Advertisement

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
Article

Abstract

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BD84.
    Buchholz, D., Doplicher, S.: Exotic infrared representations of interacting systems. Ann. Inst. H. Poincaré Phys. Théor. 40(2), 175–184 (1984). https://eudml.org/doc/76232
  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). https://www.researchgate.net/publication/222585851
  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). https://www.researchgate.net/publication/246352668
  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). https://eudml.org/doc/76250
  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). https://eudml.org/doc/143108
  18. DS82.
    Doplicher, S., Spera, M.: Representations obeying the spectrum condition. Commun. Math. Phys. 84(4), 505–513 (1982). http://projecteuclid.org/euclid.cmp/1103921286
  19. DS83.
    Doplicher, S., Spera, M.: Local normality properties of some infrared representations. Commun. Math. Phys. 89(1), 19–25(1983). http://projecteuclid.org/euclid.cmp/1103922588
  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). https://projecteuclid.org/euclid.cmp/1104286114
  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). https://www.researchgate.net/publication/281349333
  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). https://projecteuclid.org/euclid.cmp/1104114626
  23. GL92.
    Guido, D., Longo, R.: Relativistic invariance and charge conjugation in quantum field theory. Commun. Math. Phys. 148(3), 521–551 (1992). http://projecteuclid.org/euclid.cmp/1104251044
  24. GL96.
    Guido, D., Longo, R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181(1), 11–35 (1996). http://projecteuclid.org/euclid.cmp/1104287623
  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). http://www.sciencedirect.com/science/article/pii/0022123685900904
  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). http://www.mat.uniroma2.it/longo/Lecture_Notes_files/LN-Part1.pdf
  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). https://projecteuclid.org/euclid.cmp/1103858808
  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). https://arxiv.org/abs/1606.00344 ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. Wit88.
    Witten, E.: Coadjoint orbits of the Virasoro group. Commun. Math. Phys. 114(1), 1–53 (1988). https://projecteuclid.org/euclid.cmp/1104160487

Copyright information

© The Author(s) 2017

Authors and Affiliations

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

Personalised recommendations