155 research outputs found
Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case
Let be a completely rational local M\"obius covariant net on
, which describes a set of chiral observables. We show that local M\"obius
covariant nets on 2D Minkowski space which contains
as chiral left-right symmetry are in one-to-one correspondence
with Morita equivalence classes of Q-systems in the unitary modular tensor
category . The M\"obius covariant boundary
conditions with symmetry of such a net 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 -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,
-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 , i.e.,
finite-index unital inclusions of into another von Neumann algebra .
They are (special cases of) Frobenius algebras in the C* tensor category of
endomorphisms of . 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
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 -crossed
braided extension , we show that
physical considerations require that a connected \'etale algebra admit a -equivariant algebra structure for symmetry to be
preserved under condensation of . Given any categorical action
such that for all , we show there is a short exact
sequence whose splittings correspond to -equivariant algebra structures. The
non-splitting of this sequence forces spontaneous symmetry breaking under
condensation of . Furthermore, we show that if symmetry is preserved, there
is a canonically associated SETO of , 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
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