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

TL;DR

This work develops a unified mathematical framework for gapped and gapless edges of 2d topological orders, focusing on chiral gapless edges. It identifies the observable content of a chiral gapless edge with a pair $(V,\mathcal{X}^\sharp)$ where $V$ is a unitary rational VOA and $\mathcal{X}^\sharp$ is a $\mathrm{Mod}_V$-enriched unitary fusion category; the center of $\mathcal{X}^\sharp$ recovers the bulk UMTC, thereby encoding the boundary-bulk relation. The paper establishes that all observables form an enriched monoidal category, and provides explicit constructions (canonical gapless edge) and a universal fusion framework via topological Wick rotations, leading to a complete classification of chiral gapless edges and a principle of universality at RG fixed points. These results imply a deep gapped-gapless correspondence and offer a foundation for extending the theory to higher dimensions and more general boundary phenomena. The formalism hinges on deep connections between VOAs, modular tensor categories, boundary CFTs, and enriched category theory, yielding a coherent, anomaly-free picture of chiral gapless edges and their phase structure.

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.

Figures (13)

• Figure 1: This picture depicts a 2d topological order $(\EuScript{C},0)$, where $\EuScript{C}$ is a UMTC, with three gapped edges given by UFC's $\EuScript{L},\EuScript{M},\EuScript{N}$. 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: Obvious questions about gapless edges of a 2d chiral topological order.
• Figure 3: This picture depicts a 2d topological order $(\EuScript{C},c)$ on a 2-disk, together with a 1d gapless edge, propagating in time. A chiral field $\phi(z)\in U$ is depicted on the 1+1D world sheet.
• Figure 4: This picture depicts a 2d topological order $(\EuScript{C},c)$ on a 2-disk, together with a 1d gapless edge, propagating in time. At $t=0$, we move a bulk particle $a$ to the edge. The blue world line is supported on $x$.
• Figure 5: The picture (a) depicts a 2d topological order $(\EuScript{C},c)$ on a 2-disk, together with a 1d gapless edge, propagating in time. At $t=0$, a topological edge excitation $x$ is created. At $t=t_1>0$, the edge excitation is changed to $y$. This change creates a domain wall $M_{x,y}$ between OSVOA's $A_x$ and $A_y$. The picture (b) depicts the 1+1D world sheet obtained by squeezing the picture (a).

Theorems & Definitions (72)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Definition 3.1
• Remark 3.2
• Definition 3.3
• Theorem 3.4: osvoa
• Remark 3.5
• Remark 3.6