范畴扩展中的无界场算子

Categorical extensions of conformal nets

An important goal in studying the relations between unitary VOAs and conformal nets is to prove the equivalence of their ribbon categories. In this article, we prove this conjecture for many familiar examples. Our main idea is to construct new structures associated to conformal nets: the categorical extensions.

Bisognano-Wichmann property for rigid categorical extensions and non-local extensions of conformal nets

Given an (irreducible) Mobius covariant net $\mathcal A$, we prove a Bisognano-Wichmann theorem for its categorical extension $\mathscr E^{\textrm{d}}$ associated to the braided $C^*$-tensor category $\textrm{Rep}^{\textrm{d}}(\mathcal A)$ of dualizable (more precisely "dualized") Mobius covariant $\mathcal A$-modules. As a closely related result, we prove a (modified) Bisognano-Wichmann theorem for any (possibly) non-local extension of $\mathcal A$ obtained by a $C^*$-Frobenius algebra $Q$ in $\textrm{Rep}^{\textrm{d}}(\mathcal A)$. As an application, we discuss the relation between the domai…

Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case

Let $\mathcal{A}$ be a completely rational local Möbius covariant net on $S^1$, which describes a set of chiral observables. We show that local Möbius covariant nets $\mathcal{B}_2$ on 2D Minkowski space which contains $\mathcal{A}$ as chiral left-right symmetry are in one-to-one correspondence with Morita equivalence classes of Q-systems in the unitary modular tensor category $\mathrm{DHR}(\mathcal{A})$. The Möbius covariant boundary conditions with symmetry $\mathcal{A}$ of such a net $\mathcal{B}_2$ are given by the Q-systems in the Morita equivalence class or by simple objects in the modul…

Matrix product operator symmetries and intertwiners in string-nets with domain walls

We provide a description of virtual non-local matrix product operator (MPO) symmetries in projected entangled pair state (PEPS) representations of string-net models. Given such a PEPS representation, we show that the consistency conditions of its MPO symmetries amount to a set of six coupled equations that can be identified with the pentagon equations of a bimodule category. This allows us to classify all equivalent PEPS representations and build MPO intertwiners between them, synthesising and generalising the wide variety of tensor network representations of topological phases. Furthermore, w…

Vanishing of categorical obstructions for permutation orbifolds

The orbifold construction $A\mapsto A^G$ for a finite group $G$ is fundamental in rational conformal field theory. The construction of $Rep(A^G)$ from $Rep(A)$ on the categorical level, often called gauging, is also prominent in the study of topological phases of matter. Given a non-degenerate braided fusion category $\mathcal{C}$ with a $G$-action, the key step in this construction is to find a braided $G$-crossed extension compatible with the action. The extension theory of Etingof-Nikshych-Ostrik gives two obstructions for this problem, $o_3\in H^3(G)$ and $o_4\in H^4(G)$ for certain coeffi…

等变辫子张量范畴中与α-归纳相关的Frobenius代数

Frobenius algebras associated with the $α$-induction for equivariantly braided tensor categories

本研究在G-辫子多张量范畴中给出G-等变α-归纳的范畴定义，在中性double范畴中构造G-等变α-归纳Frobenius代数，还定义G-等变全中心并证明其与该代数一致，推广了已有相关定理与构造。