}$ separated by two 0d walls $\EuScript{X}$ and $\EuScript{Y}$. The 2d bulk is oriented as the usual $\mathbb{R}^2$ with the normal direction pointing out of the paper in readers' direction. The arrows indicate the induced orientation on the edge.
• Figure 2: These two picture depicts a dimensional reduction process from $(a)$ to $(b)$.
• Figure 3: The picture illustrates the complete boundary-bulk relation, which is the physical meaning of Theorem \ref{['thm:KZ']} . The arrows indicate the orientation of the edges/walls and the order of the fusion product of topological excitations on the edges/walls.
• Figure 4: This picture depicts the 1+1D world sheet of two chiral gapless edges $(V_\EuScript{A},{}^{\EuScript{A}}\EuScript{X})$ and $(V_\EuScript{B},{}^{\EuScript{B}}\EuScript{Y})$ connected by a 0+1D gapless wall (i.e. the vertical black line). The complex coordinate $z=t+ix$ is given and determines the orientation of the world sheet.
• Figure 5: This picture depicts the 0+1D world sheet of a 0d wall $(V,{}^\EuScript{Q}(\EuScript{K}\boxtimes_\EuScript{P}\EuScript{M}))$. $\EuScript{K}$ is depicted as a fictional 0D defect placed very close to another fictional 0D defect $\EuScript{M}$ such that $\EuScript{K}\boxtimes_\EuScript{P}\EuScript{M}$ should be viewed as a single fictional 0D defect that defines the category of excitations of $(V,{}^\EuScript{Q}(\EuScript{K}\boxtimes_\EuScript{P}\EuScript{M}))$.

Theorems & Definitions (88)

• Definition 2.1
• Remark 2.2
• Definition 2.3: kz1
• Definition 2.4
• Definition 2.5
• Remark 2.7
• Remark 2.8
• Remark 2.10
• Remark 2.11
• Example 2.12