Communications in Mathematical Physics

, Volume 357, Issue 1, pp 249–266 | Cite as

Rotational KMS States and Type I Conformal Nets

  • Roberto LongoEmail author
  • Yoh Tanimoto


We consider KMS states on a local conformal net on S 1 with respect to rotations. We prove that, if the conformal net is of type I, namely if it admits only type I DHR representations, then the extremal KMS states are the Gibbs states in an irreducible representation. Completely rational nets, the U(1)-current net, the Virasoro nets and their finite tensor products are shown to be of type I. In the completely rational case, we also give a direct proof that all factorial KMS states are Gibbs states.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. BD84.
    Buchholz, D., Doplicher, S.: Exotic infrared representations of interacting systems. Ann. Inst. H. Poincaré Phys. Théor. 40(2), 175–184 (1984).
  2. BDL90.
    Buchholz D., D’Antoni C., Longo R.: Nuclear maps and modular structures. I. General properties. J. Funct. Anal. 88(2), 233–250 (1990). doi: 10.1016/0022-1236(90)90104-S MathSciNetCrossRefzbMATHGoogle Scholar
  3. BDL07.
    Buchholz D., D’Antoni C., Longo R.: Nuclearity and thermal states in conformal field theory. Commun. Math. Phys. 270(1), 267–293 (2007) arXiv:math-ph/0603083 ADSMathSciNetCrossRefzbMATHGoogle 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. 5B, 20–56 (1988).
  5. BnTZ92.
    Bañados M., Teitelboim C., Zanelli J.: Black hole in three-dimensional spacetime. Phys. Rev. Lett. 69, 1849–1851 (1992) arXiv:hep-th/9204099 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. BR97.
    Bratteli O., Robinson Derek W.: Operator Algebras and Quantum Statistical Mechanics. 2. Equilibrium States. Models in Quantum Statistical Mechanics. Texts and Monographs in Physics. Springer, Berlin (1997)zbMATHGoogle Scholar
  7. BSM90.
    Buchholz, D., Schulz-Mirbach, H.: Haag duality in conformal quantum field theory. Rev. Math. Phys. 2(1), 105–125 (1990).
  8. Car04.
    Carpi S.: On the representation theory of Virasoro nets. Commun. Math. Phys. 244(2), 261–284 (2004) arXiv:math/0306425 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. CCHW13.
    Carpi S., Conti R., Hillier R., Weiner M.: Representations of conformal nets, universal \({{\rm C}^*}\) -algebras and K-theory. Commun. Math. Phys. 320(1), 275–300 (2013) arXiv:1202.2543 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. CLTW12a.
    Camassa P., Longo R., Tanimoto Y., Weiner M.: Thermal states in conformal QFT. I. Commun. Math. Phys. 309(3), 703–735 (2012) arXiv:1101.2865 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. CLTW12b.
    Camassa P., Longo R., Tanimoto Y., Weiner M.: Thermal states in conformal QFT. II.. Commun. Math. Phys. 315(3), 771–802 (2012) arXiv:1109.2064 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. CW05.
    Carpi S., Weiner M.: On the uniqueness of diffeomorphism symmetry in conformal field theory. Commun. Math. Phys. 258(1), 203–221 (2005) arXiv:math/0407190 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. CW16.
    Carpi, S., Weiner, M.: Local Energy Bounds and Representations of Conformal Nets (2016). (in preparation) Google Scholar
  14. DFG84.
    Doplicher, S., Figliolini, F., Guido, D.: Infrared representations of free Bose fields. Ann. Inst. H. Poincaré Phys. Théor. 41(1), 49–62 (1984).
  15. DFK04.
    D’Antoni C., Fredenhagen K., Köster S.: Implementation of conformal covariance by diffeomorphism symmetry. Lett. Math. Phys. 67(3), 239–247 (2004) arXiv:math-ph/0312017 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. Dix77.
    Dixmier, J.: C *-Algebras. North-Holland Publishing Co., Amsterdam (1977). (Translated from the French by Francis Jellett, North-Holland Mathematical Library, Vol. 15) Google Scholar
  17. DL84.
    Doplicher, S., Longo, R.: Standard and split inclusions of von Neumann algebras. Invent. Math. 75(3), 493–536 (1984).
  18. DS82.
    Doplicher, S., Spera, M.: Representations obeying the spectrum condition. Commun. Math. Phys. 84(4), 505–513 (1982).
  19. DS83.
    Doplicher, S., Spera, M.: Local normality properties of some infrared representations. Commun. Math. Phys. 89(1), 19–25(1983).
  20. 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).
  21. Fre90.
    Fredenhagen, K.: Generalizations of the theory of superselection sectors. In: The Algebraic Theory of Superselection Sectors (Palermo, 1989), pp. 379–387. World Scientific Publishing, River Edge, NJ (1990).
  22. GKO86.
    Goddard, P., Kent, A., Olive, D.: Unitary representations of the Virasoro and super-Virasoro algebras. Commun. Math. Phys. 103(1), 105–119 (1986).
  23. GL92.
    Guido, D., Longo, R.: Relativistic invariance and charge conjugation in quantum field theory. Commun. Math. Phys. 148(3), 521–551 (1992).
  24. GL96.
    Guido, D., Longo, R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181(1), 11–35 (1996).
  25. GL14.
    Garbarz A., Leston M.: Classification of boundary gravitons in ads3 gravity. J. High Energy Phys. 2014(5), 1–26 (2014). doi: 10.1007/JHEP05(2014)141 CrossRefGoogle Scholar
  26. GW85.
    Goodman, R., Wallach, N.R.: Projective unitary positive-energy representations of \({{\rm Diff}(S^1)}\) . J. Funct. Anal. 63(3), 299–321 (1985).
  27. Haa96.
    Haag R.: Local Quantum Physics. Texts and Monographs in Physics. Springer, Berlin (1996)CrossRefGoogle Scholar
  28. Hil15.
    Hillier R.: Super-KMS functionals for graded-local conformal nets. Ann. Henri Poincaré. 16(8), 1899–1936 (2015) arXiv:1204.5078 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. Iov15.
    Iovieno, S.: Stati KMS nella teoria dei campi conformi. Tesi di Laurea Magistrale. Sapienza Università di Roma (2015)Google Scholar
  30. Jec78.
    Jech, T.: Set Theory. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London (1978) (Pure and Applied Mathematics) Google Scholar
  31. KL04.
    Kawahigashi Y., Longo R.: Classification of local conformal nets. Case c < 1. Ann. Math. (2) 160(2), 493–522 (2004) arXiv:math-ph/0201015 MathSciNetCrossRefzbMATHGoogle Scholar
  32. KLM01.
    Kawahigashi Y., Longo R., Müger M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219((3), 631–669 (2001) arXiv:math/9903104 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 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. World Scientific Publishing Co. Inc., Teaneck, NJ (1987)zbMATHGoogle Scholar
  34. Lon08.
    Longo, R.: Real Hilbert subspaces, modular theory, \({{\rm SL}(2,R)}\) and CFT. In: Von Neumann Algebras in Sibiu: Conference Proceedings, pp. 33–91. Theta, Bucharest (2008).
  35. LX04.
    Longo R., Xu F.: Topological sectors and a dichotomy in conformal field theory. Commun. Math. Phys. 251(2), 321–364 (2004) arXiv:math/0309366 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. MTW16.
    Morinelli, V., Tanimoto, Y., Weiner. M.: Conformal Covariance and the Split Property. Commun. Math. Phys (2016) (to appear). arXiv:1609.02196
  37. MW10.
    Maloney, A., Witten, E.: Quantum gravity partition functions in three dimensions. J. High Energy Phys. (2) 029, 58 (2010). arXiv:0712.0155
  38. NS15.
    Neeb, K.-H., Salmasian, H.: Classification of positive energy representations of the Virasoro group. Int. Math. Res. Not. IMRN, (18), 8620–8656 (2015). arXiv:1402.6572
  39. Sch94.
    Schroer, B.: Some Useful Properties of Rotational Gibbs States in Chiral Conformal QFT (1994). arXiv:hep-th/9405105
  40. Sup60.
    Suppes, P.: Axiomatic Set Theory. The University Series in Undergraduate Mathematics. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York (1960)Google Scholar
  41. Tak02.
    Takesaki, M.: Theory of Operator Algebras. I, Volume 124 of Encyclopaedia of Mathematical Sciences. Springer, Berlin (2002). (Reprint of the first (1979) edition, Operator Algebras and Non-commutative Geometry, 5) Google Scholar
  42. Tak03a.
    Takesaki, M.: Theory of Operator Algebras. II, Volume 125 of Encyclopaedia of Mathematical Sciences. Springer, Berlin (2003). (Operator Algebras and Non-commutative Geometry, 6) Google Scholar
  43. Tak03b.
    Takesaki, M.: Theory of Operator Algebras. III, Volume 127 of Encyclopaedia of Mathematical Sciences. Springer, Berlin (2003). (Operator Algebras and Non-commutative Geometry, 8) Google Scholar
  44. TW73.
    Takesaki, M., Winnink, M.: Local normality in quantum statistical mechanics. Commun. Math. Phys. 30, 129–152 (1973).
  45. Wei06.
    Weiner M.: Conformal covariance and positivity of energy in charged sectors. Commun. Math. Phys. 265(2), 493–506 (2006) arXiv:math-ph/0507066 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  46. Wei16.
    Weiner M.: Local equivalence of representations of \({\mathrm{Diff}^+({S}^1)}\) corresponding to different highest weights. Commun. Math. Phys. 352(2), 759–772 (2017). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. Wit88.
    Witten, E.: Coadjoint orbits of the Virasoro group. Commun. Math. Phys. 114(1), 1–53 (1988).

Copyright information

© The Author(s) 2017

Authors and Affiliations

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

Personalised recommendations