## Abstract

We build up local, time translation covariant Boundary Quantum Field Theory nets of von Neumann algebras \({\mathcal A_V}\) on the Minkowski half-plane *M*
_{+} starting with a local conformal net \({\mathcal A}\) of von Neumann algebras on \({\mathbb R}\) and an element *V* of a unitary semigroup \({\mathcal E(\mathcal A)}\) associated with \({\mathcal A}\). The case *V* = 1 reduces to the net \({\mathcal A_+}\) considered by Rehren and one of the authors; if the vacuum character of \({\mathcal A}\) is summable, \({\mathcal A_V}\) is locally isomorphic to \({\mathcal A_+}\). We discuss the structure of the semigroup \({\mathcal E(\mathcal A)}\). By using a one-particle version of Borchers theorem and standard subspace analysis, we provide an abstract analog of the Beurling-Lax theorem that allows us to describe, in particular, all unitaries on the one-particle Hilbert space whose second quantization promotion belongs to \({\mathcal E(\mathcal A^{(0)})}\) with \({\mathcal A^{(0)}}\) the *U*(1)-current net. Each such unitary is attached to a scattering function or, more generally, to a symmetric inner function. We then obtain families of models via any Buchholz-Mack-Todorov extension of \({\mathcal A^{(0)}}\). A further family of models comes from the Ising model.

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Communicated by Y. Kawahigashi

Supported in part by the ERC Advanced Grant 227458 OACFT “Operator Algebras and Conformal Field Theory”, PRIN-MIUR, GNAMPA-INDAM and EU network “Noncommutative Geometry” MRTN-CT-2006-0031962.

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Longo, R., Witten, E. An Algebraic Construction of Boundary Quantum Field Theory.
*Commun. Math. Phys.* **303**, 213–232 (2011). https://doi.org/10.1007/s00220-010-1133-5

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DOI: https://doi.org/10.1007/s00220-010-1133-5