Skip to main content
SpringerLink
Log in

Algebraic Conformal Quantum Field Theory in Perspective

  • Chapter
  • First Online:
Advances in Algebraic Quantum Field Theory

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alazzawi, S.: Deformation of fermionic quantum field theories and integrable models. Lett. Math. Phys. 103, 37–58 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  2. Alazzawi, S.: Deformation of quantum field theories and the construction of interacting models. Ph.D. thesis, University of Vienna (2015)

    Google Scholar 

  3. Baumann, K.: There are no scalar Lie fields in three or more dimensional space-time. Commun. Math. Phys. 47, 69–74 (1976)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Bischoff, M., Tanimoto, Y.: Construction of wedge-local nets of observables through Longo-Witten endomorphisms. II. Commun. Math. Phys. 317, 667–695 (2013)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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. Bischoff, M., Kawahigashi, Y., Longo, R., Rehren, K.-H.: Phase boundaries in algebraic conformal QFT. arXiv:1405.7863

  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. Böckenhauer, J., Evans, D.: Modular invariants, graphs and \(\alpha \)-induction for nets of subfactors. II. Commun. Math. Phys. 200, 57–103 (1999)

    Article  ADS  MATH  Google Scholar 

  10. Böckenhauer, J., Evans, D., Kawahigashi, Y.: On \(\alpha \)-induction, chiral projectors and modular invariants for subfactors. Commun. Math. Phys. 208, 429–487 (1999)

    Article  ADS  MATH  Google Scholar 

  11. Böckenhauer, J., Evans, D., Kawahigashi, Y.: Chiral structure of modular invariants for subfactors. Commun. Math. Phys. 210, 733–784 (2000)

    Article  ADS  MATH  Google Scholar 

  12. Borchers, H.-J.: The CPT-theorem in two-dimensional theories of local observables. Commun. Math. Phys. 143, 315–332 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Brunetti, R., Guido, D., Longo, R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys. 156, 201–219 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Brunetti, R., Guido, D., Longo, R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–785 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Buchholz, D., Fredenhagen, K.: Locality and the structure of particle states. Commun. Math. Phys. 84, 1–54 (1982)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Buchholz, D., Haag, R.: The quest for understanding in relativistic quantum physics. J. Math. Phys. 41, 3674–3697 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Buchholz, D., Roberts, J.E.: New light on infrared problems: sectors, statistics, symmetries and spectrum. Commun. Math. Phys. 330, 935–972 (2014)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Buchholz, D., Wichmann, E.: Causal independence and energy-level density in quantum field theory. Commun. Math. Phys. 106, 321–344 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Carpi, S., Kawahigashi, Y., Longo, R.: How to add a boundary condition. Commun. Math. Phys. 322, 149–166 (2013)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Carpi, S., Kawahigashi, Y., Longo, R., Weiner, M.: From vertex operator algebras to conformal nets and back. arXiv:1503.01260

  24. Dolan, F.A., Osborn, H.: Conformal four point functions and the operator product expansion. Nucl. Phys. B 599, 459–496 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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. 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. Frenkel, J.B., Kac, V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62, 23–66 (1980)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. Friedan, D., Qiu, Z., Shenker, S.: Conformal invariance, unitarity, and critical exponents in two dimensions. Phys. Rev. Lett. 52, 1575–1578 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  30. Fröhlich, J., Fuchs, J., Runkel, I., Schweigert, C.: Kramers-Wannier duality from conformal defects. Phys. Rev. Lett. 93, 070601 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  31. Fröhlich, J., Fuchs, J., Runkel, I., Schweigert, C.: Correspondences of ribbon categories. Ann. Math. 199, 192–329 (2006)

    MATH  Google Scholar 

  32. Goddard, P., Kent, A., Olive, D.: Unitary representations of the Virasoro and super-Virasoro algebras. Commun. Math. Phys. 103, 105–119 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Guido, D., Longo, R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181, 11–35 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Guido, D., Longo, R., Wiesbrock, H.-W.: Extensions of conformal nets and superselection structures. Commun. Math. Phys. 192, 217–244 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Haag, R.: Local Quantum Physics. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  36. Izumi, M., Kosaki, H.: On a subfactor analogue of the second cohomology. Rev. Math. Phys. 14, 733–737 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kac, V.G.: Infinite Dimensional Lie Algebras. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  38. Kähler, R., Wiesbrock, H.-W.: Modular theory and the reconstruction of four-dimensional quantum field theories. J. Math. Phys. 42, 74–86 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. Kawahigashi, Y., Longo, R.: Classification of local conformal nets. Case \(c<1\). Ann. Math. 160, 493–522 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  41. Kawahigashi, Y., Longo, R.: Local conformal nets arising from framed vertex operator algebras. Adv. Math. 206, 729–751 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  44. Kong, L., Runkel, I.: Morita classes of algebras in modular tensor categories. Adv. Math. 219, 1548–1576 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  45. Köhler, Ch.: On the localization properties of quantum fields with zero mass and infinite spin, Ph.D. thesis, Vienna (2015)

    Google Scholar 

  46. Lechner, G.: Construction of quantum field theories with factorizing S-matrices. Commun. Math. Phys. 277, 821–860 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. Longo, R., Morinelli, V., Rehren, K.-H.: Where infinite-spin particles are localizable. Commun. Math. Phys. arXiv:1505.01759 (to appear)

  49. Longo, R.: Conformal subnets and intermediate subfactors. Commun. Math. Phys. 237, 7–30 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  50. Longo, R., Rehren, K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567–597 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  51. Longo, R., Rehren, K.-H.: Local fields in boundary CFT. Rev. Math. Phys. 16, 909–960 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  52. Longo, R., Rehren, K.-H.: How to remove the boundary in CFT—an operator algebraic procedure. Commun. Math. Phys. 285, 1165–1182 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  53. Longo, R., Witten, E.: An algebraic construction of boundary quantum field theory. Commun. Math. Phys. 303, 213–232 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  54. Longo, R., Xu, F.: Topological sectors and a dichotomy in conformal field theory. Commun. Math. Phys. 251, 321–364 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  55. Lüscher, M., Mack, G.: Global conformal invariance in quantum field theory. Commun. Math. Phys. 41, 203–234 (1975)

    Article  ADS  Google Scholar 

  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. Mund, J., Schroer, B., Yngvason, J.: String-localized quantum fields and modular localization. Commun. Math. Phys. 268, 621–672 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  ADS  Google Scholar 

  60. Pressley, A., Segal, G.: Loop Groups. Oxford University Press, Oxford (1986)

    MATH  Google Scholar 

  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. Rehren, K.-H.: Canonical tensor product subfactors. Commun. Math. Phys. 211, 395–406 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  63. Rehren, K.-H.: Algebraic holography. Ann. H. Poinc. 1, 607–623 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  64. Schellekens, A., Warner, N.: Conformal subalgebras of Kac-Moody algebras. Phys. Rev. D 34, 3092–3096 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  65. Schroer, B.: Braided structure in 4-dimensional conformal quantum field theory. Phys. Lett. B 506, 337–343 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  67. Staszkiewicz, C.-P.: Die lokale Struktur abelscher Stromalgebren auf dem Kreis. Ph.D. thesis (in German), Freie Universität Berlin (1995)

    Google Scholar 

  68. Takesaki, M.: Theory of Operator Algebras II. Springer Encyclopedia of Mathematical Sciences, vol. 125 (2003)

    Google Scholar 

  69. Tanimoto, Y.: Construction of wedge-local nets of observables through Longo-Witten endomorphisms. Commun. Math. Phys. 314, 443–469 (2012)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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. Verlinde, E.: Fusion rules and modular transformations in 2D conformal field theory. Nucl. Phys. B 300, 360–376 (1988)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  72. Wassermann, A.: Kac-Moody and Virasoro Algebras. arXiv:1004.1287

  73. Xu, F.: Mirror extensions of local nets. Commun. Math. Phys. 270, 835–847 (2007)

    Article  ADS  MATH  Google Scholar 

Download references

Acknowledgments

I thank Yoh Tanimoto and Jakob Yngvason for a critical reading of the manuscript. Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen. The hospitality and support of the Erwin Schrödinger International Institute for Mathematical Physics, Vienna, is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Institut für Theoretische Physik, Universität Göttingen, Friedrich-Hund-Platz 1, D-37077, Göttingen, Germany

    Karl-Henning Rehren

Authors
  1. Karl-Henning Rehren
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karl-Henning Rehren .

Editor information

Editors and Affiliations

  1. University of Trento, Trento, Italy

    Romeo Brunetti

  2. University of Pavia, Pavia, Italy

    Claudio Dappiaggi

  3. University of Hamburg, Hamburg, Germany

    Klaus Fredenhagen

  4. University of Vienna, Vienna, Austria

    Jakob Yngvason

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Rehren, KH. (2015). Algebraic Conformal Quantum Field Theory in Perspective. In: Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J. (eds) Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-21353-8_8

Download citation

Publish with us

Policies and ethics

Search

Navigation