Communications in Mathematical Physics

, Volume 315, Issue 3, pp 771–802

Thermal States in Conformal QFT. II

  • Paolo Camassa
  • Roberto Longo
  • Yoh Tanimoto
  • Mihály Weiner
Open Access


We continue the analysis of the set of locally normal KMS states w.r.t. the translation group for a local conformal net \({\mathcal{A}}\) of von Neumann algebras on \({\mathbb{R}}\) . In the first part we have proved the uniqueness of the KMS state on every completely rational net. In this second part, we exhibit several (non-rational) conformal nets which admit continuously many primary KMS states. We give a complete classification of the KMS states on the U(1)-current net and on the Virasoro net Vir1 with the central charge c = 1, whilst for the Virasoro net Vir c with c > 1 we exhibit a (possibly incomplete) list of continuously many primary KMS states. To this end, we provide a variation of the Araki-Haag-Kastler-Takesaki theorem within the locally normal system framework: if there is an inclusion of split nets \({\mathcal{A}\subset \mathcal{B}}\) and \({\mathcal{A}}\) is the fixed point of \({\mathcal{B}}\) w.r.t. a compact gauge group, then any locally normal, primary KMS state on \({\mathcal{A}}\) extends to a locally normal, primary state on \({\mathcal{B}}\) , KMS w.r.t. a perturbed translation. Concerning the non-local case, we show that the free Fermi model admits a unique KMS state.


  1. 1.
    Araki H., Kastler D., Takesaki M., Haag R.: Extension of KMS states and chemical potential. Commun. Math. Phys. 53(2), 97–134 (1977)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Böckenhauer J.: Localized endomorphisms of the chiral Ising model. Commun. Math. Phys. 177, 265–304 (1996)ADSMATHCrossRefGoogle Scholar
  3. 3.
    Bratteli, O., Robinson, D.: Operator algebras and quantum statistical mechanics. Vol. 2, Berlin: Springer-Verlag, 1997Google Scholar
  4. 4.
    Buchholz D., D’Antoni C., Longo R.: Nuclearity and thermal states in conformal field theory. Commun. Math. Phys. 270, 267–293 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    Buchholz D., Mack G., Todorov I.: The current algebra on the circle as a germ of local field theories. Nucl. Phys. B (Proc. Suppl.) 5B, 20–56 (1988)MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Buchholz D., Schulz-Mirbach H.: Haag duality in conformal quantum field theory. Rev. Math. Phys. 2, 105–125 (1990)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Camassa P., Longo R., Tanimoto Y., Weiner M.: Thermal states in conformal QFT. I. Commun. Math. Phys. 309, 703–735 (2011)MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Carpi S.: Classification of subsystems for the Haag-Kastler nets generated by c = 1 chiral current algebras. Lett. Math. Phys. 47, 353–364 (1999)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Carpi S.: The Virasoro algebra and sectors with infinite statistical dimension. Ann. Henri Poincaré 4, 601–611 (2003)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Carpi S.: On the representation theory of Virasoro nets. Commun. Math. Phys. 244, 261–284 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Carpi S., Kawahigashi Y., Longo R.: Structure and classification of superconformal nets. Ann. Henri Poincaré. 9, 1069–1121 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    D’Antoni C., Longo R., Radulescu F.: Conformal nets, maximal temperature and models from free probability. J. Op. Theory 45, 195–208 (2001)MathSciNetMATHGoogle Scholar
  13. 13.
    Dixmier, J.: Von Neumann algebras. Amsterdam: North-Holland, 1981Google Scholar
  14. 14.
    Dixmier, J.: C *-algebras. Amsterdam: North-Holland, 1982Google Scholar
  15. 15.
    Epstein, D.B.A.: Commutators of C -diffeomorphisms. Appendix to: “A curious remark concerning the geometric transfer map” by John N. Mather [Comment. Math. Helv. 59(1), 86–110 (1984)] Comment. Math. Helv. 59, 111–122 (1984)Google Scholar
  16. 16.
    Fewster C.J., Hollands S.: Quantum energy inequalities in two-dimensional conformal field theory. Rev. Math. Phys. 17, 577–612 (2005)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Fröhlich J., Gabbiani F.: Operator algebras and conformal field theory. Commun. Math. Phys. 155(3), 569–640 (1993)ADSMATHCrossRefGoogle Scholar
  18. 18.
    Haag, R.:Local quantum physics. Fields, particles, algebras. Berlin: Springer-Verlag, 1996Google Scholar
  19. 19.
    Hardy, G.H.: Asymptotic theory of partitions. In: Ramanujan: Twelve lectures on subjects suggested by his life and work, 3rd ed., New York: Chelsea, 1999, pp. 113–131Google Scholar
  20. 20.
    Kastler, D.: Equilibrium states of matter and operator algebras, Symposia Mathematica, Vol. XX. London-New York: Academic Press, 1976Google Scholar
  21. 21.
    Kawahigashi Y., Longo R.: Classification of local conformal nets. Case c < 1 Ann. of Math. 160, 493–522 (2004)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Kawahigashi Y., Longo R., Müger M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001)ADSMATHCrossRefGoogle Scholar
  23. 23.
    Knapp, A.: Representation Theory of Semisimple Groups: An Overview Based on Examples. Princeton, NJ: Princeton University Press, 2001Google Scholar
  24. 24.
    Longo, R.: Real Hilbert subspaces, modular theory, \({\rm SL(2,\mathbb{R})}\) and CFT. In: Von Neumann algebas in Sibiu: Conference Proceedings. Bucharest: Theta, 2008, pp. 33–91Google Scholar
  25. 25.
    Longo R., Peligrad C.: Noncommutative topological dynamics and compact actions on C *-algebras. J. Funct. Anal. 58(2), 157–174 (1984)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Longo R., Rehren K.-H.: Nets of subfactors. Rev. Math. Phys. 7(4), 567–597 (1995)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Longo, R., Rehren, K.-H.: Boundary quantum field theory on the interior of the Lorentz hyperboloid. Commun. Math. Phys., to appearGoogle Scholar
  28. 28.
    Longo R., Xu F.: Topological sectors and a dichotomy in conformal field theory. Commun. Math. Phys. 251(2), 321–364 (2004)MathSciNetADSMATHCrossRefGoogle Scholar
  29. 29.
    Manuceau J.: Étude de quelques automorphismes de la C *-algèbre du champ de bosons libres. Ann. Inst. H. Poincarè Sect. A (N.S.) 8, 117–138 (1968)MathSciNetMATHGoogle Scholar
  30. 30.
    Mack G., Schomerus V.: Conformal field algebras with quantum symmetry from the theory of superselection sectors. Commun. Math. Phys. 134, 139–196 (1990)MathSciNetADSMATHCrossRefGoogle Scholar
  31. 31.
    Mather J.N.: Commutators of diffeomorphisms. Comment. Math. Helv. 49, 512–528 (1974)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Meisters G.H.: Translation-invariant linear forms and a formula for the Dirac measure. J. Funct. Anal. 8, 173–188 (1971)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Ottesen, J.T.: Infinite-dimensional groups and algebras in quantum physics. Berlin: Springer-Verlag, 1995Google Scholar
  34. 34.
    Reed, M., Simons, B.: Methods of modern mathematical physics. Vol. 2: Fourier analysis, self-adjointness. London-New York: Academic Press, 1975Google Scholar
  35. 35.
    Rehren K.-H.: A new view of the Virasoro algebra. Lett. Math. Phys. 30, 125–130 (1994)MathSciNetADSMATHCrossRefGoogle Scholar
  36. 36.
    Rocca F., Sirigue M., Testard D.: On a class of equilibrium states under the Kubo-Martin-Schwinger condition II. Bosons. Commun. Math. Phys. 19, 119–141 (1970)ADSCrossRefGoogle Scholar
  37. 37.
    Schmüdgen K.: Strongly commuting selfadjoint operators and commutants of unbounded operator algebras. Proc. Amer. Math. Soc. 102, 365–372 (1988)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Schroer B., Wiesbrock H.-W.: Looking beyond the thermal horizon: hidden symmetries in chiral models. Rev. Math. Phys. 12(3), 461–473 (2000)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Takesaki M., Winnink M.: Local normality in quantum statistical mechanics. Commun. Math. Phys. 30, 129–152 (1973)MathSciNetADSCrossRefGoogle Scholar
  40. 40.
    Wang, Y.: Locally normal KMS states of diffeomorphism covariant nets w.r.t. translation subgroups. Ph.D. thesis Univ. of Roma “Tor Vergata”, 2008Google Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  • Paolo Camassa
    • 1
  • Roberto Longo
    • 1
  • Yoh Tanimoto
    • 1
  • Mihály Weiner
    • 1
    • 2
  1. 1.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly
  2. 2.Department of AnalysesBudapest University of Technology and EconomicsBudapestHungary

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