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Communications in Mathematical Physics

, Volume 315, Issue 3, pp 827–858 | Cite as

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

  • Marcel Bischoff
Article

Abstract

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 \({\mathcal{A}}\) on the real line together with an element of a unitary semigroup associated with \({\mathcal{A}}\). 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 \({\mathsf{Spin}(n)}\) at level 2 using the orbifold construction.

Keywords

Vertex Operator Algebra Loop Group Standard Pair Haag Duality Positive Energy Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. BGL02.
    Brunetti R., Guido D., Longo R.: Modular Localization and Wigner Particles. Rev. Math. Phys. 14, 759–785 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  2. BGL93.
    Brunetti R., Guido D., Longo R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys. 156, 201–219 (1993)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. BMRW09.
    Bischoff M., Meise D., Rehren K.-H., Wagner I.: Conformal quantum field theory in various dimensions. Bulg. J. Phys. 36(3), 170–185 (2009)MathSciNetzbMATHGoogle Scholar
  4. BMT88.
    Buchholz D., Mack G., Todorov I.: The current algebra on the circle as a germ of local field theories. Nucl. Phys., B, Proc. Suppl. 5(2), 20–56 (1988)MathSciNetADSCrossRefGoogle Scholar
  5. Con73.
    Connes A.: Une classification des facteurs de type III. Ann. Sci. École Norm. Sup. 6(4), 133–252 (1973)MathSciNetzbMATHGoogle Scholar
  6. CS98.
    Conway, J.H., Sloane, N.J.A.: Sphere packings, lattices, and groups. 3rd ed., Grundlagen der mathematischen Wissenschaften, Vol. 290, New York: Springer-Verlag, 1998Google Scholar
  7. DHR69.
    Doplicher S., Haag R., Roberts J.E.: Fields, observables and gauge transformations II. Commun. Math. Phys. 15, 173–200 (1969)MathSciNetADSzbMATHCrossRefGoogle Scholar
  8. DX06.
    Dong C., Xu F.: Conformal nets associated with lattices and their orbifolds. Adv. Math. 206(1), 279–306 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. FJ96.
    Fredenhagen K., Jörß M.: Conformal Haag-Kastler nets, pointlike localized fields and the existence of operator product expansions. Commun. Math. Phys. 176(3), 541–554 (1996)ADSzbMATHCrossRefGoogle Scholar
  10. GF93.
    Gabbiani F., Fröhlich J.: Operator algebras and conformal field theory. Commun. Math. Phys. 155(3), 569–640 (1993)ADSzbMATHCrossRefGoogle Scholar
  11. Gui11.
    Guido, D.: Modular Theory for the Von Neumann Algebras of Local Quantum Physics. In: Contemporary Mathematics, Vol. 534, Providence, RI: Amer. Math. Soc., 2011, pp. 97–120Google Scholar
  12. Haa96.
    Haag R.: Local quantum physics. Springer, Berlin (1996)zbMATHCrossRefGoogle Scholar
  13. Izu00.
    Izumi M.: The Structure of Sectors Associated with Longo–Rehren Inclusions I. General Theory. Commun. Math. Phys. 213, 127–179 (2000)MathSciNetADSzbMATHGoogle Scholar
  14. Kac98.
    Kac, V. G.:Vertex algebras for beginners. Providence, RI: Amer Math. Soc., 1998Google Scholar
  15. Kaw01.
    Kawahigashi, Y.: Braiding and extensions of endomorphisms of subfactors. In: Mathematical physics in mathematics and physics: quantum and operator algebraic aspects. Fields Inst. Pub. 30, Providence, RI: Amer. Math. Soc., 2001, pp. 261–269Google Scholar
  16. KL06.
    Kawahigashi Y., Longo R.: Local conformal nets arising from framed vertex operator algebras. Adv. Math. 206(2), 729–751 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  17. KLM01.
    Kawahigashi Y., Longo R., Müger M.: Multi-Interval Subfactors and Modularity of Representations in Conformal Field Theory. Commun. Math. Phys. 219, 631–669 (2001)ADSzbMATHCrossRefGoogle Scholar
  18. Kos98.
    Kosaki, H.: Type III factors and index theory. Lecture Notes Series, Vol. 43, Seoul: Seoul National University Research Institute of Mathematics Global Analysis Research Center, 1998Google Scholar
  19. Lec08.
    Lechner G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. Lon03.
    Longo R.: Conformal Subnets and Intermediate Subfactors. Commun. Math. Phys. 237, 7–30 (2003)MathSciNetADSzbMATHGoogle Scholar
  21. Lon08a.
    Longo, R.: Lecture notes on conformal nets (2008), available at http://www.mat.uniroma2.it/longo/Lecture_Notes_files/LN-Part2.pdf, 2008
  22. Lon08b.
    Longo, R.:Real Hilbert subspaces, modular theory, SL(2, \({\mathbb{R}}\)) and CFT, In: Von Neumann algebras in Sibiu, Theta Ser. Adv. Math. 10, Bucharest: Theta, 2008, pp. 33–91Google Scholar
  23. LR04.
    Longo R., Rehren K.-H.: Local Fields in Boundary Conformal QFT. Rev. Math. Phys. 16, 909–960 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  24. LR11.
    Longo R., Rehren K.-H.: Boundary Quantum Field Theory on the Interior of the Lorentz Hyperboloid. Commun. Math. Phys. 311, 769–785 (2012)MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. LR95.
    Longo R., Rehren K.-H.: Nets of Subfactors. Rev. Math. Phys. 7, 567–597 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  26. LW11.
    Longo R., Witten E.: An Algebraic Construction of Boundary Quantum Field Theory. Commun. Math. Phys. 303, 213–232 (2011)MathSciNetADSzbMATHCrossRefGoogle Scholar
  27. PS86.
    Pressley A., Segal G.: Loop groups. Clarendon Press, Oxford (1986)zbMATHGoogle Scholar
  28. Seg81.
    Segal G.: Unitary representations of some infinite dimensional groups. Commun. Math. Phys. 80(3), 301–342 (1981)ADSzbMATHCrossRefGoogle Scholar
  29. Sta95.
    Staszkiewicz, C.P.: Die lokale Struktur abelscher Stromalgebren auf dem Kreis Ph.D. Thesis, Freie Universität Berlin, 1995Google Scholar
  30. Tak03.
    Takesaki, M.: Theory of Operator Algebras II In: Encyclopaedia of Mathematical Sciences, Vol. 125, Berlin: Springer-Verlag, 2003.Google Scholar
  31. Tan11.
    Tanimoto, Y.: Construction of wedge-local nets of observables through Longo-Witten endomorphisms. Commun. Math. Phys. (2011). doi: 10.1007/s00220-012-1462-7
  32. Xu00.
    Xu F.: Algebraic orbifold conformal field theories. Proc. Nat. Acad. Sci. U.S.A. 97(26), 14069 (2000)ADSzbMATHCrossRefGoogle Scholar
  33. Xu09.
    Xu F.: On affine orbifold nets associated with outer automorphisms. Commun. Math. Phys. 291, 845–861 (2009)ADSzbMATHCrossRefGoogle Scholar
  34. ZZ79.
    Zamolodchikov A.B., Zamolodchikov A.B.: Factorized S-Matrices in Two Dimensions as the Exact Solutions of Certain Relativistic Quantum Field Theory Models. Ann. Phys. 120(2), 253–291 (1979)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly

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