Para-fusion Category and Topological Defect Lines in $\mathbb Z_N$-parafermionic CFTs

TL;DR

This work introduces a parafermionic extension of fusion theory by proposing a $\mathbb Z_N$ para-fusion category to describe topological defect lines in two-dimensional $\mathbb Z_N$-parafermionic CFTs, including parafermionic defect operators living on TDLs with fractional statistics. It develops a 3d-bulk perspective via parafermionic anyon condensation and constructs a functor from bosonic fusion categories to para-fusion categories, governed by para-pentagon relations. The paper provides detailed examples, notably pf-$\mathbb Z_N$ Verlinde lines and parafermionic Tambara-Yamagami categories $pf-\mathrm{TY}_{\mathbb Z_N}^{t,\kappa,\beta}$, and solves the associated para-pentagon equations to classify these structures for general $N$. The results illuminate non-invertible symmetries in parafermionic CFTs and reveal domain-wall connections to 3d topological orders, broadening the toolkit for parafermionic topological phases. Overall, the work lays a comprehensive framework linking 2d parafermionic defect data to 3d bulk/topological-order phenomena.

Abstract

We study topological defect lines (TDLs) in two-dimensional $\mathbb Z_N$-parafermoinic CFTs. Different from the bosonic case, in the 2d parafermionic CFTs, there exist parafermionic defect operators that can live on the TDLs and satisfy interesting fractional statistics. We propose a categorical description for these TDLs, dubbed as ``para-fusion category", which contains various novel features, including $\mathbb Z_M$ $q$-type objects for $M\vert N$, and parafermoinic defect operators as a type of specialized 1-morphisms of the TDLs. The para-fusion category in parafermionic CFTs can be regarded as a natural generalization of the super-fusion category for the description of TDLs in 2d fermionic CFTs. We investigate these distinguishing features in para-fusion category from both a 2d pure CFT perspective, and also a 3d anyon condensation viewpoint. In the latter approach, we introduce a generalized parafermionic anyon condensation, and use it to establish a functor from the parent fusion category for TDLs in bosonic CFTs to the para-fusion category for TDLs in the parafermionized ones. At last, we provide many examples to illustrate the properties of the proposed para-fusion category, and also give a full classification for a universal para-fusion category obtained from parafermionic condensation of Tambara-Yamagami $\mathbb Z_N$ fusion category.

Figures (3)

• Figure 1: Bosonic anyon condensation in 3d topological order of UMBC $\mathcal{B}$. $A$ is a connected commutative separable algebra in $\mathcal{B}$. The red domain wall that separates the topological orders before and after the condensation is described by the fusion category $\mathcal{B}_A$ of the right $A$-modules. Subsequently, the condensed phase is characterized by the UMTC $\mathcal{B}_A^0$ which consists of local (or dyslectic) right $A$-modules in $\mathcal{B}$.
• Figure 2: Two types of bosonic anyon condensation in UFC $\mathcal{C}$ corresponding to a 2d boundary or UMTC $Z(\mathcal{C})$ associated with a 3d bulk topological order. The 2d boundary can be understood from two different perspectives: as a 2d spatial manifold used to define the Hamiltonian, or as a 1+1d boundary through a Wick rotation. (a) $A$ is an algebra in fusion category $\mathcal{C}$. The domain wall $\mathcal{C}_A$ has no monoidal/tensor structure in general. The original fusion category $\mathcal{C}$ and the bimodule category $_A\mathcal{C}_A$ are Morita equivalent, i.e., $Z(\mathcal{C})\simeq Z(_A\mathcal{C}_A)$. So the red domain wall between them is invertible or an equivalence of bulk UMTC. A prototypical example is $\mathcal{C}=\mathrm{Vec}_G$, $A=\mathbb C[G]$, $\mathcal{C}_A=\mathrm{Vec}$ and $_A\mathcal{C}_A=\mathrm{Rep}(G)$, where the two sides are related by gauging/condensing $A$. (b) $A$ is a central commutative algebra, i.e., a commutative algebra in $Z(\mathcal{C})$. The category of right-$A$ modules already has a monoidal/tensor structure. The bulk of $\mathcal{C}$ and $\mathcal{C}_A$ are related by $Z(\mathcal{C}_A)=[Z(\mathcal{C})]_A^0$schauenburg_monoidal_2001. So the 3d red domain wall corresponds to anyon condensation of topological orders from UMTC $Z(\mathcal{C})$ to $[Z(\mathcal{C})]_A^0$.
• Figure 3: Physical interpretation of the fusion category equivalence $_A\mathcal{C}_A \simeq \mathrm{Fun}_\mathcal{C}(\mathcal{C}_A,\mathcal{C}_A)$. The fusion category $\mathcal{C}$ and $_A\mathcal{C}_A$ describe the TDLs of the left original theory and the right $A$-condensed theory in 2d, respectively. They are separated by a (red) domain wall in the middle. The domain wall is labeled by objects in the module category $\mathcal{C}_A$. When moving a (yellow) TDL $M\in {}_A\mathcal{C}_A$ from the right condensed theory to the middle (red) domain wall labeled by object in $\mathcal{C}_A$, the domain wall is changed to another (orange) one in $\mathcal{C}_A$. This process of fusing $M\in {}_A\mathcal{C}_A$ provides a functor from $\mathcal{C}_A$ to itself, establishing an equivalence between the TDLs in $_A\mathcal{C}_A$ and the boundary condition changing processes in $\mathrm{Fun}_\mathcal{C}(\mathcal{C}_A,\mathcal{C}_A)$ from $\mathcal{C}_A$ to itself.

Theorems & Definitions (1)

• Theorem 1