Gapless edges of 2d topological orders and enriched monoidal categories

TL;DR

The paper develops a rigorous, unified framework for describing fully chiral gapless edges of 2d topological orders, expressing edge observables as a pair $(V,\mathcal{X}^{\sharp})$ with $V$ a unitary rational VOA and $\mathcal{X}^{\sharp}$ a $\mathrm{Mod}_V$-enriched monoidal category. It identifies a canonical gapless edge, demonstrates boundary-bulk duality via the Drinfeld center, and shows how general gapless edges arise by fusing the canonical edge with gapped walls, all within a common Langragian-algebra/center formalism. The authors propose a classification of all gapped and chiral gapless edges in terms of triples $(V,\mathcal{B},\mathcal{X})$ and equivalently Lagrangian algebras in $\overline{\mathcal{B}}\boxtimes \mathcal{C}$, reducing the problem to Witt-equivalence questions for VOAs. They also connect 0d defects and bulk CFTs to modular-invariant bulk theories, providing a holographic-like perspective and setting the stage for further exploration of symmetry-enriched cases and factorization-homology computations.

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.

Figures (5)

• Figure 1: The picture (a) depicts a 2d topological order $(\EuScript{C},c)$ on a 2-disk, together with a 1d gapless edge, propagating in time. When a topological bulk excitation $a\in \EuScript{C}$ is moved to the edge at $t=0$, it creates a topological edge excitation $x$ or a boundary condition $M_x$ for the OSVOA $A_x$ living on the $t>0$ part of the world line. At $t=t_1>0$, the topological edge excitation $x$ is changed to another topological edge excitation $y$. This change creates a wall $M_{x,y}$ between $A_x$ and $A_y$. The picture (b) depicts the quasi-1+1D world sheet obtained by stretching the picture (a) along the dotted arrow from $a$ to $x$.
• Figure 2: This picture depicts the fusion of the chiral fields $\phi(z)$ in $U$ into those in $A_x$ along different paths. Chiral fields $\Psi(t)$ in the boundary CFT $A_x$ (or $A_y$) are restricted on the world line (the $t$-axis), which is also the blue line in Figure \ref{['fig:cylinder']}. Defect fields $\Phi(t_1)$ in $M_{x,y}$ are restricted on the point $t=t_1$.
• Figure 3: This picture depicts how to fuse horizontally (on the same time slide) two topological edge excitations (or boundary conditions of boundary CFT's) $x$ and $y$, together with boundary CFT's $A_x$, $A_y$, $A_{x'}$, $A_{y'}$ and walls $M_{x,y}$, $M_{x',y'}$. For convenience, we abbreviate $x'\otimes x$ to $x'x$ in the picture.
• Figure 4: This picture illustrate a wall $(U,\EuScript{A},\EuScript{M})$ between $(\EuScript{C},c_1)$ and $(\EuScript{D},c_1+c_2)$ and a wall $(V,\EuScript{B},\EuScript{N})$ between $(\EuScript{D},c_1+c_2)$ and $(\EuScript{E},c_1+c_2+c_3)$. The fusion of these two walls is defined by Eq. (\ref{['eq:fusing-walls']}).
• Figure 5: (a) depicts a two 2+1D bulk phases $(\EuScript{C},c)$ and $(\EuScript{C},c)$, equipped with the canonical edge and its time reverse, respectively, and separated by a gapped domain wall $\EuScript{M}$. $x,z$ are objects in $\EuScript{M}$, and the internal hom $[x,z] \in \EuScript{M}$ is defined by $[x,z]=z\otimes x^\ast$. These internal homs form a canonical 0d edge of the 1d wall $\EuScript{M}$ (more details will be given in kz3). (b) depicts a 1+1D world sheet obtained by squeezing the entire 2-disk in (a) to the 1d wall $\EuScript{M}$.

Theorems & Definitions (40)

• Remark 2.1
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Remark 3.5
• Remark 3.6
• Remark 4.1
• Remark 4.2
• Remark 4.3