Algebraic Conformal Quantum Field Theory in Perspective



Conformal quantum field theory is reviewed in the perspective of Axiomatic, notably Algebraic QFT. This theory is particularly developed in two spacetime dimensions, where many rigorous constructions are possible, as well as some complete classifications. The structural insights, analytical methods and constructive tools are expected to be useful also for four-dimensional QFT.


  1. 1.
    Alazzawi, S.: Deformation of fermionic quantum field theories and integrable models. Lett. Math. Phys. 103, 37–58 (2012)MathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Alazzawi, S.: Deformation of quantum field theories and the construction of interacting models. Ph.D. thesis, University of Vienna (2015)Google Scholar
  3. 3.
    Baumann, K.: There are no scalar Lie fields in three or more dimensional space-time. Commun. Math. Phys. 47, 69–74 (1976)MathSciNetCrossRefADSMATHGoogle Scholar
  4. 4.
    Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333–380 (1984)MathSciNetCrossRefADSMATHGoogle Scholar
  5. 5.
    Bischoff, M., Tanimoto, Y.: Construction of wedge-local nets of observables through Longo-Witten endomorphisms. II. Commun. Math. Phys. 317, 667–695 (2013)MathSciNetCrossRefADSMATHGoogle Scholar
  6. 6.
    Bischoff, M., Kawahigashi, Y., Longo, R.: Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case. arXiv:1410.8848
  7. 7.
    Bischoff, M., Kawahigashi, Y., Longo, R., Rehren, K.-H.: Phase boundaries in algebraic conformal QFT. arXiv:1405.7863
  8. 8.
    Bischoff, M., Kawahigashi, Y., Longo, R., Rehren, K.-H.: Tensor categories and endomorphisms of von Neumann algebras, with applications to quantum field theory. SpringerBriefs in Mathematical Physics, vol. 3 (2015). arXiv:1407.4793v3
  9. 9.
    Böckenhauer, J., Evans, D.: Modular invariants, graphs and \(\alpha \)-induction for nets of subfactors. II. Commun. Math. Phys. 200, 57–103 (1999)CrossRefADSMATHGoogle Scholar
  10. 10.
    Böckenhauer, J., Evans, D., Kawahigashi, Y.: On \(\alpha \)-induction, chiral projectors and modular invariants for subfactors. Commun. Math. Phys. 208, 429–487 (1999)CrossRefADSMATHGoogle Scholar
  11. 11.
    Böckenhauer, J., Evans, D., Kawahigashi, Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys. 210, 733–784 (2000)CrossRefADSMATHGoogle Scholar
  12. 12.
    Borchers, H.-J.: The CPT-theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992)MathSciNetCrossRefADSMATHGoogle Scholar
  13. 13.
    Brunetti, R., Guido, D., Longo, R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys. 156, 201–219 (1993)MathSciNetCrossRefADSMATHGoogle Scholar
  14. 14.
    Brunetti, R., Guido, D., Longo, R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–785 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)MathSciNetCrossRefADSMATHGoogle Scholar
  16. 16.
    Buchholz, D., Haag, R.: The quest for understanding in relativistic quantum physics. J. Math. Phys. 41, 3674–3697 (2000)MathSciNetCrossRefADSMATHGoogle Scholar
  17. 17.
    Buchholz, D., Roberts, J.E.: New light on infrared problems: sectors, statistics, symmetries and spectrum. Commun. Math. Phys. 330, 935–972 (2014)MathSciNetCrossRefADSMATHGoogle Scholar
  18. 18.
    Buchholz, D., Wichmann, E.: Causal independence and energy-level density in quantum field theory. Commun. Math. Phys. 106, 321–344 (1986)MathSciNetCrossRefADSMATHGoogle Scholar
  19. 19.
    Buchholz, D., Lechner, G., Summers, S.J.: Warped convolutions, Rieffel deformations and the construction of quantum field theories. Commun. Math. Phys. 304, 95–123 (2011)MathSciNetCrossRefADSMATHGoogle Scholar
  20. 20.
    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)MathSciNetCrossRefADSMATHGoogle Scholar
  21. 21.
    Cappelli, A., Itzykson, C., Zuber, J.-B.: The A-D-E classification of minimal and \(A^{(1)}_1\) conformal invariant theories. Commun. Math. Phys. 113, 1–26 (1987)MathSciNetCrossRefADSMATHGoogle Scholar
  22. 22.
    Carpi, S., Kawahigashi, Y., Longo, R.: How to add a boundary condition. Commun. Math. Phys. 322, 149–166 (2013)MathSciNetCrossRefADSMATHGoogle Scholar
  23. 23.
    Carpi, S., Kawahigashi, Y., Longo, R., Weiner, M.: From vertex operator algebras to conformal nets and back. arXiv:1503.01260
  24. 24.
    Dolan, F.A., Osborn, H.: Conformal four point functions and the operator product expansion. Nucl. Phys. B 599, 459–496 (2001)MathSciNetCrossRefADSMATHGoogle Scholar
  25. 25.
    Doplicher, S., Roberts, J.E.: Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics. Commun. Math. Phys. 131, 51–207 (1990)MathSciNetCrossRefADSMATHGoogle Scholar
  26. 26.
    Doplicher, S., Haag, R., Roberts, J.E.: Local observables and particle statistics. I, Commun. Math. Phys. 23, 199–230 (1971), and II, Commun. Math. Phys. 35, 49–85 (1974)Google Scholar
  27. 27.
    Fredenhagen, K., Rehren, K.-H., Schroer, B.: Superselection sectors with braid group statistics and exchange algebras. I, Commun. Math. Phys. 125, 201–226 (1989), and II, Rev. Math. Phys. SI1 (Special Issue) 113–157 (1992)Google Scholar
  28. 28.
    Frenkel, J.B., Kac, V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62, 23–66 (1980)MathSciNetCrossRefADSMATHGoogle Scholar
  29. 29.
    Friedan, D., Qiu, Z., Shenker, S.: Conformal invariance, unitarity, and critical exponents in two dimensions. Phys. Rev. Lett. 52, 1575–1578 (1984)MathSciNetCrossRefADSGoogle Scholar
  30. 30.
    Fröhlich, J., Fuchs, J., Runkel, I., Schweigert, C.: Kramers-Wannier duality from conformal defects. Phys. Rev. Lett. 93, 070601 (2004)MathSciNetCrossRefADSGoogle Scholar
  31. 31.
    Fröhlich, J., Fuchs, J., Runkel, I., Schweigert, C.: Correspondences of ribbon categories. Ann. Math. 199, 192–329 (2006)MATHGoogle Scholar
  32. 32.
    Goddard, P., Kent, A., Olive, D.: Unitary representations of the Virasoro and super-Virasoro algebras. Commun. Math. Phys. 103, 105–119 (1986)MathSciNetCrossRefADSMATHGoogle Scholar
  33. 33.
    Guido, D., Longo, R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181, 11–35 (1996)MathSciNetCrossRefADSMATHGoogle Scholar
  34. 34.
    Guido, D., Longo, R., Wiesbrock, H.-W.: Extensions of conformal nets and superselection structures. Commun. Math. Phys. 192, 217–244 (1998)MathSciNetCrossRefADSMATHGoogle Scholar
  35. 35.
    Haag, R.: Local Quantum Physics. Springer, Berlin (1996)CrossRefMATHGoogle Scholar
  36. 36.
    Izumi, M., Kosaki, H.: On a subfactor analogue of the second cohomology. Rev. Math. Phys. 14, 733–737 (2002)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Kac, V.G.: Infinite Dimensional Lie Algebras. Cambridge University Press, Cambridge (1985)MATHGoogle Scholar
  38. 38.
    Kähler, R., Wiesbrock, H.-W.: Modular theory and the reconstruction of four-dimensional quantum field theories. J. Math. Phys. 42, 74–86 (2001)MathSciNetCrossRefADSMATHGoogle Scholar
  39. 39.
    Kawahigashi, Y., Longo, R.: Classification of local conformal nets. Case \(c<1\). Ann. Math. 160, 493–522 (2004)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Kawahigashi, Y., Longo, R.: Classification of two-dimensional local conformal nets with \(c<1\) and 2-cohomology vanishing for tensor categories. Commun. Math. Phys. 244, 63–97 (2004)MathSciNetCrossRefADSMATHGoogle Scholar
  41. 41.
    Kawahigashi, Y., Longo, R.: Local conformal nets arising from framed vertex operator algebras. Adv. Math. 206, 729–751 (2006)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Kawahigashi, Y., Longo, R., Müger, M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001)CrossRefADSMATHGoogle Scholar
  43. 43.
    Kirillov Jr, A., Ostrik, V.: On \(q\)-analog of McKay correspondence and ADE classification of sl(2) conformal field theories. Adv. Math. 171, 183–227 (2002)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Kong, L., Runkel, I.: Morita classes of algebras in modular tensor categories. Adv. Math. 219, 1548–1576 (2008)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Köhler, Ch.: On the localization properties of quantum fields with zero mass and infinite spin, Ph.D. thesis, Vienna (2015)Google Scholar
  46. 46.
    Lechner, G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008)MathSciNetCrossRefADSMATHGoogle Scholar
  47. 47.
    Lechner, G., Schlemmer, J., Tanimoto, Y.: On the equivalence of two deformation schemes in quantum field theory. Lett. Math. Phys. 103, 421–437 (2013)MathSciNetCrossRefADSMATHGoogle Scholar
  48. 48.
    Longo, R., Morinelli, V., Rehren, K.-H.: Where infinite-spin particles are localizable. Commun. Math. Phys. arXiv:1505.01759 (to appear)
  49. 49.
    Longo, R.: Conformal subnets and intermediate subfactors. Commun. Math. Phys. 237, 7–30 (2003)MathSciNetCrossRefADSMATHGoogle Scholar
  50. 50.
    Longo, R., Rehren, K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567–597 (1995)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Longo, R., Rehren, K.-H.: Local fields in boundary CFT. Rev. Math. Phys. 16, 909–960 (2004)MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Longo, R., Rehren, K.-H.: How to remove the boundary in CFT—an operator algebraic procedure. Commun. Math. Phys. 285, 1165–1182 (2009)MathSciNetCrossRefADSMATHGoogle Scholar
  53. 53.
    Longo, R., Witten, E.: An algebraic construction of boundary quantum field theory. Commun. Math. Phys. 303, 213–232 (2011)MathSciNetCrossRefADSMATHGoogle Scholar
  54. 54.
    Longo, R., Xu, F.: Topological sectors and a dichotomy in conformal field theory. Commun. Math. Phys. 251, 321–364 (2004)MathSciNetCrossRefADSMATHGoogle Scholar
  55. 55.
    Lüscher, M., Mack, G.: Global conformal invariance in quantum field theory. Commun. Math. Phys. 41, 203–234 (1975)CrossRefADSGoogle Scholar
  56. 56.
    Mack, G.: Introduction to conformal invariant quantum field theory in two and more dimensions. In: ’t Hooft, G., et al. (eds.) Nonperturbative QFT, pp. 353–383. Plenum Press, New York (1988)Google Scholar
  57. 57.
    Mund, J., Schroer, B., Yngvason, J.: String-localized quantum fields and modular localization. Commun. Math. Phys. 268, 621–672 (2006)MathSciNetCrossRefADSMATHGoogle Scholar
  58. 58.
    Neumann, C., Rehren, K.-H., Wallenhorst, L.: New methods in conformal partial wave analysis. In: Dobrev, V. (ed.) Lie Theory and Its Applications in Physics: IX International Workshop, Springer Proceedings in Mathematics and Statistics, vol. 36, pp. 109–126 (2013)Google Scholar
  59. 59.
    Nikolov, N.M., Rehren, K.-H., Todorov, I.T.: Partial wave expansion and Wightman positivity in conformal field theory. Nucl. Phys. B 722 [PM], 266–296 (2005)MathSciNetCrossRefADSGoogle Scholar
  60. 60.
    Pressley, A., Segal, G.: Loop Groups. Oxford University Press, Oxford (1986)MATHGoogle Scholar
  61. 61.
    Rehren, K.-H.: Weak C* Hopf symmetry. In: Doebner, H.-D. et al. (eds.) Quantum Groups Symposium at “Group21”, Goslar 1996 Proceedings. Heron Press, Sofia, pp. 62–69. arXiv:q-alg/9611007 (1997)
  62. 62.
    Rehren, K.-H.: Canonical tensor product subfactors. Commun. Math. Phys. 211, 395–406 (2000)MathSciNetCrossRefADSMATHGoogle Scholar
  63. 63.
    Rehren, K.-H.: Algebraic holography. Ann. H. Poinc. 1, 607–623 (2000)MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Schellekens, A., Warner, N.: Conformal subalgebras of Kac-Moody algebras. Phys. Rev. D 34, 3092–3096 (1986)MathSciNetCrossRefADSGoogle Scholar
  65. 65.
    Schroer, B.: Braided structure in 4-dimensional conformal quantum field theory. Phys. Lett. B 506, 337–343 (2001)MathSciNetCrossRefADSMATHGoogle Scholar
  66. 66.
    Schroer, B., Truong, T.T.: The order/disorder quantum field operators associated with the two-dimensional Ising model in the continuum limit. Nucl. Phys. B 144, 80–122 (1978)MathSciNetCrossRefADSGoogle Scholar
  67. 67.
    Staszkiewicz, C.-P.: Die lokale Struktur abelscher Stromalgebren auf dem Kreis. Ph.D. thesis (in German), Freie Universität Berlin (1995)Google Scholar
  68. 68.
    Takesaki, M.: Theory of Operator Algebras II. Springer Encyclopedia of Mathematical Sciences, vol. 125 (2003)Google Scholar
  69. 69.
    Tanimoto, Y.: Construction of wedge-local nets of observables through Longo-Witten endomorphisms. Commun. Math. Phys. 314, 443–469 (2012)MathSciNetCrossRefADSMATHGoogle Scholar
  70. 70.
    Tanimoto, Y.: Construction of two-dimensional quantum field theory models through Longo-Witten endomorphisms. Forum Math. Sigma 2, e7 (2014). arXiv:1301.6090
  71. 71.
    Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300, 360–376 (1988)MathSciNetCrossRefADSMATHGoogle Scholar
  72. 72.
    Wassermann, A.: Kac-Moody and Virasoro Algebras. arXiv:1004.1287
  73. 73.
    Xu, F.: Mirror extensions of local nets. Commun. Math. Phys. 270, 835–847 (2007)CrossRefADSMATHGoogle Scholar

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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut für Theoretische Physik, Universität GöttingenGöttingenGermany

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