Communications in Mathematical Physics

, Volume 317, Issue 3, pp 667–695 | Cite as

Construction of Wedge-Local Nets of Observables through Longo-Witten Endomorphisms. II

  • Marcel Bischoff
  • Yoh TanimotoEmail author
Open Access


In the first part, we have constructed several families of interacting wedge-local nets of von Neumann algebras. In particular, we discovered a family of models based on the endomorphisms of the U(1)-current algebra \({\mathcal{A} ^{(0)}}\) of Longo-Witten.

In this second part, we further investigate endomorphisms and interacting models. The key ingredient is the free massless fermionic net, which contains the U(1)-current net as the fixed point subnet with respect to the U(1) gauge action. Through the restriction to the subnet, we construct a new family of Longo-Witten endomorphisms on \({\mathcal{A} ^{(0)}}\) and accordingly interacting wedge-local nets in two-dimensional spacetime. The U(1)-current net admits the structure of particle numbers and the S-matrices of the models constructed here do mix the spaces with different particle numbers of the bosonic Fock space.


Gauge Action Asymptotic Completeness Conformal Character Positive Energy Representation Standard Subspace 
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.



We thank our supervisor Roberto Longo for his constant support and useful suggestions. Y. T. thanks Gandalf Lechner and Jan Schlemmer for discussions on the relation between the present construction and the deformation of [Lec11].

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


  1. Aks65.
    Aks S.O.: Proof that scattering implies production in quantum field theory. J. Math. Phys. 6, 516–532 (1965)MathSciNetADSCrossRefGoogle Scholar
  2. Apo76.
    Apostol, T.M.: Introduction to analytic number theory. Undergraduate Texts in Mathematics. New York: Springer-Verlag, 1976Google Scholar
  3. BF77.
    Buchholz D., Fredenhagen K.: Dilations and interaction. J. Math. Phys. 18(5), 1107–1111 (1977)ADSCrossRefGoogle Scholar
  4. BLS11.
    Buchholz D., Lechner G., Summers S.J.: Warped convolutions, Rieffel deformations and the construction of quantum field theories. Commun. Math. Phys. 304, 95–123 (2011)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. Bor92.
    Borchers H.-J.: The CPT-theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143(2), 315–332 (1992)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. BS08.
    Buchholz, D., Summers, S.J.: Warped convolutions: a novel tool in the construction of quantum field theories. In: Quantum field theory and beyond. Hackensack, NJ: World Sci. Publ. 2008, pp. 107–121Google Scholar
  7. BSM90.
    Buchholz D., Schulz-Mirbach H.: Haag duality in conformal quantum field theory. Rev. Math. Phys. 2(1), 105–125 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  8. Buc75.
    Buchholz D.: Collision theory for waves in two dimensions and a characterization of models with trivial S-matrix. Commun. Math. Phys. 45(1), 1–8 (1975)MathSciNetADSCrossRefGoogle Scholar
  9. CKL08.
    Carpi S., Kawahigashi Y., Longo R.: Structure and classification of superconformal nets. Ann. Henri Poincaré 9(6), 1069–1121 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  10. DF77.
    Driessler W., Fröhlich J.: The reconstruction of local observable algebras from the euclidean green’s functions of relativistic quantum field theory. Ann. L’Inst. H. Poincare Section Phys. Theor. 27, 221–236 (1977)ADSzbMATHGoogle Scholar
  11. DT11.
    Dybalski W., Tanimoto Y.: Asymptotic completeness in a class of massless relativistic quantum field theories. Commun. Math. Phys. 305, 427–440 (2011)MathSciNetADSzbMATHCrossRefGoogle Scholar
  12. GL07.
    Grosse H., Lechner G.: Wedge-local quantum fields and noncommutative Minkowski space. J. High Energy Phys. 0711, 012 (2007)MathSciNetADSCrossRefGoogle Scholar
  13. GL08.
    Grosse H., Lechner G.: Noncommutative deformations of Wightman quantum field theories. J. High Energy Phys. 0809, 131 (2008)MathSciNetADSCrossRefGoogle Scholar
  14. Kac98.
    Kac, V.G.: Vertex algebras for beginners. Providence, RI: Amer. Math. Soc., 1998Google Scholar
  15. KR87.
    Kac, V.G., Raina, A.K.: Bombay lectures on highest weight representations of infinite-dimensional Lie algebras. Volume 2 of Advanced Series in Mathematical Physics. Teaneck, NJ: World Scientific Publishing Co. Inc., 1987Google Scholar
  16. Lec08.
    Lechner G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277(3), 821–860 (2008)MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. Lec11.
    Lechner G.: Deformations of quantum field theories and integrable models. Commun. Math. Phys. 305, 99–130 (2011)MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. Lon08.
    Longo, R.: Real Hilbert subspaces, modular theory, SL(2, R) and CFT. In: Von Neumann algebas in Sibiu: Conference Proceedings. Bucharest: Theta, 2008, pp. 33–91Google Scholar
  19. LW11.
    Longo R., Witten E.: An algebraic construction of boundary quantum field theory. Commun. Math. Phys. 303, 213–232 (2011)MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. Reh98.
    Rehren, K.-H.: Konforme quantenfeldtheorie. Lecture note available at, 1997
  21. RS75.
    Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1975Google Scholar
  22. Tak03.
    Takesaki M.: Theory of operator algebras. II. Volume 125 of Encyclopaedia of Mathematical Sciences. Berlin: Springer-Verlag, 2003Google Scholar
  23. Tan11a.
    Tanimoto Y.: Construction of wedge-local nets of observables through longo-witten endomorphisms. Commun. Math. Phys. 314(2), 443–469 (2012)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. Tan11b.
    Tanimoto Y.: Noninteraction of waves in two-dimensional conformal field theory. Commun. Math. Phys. 314(2), 419–441 (2012)MathSciNetADSzbMATHCrossRefGoogle Scholar
  25. Was98.
    Wassermann A.: Operator algebras and conformal field theory. III. Fusion of positive energy representations of LSU (N) using bounded operators. Invent. Math. 133(3), 467–538 (1998)MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly
  2. 2.Institute für Theoretische PhysikUniversität GöttingenGoettingenGermany

Personalised recommendations