arXiv subjects

Marcel Bischoff

Publications and source records attributed to Marcel Bischoff.

23 records · Page 2Linked to original sources

arXiv2015

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 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, $α$-induction construction) coincides with the categorical full center. This gives a new view and new results for the study of braided subfactors.

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arXiv2015

A Remark on CFT Realization of Quantum Doubles of Subfactors. Case Index < 4

It is well-known that the quantum double $D(N\subset M)$ of a finite depth subfactor $N\subset M$, or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. Thus should arise in conformal field theory. We show that for every subfactor $N\subset M$ with index $[M:N]<4$ the quantum double $D(N\subset M)$ is realized as the representation category of a completely rational conformal net. In particular, the quantum double of $E_6$ can be realized as a $\mathbb Z_2$-simple current extension of $\mathrm{SU}(2)_{10}\times \mathrm{Spin}(11)_1$ and thus is not exotic in any sense. As a byproduct we obtain a vertex operator algebra for every such subfactor. We obtain the result by showing that if a subfactor $N\subset M $ arises from $α$-induction of completely rational nets $\mathcal A\subset \mathcal B$ and there is a net $\tilde{\mathcal A}$ with the opposite braiding, then the quantum $D(N\subset M)$ is realized by completely rational net. We construct completely rational nets with the opposite braiding of $\mathrm{SU}(2)_k$ and use the well-known fact that all subfactors with index $[M:N]<4$ arise by $α$-induction from $\mathrm{SU}(2)_k$.

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arXiv2015

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].

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arXiv2012

Models in Boundary Quantum Field Theory Associated with Lattices and Loop Group Models

In this article we give new examples of models in boundary quantum field theory, i.e. local time-translation covariant nets of von Neumann algebras, using a recent construction of Longo and Witten, which uses a local conformal net A on the real line together with an element of a unitary semigroup associated with A. Namely, we compute elements of this semigroup coming from Hölder continuous symmetric inner functions for a family of (completely rational) conformal nets which can be obtained by starting with nets of real subspaces, passing to its second quantization nets and taking local extensions of the former. This family is precisely the family of conformal nets associated with lattices, which as we show contains as a special case the level 1 loop group nets of simply connected, simply laced groups. Further examples come from the loop group net of Spin(n) at level 2 using the orbifold construction.

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arXiv2009

Conformal quantum field theory in various dimensions

Various relations between conformal quantum field theories in one, two and four dimensions are explored. The intention is to obtain a better understanding of 4D CFT with the help of methods from lower dimensional CFT.

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