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Annales Henri Poincaré

, Volume 16, Issue 8, pp 1899–1936 | Cite as

Super-KMS Functionals for Graded-Local Conformal Nets

  • Robin HillierEmail author
Article

Abstract

Motivated by a few preceding papers and a question of R. Longo, we introduce super-KMS functionals for graded translation-covariant nets over \({\mathbb{R}}\) with superderivations, roughly speaking as a certain supersymmetric modification of classical KMS states on translation-covariant nets over \({\mathbb{R}}\), fundamental objects in chiral algebraic quantum field theory. Although we are able to make a few statements concerning their general structure, most properties will be studied in the setting of specific graded-local (super-) conformal models. In particular, we provide a constructive existence and partial uniqueness proof of super-KMS functionals for the supersymmetric free field, for certain subnets, and for the super-Virasoro net with central charge \({c\ge 3/2}\). Moreover, as a separate result, we classify bounded super-KMS functionals for graded-local conformal nets over S 1 with respect to rotations.

Keywords

Irreducible Representation Canonical Commutation Relation Cyclic Cocycles Irreducible General Representation Conformal Subnet 
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.

References

  1. 1.
    Araki H.: On quasi-free states of CAR and Bogoliubov automorphisms. Publ. Res. Inst. Math. Sci. 6, 385–442 (1970)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Araki H.: On quasi-free states of the canonical commutation relations. II. Publ. Res. Inst. Math. Sci. 7, 121–152 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Araki H., Masafumi S.: On quasi-free states of the canonical commutation relations. I. Publ. Res. Inst. Math. Sci. 7, 105–120 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Böckenhauer J.: Localized endomorphisms of the chiral Ising model. Commun. Math. Phys. 177, 265–304 (1996)ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Bratteli O., Robinson D.: Operator Algebras and Quantum Statistical Mechanics. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  6. 6.
    Buchholz D., Grundling H.: Algebraic supersymmetry: a case study. Commun. Math. Phys. 272, 699–750 (2007)MathSciNetADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Buchholz D., Junglas P.: On the existence of equilibrium states in local quantum field theory. Commun. Math. Phys. 121, 255–270 (1989)MathSciNetADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Buchholz, D., Longo, R.: Graded KMS functionals and the breakdown of supersymmetry. Adv. Theor. Math. Phys. 3, 615–626 (2000). Addendum: Adv. Theor. Math. Phys. 6, 1909–1910 (2000)Google Scholar
  9. 9.
    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, 20–56 (1988)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Buchholz D., Schulz-Mirbach H.: Haag duality in conformal quantum field theory. Rev. Math. Phys. 2, 105–125 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Camassa P., Longo R., Tanimoto Y., Weiner M.: Thermal states in conformal QFT. I. Commun. Math. Phys. 309, 703–735 (2012)MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Camassa P., Longo R., Tanimoto Y., Weiner M.: Thermal states in conformal QFT. II. Commun. Math. Phys. 315, 771–802 (2012)MathSciNetADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Carpi S.: On the representation theory of Virasoro nets. Commun. Math. Phys. 244, 261–284 (2004)MathSciNetADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Carpi S., Conti R., Hillier R., Weiner M.: Representations of conformal nets, universal C*-algebras and K-theory. Commun. Math. Phys. 320, 275–300 (2013)MathSciNetADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Carpi S., Hillier R., Kawahigashi Y., Longo R.: Spectral triples and the super-Virasoro algebra. Commun. Math. Phys. 295, 71–97 (2010)MathSciNetADSCrossRefzbMATHGoogle Scholar
  16. 16.
    Carpi, S., Hillier, R., Kawahigashi, Y., Longo, R., Xu, F.: N = 2 superconformal nets. arXiv:1207.2398v3 [math.OA] (2013)
  17. 17.
    Carpi, S., Hillier, R., Longo, R.: Superconformal nets and noncommutative geometry. J. Noncomm. Geom., to appear. arXiv:1304.4062v2 [math.OA] (2013)
  18. 18.
    Carpi S., Kawahigashi Y., Longo R.: Structure and classification of superconformal nets. Ann. Henri Poincaré 9, 1069–1121 (2008)MathSciNetADSCrossRefzbMATHGoogle Scholar
  19. 19.
    Connes A.: Noncommutative Geometry. Academic Press, New York (1994)zbMATHGoogle Scholar
  20. 20.
    Fröhlich J., Gabbiani F.: Operator algebras and conformal field theory. Commun. Math. Phys. 155, 569–640 (1993)ADSCrossRefzbMATHGoogle Scholar
  21. 21.
    Fredenhagen, K., Rehren, K.H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras II. Geometric aspects and conformal covariance. Rev. Math. Phys. (Special Issue), 113–157 (1992)Google Scholar
  22. 22.
    Haag R.: Local Quantum Physics. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  23. 23.
    Hillier R.: Local-entire cyclic cocycles for graded quantum field nets. Lett. Math. Phys. 104, 271–298 (2014)MathSciNetADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Jaffe A., Lesniewski A., Wisniowski M.: Deformations of super-KMS functionals. Commun. Math. Phys. 121, 527–540 (1989)MathSciNetADSCrossRefzbMATHGoogle Scholar
  25. 25.
    Kac V.G., Todorov I.T.: Superconformal current algebras and their unitary representations. Commun. Math. Phys. 102, 337–347 (1985)MathSciNetADSCrossRefzbMATHGoogle Scholar
  26. 26.
    Kadison R.V., Ringrose J.R.: Fundamentals of the Theory of Operator Algebras. Academic Press, New York (1986)zbMATHGoogle Scholar
  27. 27.
    Kastler D.: Cyclic cocycles from graded KMS functionals. Commun. Math. Phys. 121, 345–350 (1989)MathSciNetADSCrossRefGoogle Scholar
  28. 28.
    Kawahigashi Y., Longo R.: Noncommutative spectral invariants and black hole entropy. Commun. Math. Phys. 257, 193–225 (2005)MathSciNetADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Kawahigashi Y., Longo R., Müger M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001)ADSCrossRefzbMATHGoogle Scholar
  30. 30.
    Longo R.: Notes for a quantum index theorem. Commun. Math. Phys. 222, 45–96 (2001)MathSciNetADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Martin P., Schwinger J.: Theory of many-particle systems. I. Phys. Rev. 115, 1342–1373 (1959)MathSciNetADSCrossRefzbMATHGoogle Scholar
  32. 32.
    Moriya, H.: Supersymmetric C*-dynamical systems. arXiv:1001.2622 [math.OA] (2010)
  33. 33.
    Moriya H.: On GNS representation of supersymmetric states in C*-dynamical systems. Mathematical quantum field theory and renormalization theory. COE Lect. Note 30, 39–47 (2011)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Powers R.T., Størmer E.: Free states of the canonical anticommutation relations. Commun. Math. Phys. 16, 1–33 (1970)ADSCrossRefzbMATHGoogle Scholar
  35. 35.
    Remmert R.: Theory of Complex Functions. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  36. 36.
    Rocca F., Sirugue M., Testard D.: On a class of equilibrium states under the Kubo-Martin-Schwinger boundary condition. Fermions. Commun. Math. Phys. 13, 317–334 (1969)MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    Rocca F., Sirugue M., Testard D.: On a class of equilibrium states under the Kubo-Martin-Schwinger boundary condition. Bosons. Commun. Math. Phys. 19, 119–141 (1970)MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    Stoytchev O.: Modular conjugation and the implementation of supersymmetry. Lett. Math. Phys. 79, 235–249 (2007)MathSciNetADSCrossRefzbMATHGoogle Scholar
  39. 39.
    Takesaki M., Winnink M.: Local normality in quantum statistical mechanics. Commun. Math. Phys. 30, 129–152 (1973)MathSciNetADSCrossRefzbMATHGoogle Scholar
  40. 40.
    Takesaki M.: Theory of operator algebras I. Springer, Berlin (1979)CrossRefGoogle Scholar
  41. 41.
    Weiner, M.: Conformal covariance and related properties of chiral QFT. PhD Thesis. Università di Roma “Tor Vergata”, arXiv:math/0703336 [math.OA] (2005)
  42. 42.
    Xu F.: Strong additivity and conformal nets. Pac. J. Math. 221, 167–199 (2005)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLancaster UniversityLancasterUK

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