## Abstract

We construct local, boost covariant boundary QFT nets of von Neumann algebras on the interior of the Lorentz hyperboloid \({\mathfrak{H}_R}\), *x*
^{2} − *t*
^{2} > *R*
^{2}, *x* > 0, in the two-dimensional Minkowski spacetime. Our first construction is canonical, starting with a local conformal net on \({\mathbb{R}}\), and is analogous to our previous construction of local boundary CFT nets on the Minkowski half-space. This net is in a thermal state at Hawking temperature. Then, inspired by a recent construction by E. Witten and one of us, we consider a unitary semigroup that we use to build up infinitely many nets. Surprisingly, the one-particle semigroup is again isomorphic to the semigroup of symmetric inner functions of the disk. In particular, by considering the *U*(1)-current net, we can associate with any given symmetric inner function a local, boundary QFT net on \({\mathfrak{H}_R}\). By considering different states, we shall also have nets in a ground state, rather than in a KMS state.

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## Acknowledgements

R.L. is grateful to E. Witten for stimulating comments and D. Voiculescu for pointing out ref. [14].

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Communicated by Y. Kawahigashi

Supported 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.

Supported in part by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen.

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**Open Access** This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Longo, R., Rehren, KH. Boundary Quantum Field Theory on the Interior of the Lorentz Hyperboloid.
*Commun. Math. Phys.* **311**, 769–785 (2012). https://doi.org/10.1007/s00220-011-1381-z

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DOI: https://doi.org/10.1007/s00220-011-1381-z