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

Let $\mathcal{A}$ be a completely rational local M\"obius covariant net on $S^1$, which describes a set of chiral observables. We show that local M\"obius 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 M\"obius 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 module category modulo automorphisms of the dual category. We generalize to reducible boundary conditions. To establish this result we define the notion of Morita equivalence for Q-systems (special symmetric $\ast$-Frobenius algebra objects) and non-degenerately braided subfactors. We prove a conjecture by Kong and Runkel, namely that Rehren's construction (generalized Longo-Rehren construction, $\alpha$-induction construction) coincides with the categorical full center. This gives a new view and new results for the study of braided subfactors.Comment: 44 pages, many tikz figures. Some improvements. Some typos fixe

Tensor categories and endomorphisms of von Neumann algebras (with applications to Quantum Field Theory)

Q-systems describe "extensions" of an infinite von Neumann factor $N$, i.e., finite-index unital inclusions of $N$ into another von Neumann algebra $M$. They are (special cases of) Frobenius algebras in the C* tensor category of endomorphisms of $N$. We review the relation between Q-systems, their modules and bimodules as structures in a category on one side, and homomorphisms between von Neumann algebras on the other side. We then elaborate basic operations with Q-systems (various decompositions in the general case, and the centre, the full centre, and the braided product in braided categories), and illuminate their meaning in the von Neumann algebra setting. The main applications are in local quantum field theory, where Q-systems in the subcategory of DHR endomorphisms of a local algebra encode extensions $A(O)\subset B(O)$ of local nets. These applications, notably in conformal quantum field theories with boundaries, are briefly exposed, and are discussed in more detail in two separate papers [arXiv:1405.7863, 1410.8848].Comment: v1: 54 pages. v2: 90 pages. Title changed by request of editor, numerous material added, especially an overview of applications in Quantum Field Theory; some corrections. v3: minor corrections to match the published version; Springer Briefs in Mathematical Physics, vol. 3, 201

Spontaneous symmetry breaking from anyon condensation

In a physical system undergoing a continuous quantum phase transition, spontaneous symmetry breaking occurs when certain symmetries of the Hamiltonian fail to be preserved in the ground state. In the traditional Landau theory, a symmetry group can break down to any subgroup. However, this no longer holds across a continuous phase transition driven by anyon condensation in symmetry enriched topological orders (SETOs). For a SETO described by a $G$-crossed braided extension $\mathcal{C}\subseteq \mathcal{C}^{\times}_{G}$, we show that physical considerations require that a connected \'etale algebra $A\in \mathcal{C}$ admit a $G$-equivariant algebra structure for symmetry to be preserved under condensation of $A$. Given any categorical action $\underline{G}\rightarrow \underline{\sf Aut}_{\otimes}^{\sf br}(\mathcal{C})$ such that $g(A)\cong A$ for all $g\in G$, we show there is a short exact sequence whose splittings correspond to $G$-equivariant algebra structures. The non-splitting of this sequence forces spontaneous symmetry breaking under condensation of $A$. Furthermore, we show that if symmetry is preserved, there is a canonically associated SETO of $\mathcal{C}^{\operatorname{loc}}_{A}$, and gauging this symmetry commutes with anyon condensation.Comment: 35 pages, comments welcome. To appear in Journal of High Energy Physic

Phase boundaries in algebraic conformal QFT

We study the structure of local algebras in relativistic conformal quantum field theory with phase boundaries. Phase boundaries are instances of a more general notion of boundaries that give rise to a variety of algebraic structures. These can be formulated in a common framework originating in Algebraic QFT, with the principle of Einstein Causality playing a prominent role.We classify the phase boundary conditions by the centre of a certain universal construction, which produces a reducible representation in which all possible boundary conditions are realized. For a large class of models, the classification reproduces results obtained in a different approach by Fuchs et al. before.Comment: 40 pages, v3: several corrections, matches published versio