Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Araki H., Kastler D., Takesaki M., Haag R.: Extension of KMS states and chemical potential. Commun. Math. Phys. 53(2), 97–134 (1977)
Böckenhauer J.: Localized endomorphisms of the chiral Ising model. Commun. Math. Phys. 177, 265–304 (1996)
Bratteli, O., Robinson, D.: Operator algebras and quantum statistical mechanics. Vol. 2, Berlin: Springer-Verlag, 1997
Buchholz D., D’Antoni C., Longo R.: Nuclearity and thermal states in conformal field theory. Commun. Math. Phys. 270, 267–293 (2007)
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)
Buchholz D., Schulz-Mirbach H.: Haag duality in conformal quantum field theory. Rev. Math. Phys. 2, 105–125 (1990)
Camassa P., Longo R., Tanimoto Y., Weiner M.: Thermal states in conformal QFT. I. Commun. Math. Phys. 309, 703–735 (2011)
Carpi S.: Classification of subsystems for the Haag-Kastler nets generated by c = 1 chiral current algebras. Lett. Math. Phys. 47, 353–364 (1999)
Carpi S.: The Virasoro algebra and sectors with infinite statistical dimension. Ann. Henri Poincaré 4, 601–611 (2003)
Carpi S.: On the representation theory of Virasoro nets. Commun. Math. Phys. 244, 261–284 (2004)
Carpi S., Kawahigashi Y., Longo R.: Structure and classification of superconformal nets. Ann. Henri Poincaré. 9, 1069–1121 (2008)
D’Antoni C., Longo R., Radulescu F.: Conformal nets, maximal temperature and models from free probability. J. Op. Theory 45, 195–208 (2001)
Dixmier, J.: Von Neumann algebras. Amsterdam: North-Holland, 1981
Dixmier, J.: C *-algebras. Amsterdam: North-Holland, 1982
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)
Fewster C.J., Hollands S.: Quantum energy inequalities in two-dimensional conformal field theory. Rev. Math. Phys. 17, 577–612 (2005)
Fröhlich J., Gabbiani F.: Operator algebras and conformal field theory. Commun. Math. Phys. 155(3), 569–640 (1993)
Haag, R.:Local quantum physics. Fields, particles, algebras. Berlin: Springer-Verlag, 1996
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. 113–131
Kastler, D.: Equilibrium states of matter and operator algebras, Symposia Mathematica, Vol. XX. London-New York: Academic Press, 1976
Kawahigashi Y., Longo R.: Classification of local conformal nets. Case c < 1 Ann. of Math. 160, 493–522 (2004)
Kawahigashi Y., Longo R., Müger M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001)
Knapp, A.: Representation Theory of Semisimple Groups: An Overview Based on Examples. Princeton, NJ: Princeton University Press, 2001
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. 33–91
Longo R., Peligrad C.: Noncommutative topological dynamics and compact actions on C *-algebras. J. Funct. Anal. 58(2), 157–174 (1984)
Longo R., Rehren K.-H.: Nets of subfactors. Rev. Math. Phys. 7(4), 567–597 (1995)
Longo, R., Rehren, K.-H.: Boundary quantum field theory on the interior of the Lorentz hyperboloid. Commun. Math. Phys., to appear
Longo R., Xu F.: Topological sectors and a dichotomy in conformal field theory. Commun. Math. Phys. 251(2), 321–364 (2004)
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)
Mack G., Schomerus V.: Conformal field algebras with quantum symmetry from the theory of superselection sectors. Commun. Math. Phys. 134, 139–196 (1990)
Mather J.N.: Commutators of diffeomorphisms. Comment. Math. Helv. 49, 512–528 (1974)
Meisters G.H.: Translation-invariant linear forms and a formula for the Dirac measure. J. Funct. Anal. 8, 173–188 (1971)
Ottesen, J.T.: Infinite-dimensional groups and algebras in quantum physics. Berlin: Springer-Verlag, 1995
Reed, M., Simons, B.: Methods of modern mathematical physics. Vol. 2: Fourier analysis, self-adjointness. London-New York: Academic Press, 1975
Rehren K.-H.: A new view of the Virasoro algebra. Lett. Math. Phys. 30, 125–130 (1994)
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)
Schmüdgen K.: Strongly commuting selfadjoint operators and commutants of unbounded operator algebras. Proc. Amer. Math. Soc. 102, 365–372 (1988)
Schroer B., Wiesbrock H.-W.: Looking beyond the thermal horizon: hidden symmetries in chiral models. Rev. Math. Phys. 12(3), 461–473 (2000)
Takesaki M., Winnink M.: Local normality in quantum statistical mechanics. Commun. Math. Phys. 30, 129–152 (1973)
Wang, Y.: Locally normal KMS states of diffeomorphism covariant nets w.r.t. translation subgroups. Ph.D. thesis Univ. of Roma “Tor Vergata”, 2008
Acknowledgments
We would like to thank the referee for pointing out imprecise statements in Appendix A.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Dedicated to Rudolf Haag on the occasion of his 90th birthday
Research supported in part by the ERC Advanced Grant 227458 OACFT “Operator Algebras and Conformal Field Theory”, PRIN-MIUR, GNAMPA-INDAM and EU network “Noncommutative Geometry” MRTN-CT-2006-0031962.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Camassa, P., Longo, R., Tanimoto, Y. et al. Thermal States in Conformal QFT. II. Commun. Math. Phys. 315, 771–802 (2012). https://doi.org/10.1007/s00220-012-1514-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-012-1514-z