Communications in Mathematical Physics

, Volume 342, Issue 1, pp 1–45 | Cite as

Phase Boundaries in Algebraic Conformal QFT

  • Marcel Bischoff
  • Yasuyuki Kawahigashi
  • Roberto Longo
  • Karl-Henning Rehren


We study the structure of local algebras in relativistic conformal quantum field theory with phase boundaries. Phase boundaries are instances of a more general notion of boundaries that give rise to a variety of algebraic structures. These can be formulated in a common framework originating in Algebraic QFT, with the principle of Einstein Causality playing a prominent role. We classify the phase boundary conditions by the centre of a certain universal construction, which produces a reducible representation in which all possible boundary conditions are realized. For a large class of models, the classification reproduces results obtained in a different approach by Fuchs et al. before.


Fusion Rule Tensor Category Local Extension Local Algebra Superselection Sector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bartels, A., Douglas, C.L., Henriques, A.: Conformal nets III: fusion of defects. arXiv:1310.8263v2
  2. 2.
    Bischoff M., Kawahigashi Y., Longo R.: Characterization of 2D rational local conformal nets and its boundary conditions: the maximal case. Documenta Math. 20, 1137–1184 (2015)MathSciNetGoogle Scholar
  3. 3.
    Bischoff, M., Kawahigashi,Y., Longo, R., Rehren, K.-H.: Tensor categories and endomorphisms of von Neumann algebras. Springer Briefs in Mathematical Physics, vol. 3 (2015). arXiv:1407.4793v3
  4. 4.
    Borchers H.-J.: On revolutionizing QFT with modular theory. J. Math. Phys. 41, 3604–3673 (2000)CrossRefADSMathSciNetMATHGoogle Scholar
  5. 5.
    Böckenhauer J., Evans D., Kawahigashi Y.: On α-induction, chiral generators and modular invariants for subfactors. Commun. Math. Phys. 208, 429–487 (1999)CrossRefADSMATHMathSciNetGoogle Scholar
  6. 6.
    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)CrossRefADSMathSciNetMATHGoogle Scholar
  7. 7.
    Carpi S., Kawahigashi Y., Longo R.: How to add a boundary condition. Commun. Math. Phys. 322, 149–166 (2013)CrossRefADSMathSciNetMATHGoogle Scholar
  8. 8.
    Davydov, A., Kong, L., Runkel, I.: Field theories with defects and the centre functor. In: Mathematical foundations of quantum field theory and perturbative string theory, pp. 71–128, Proc. Sympos. Pure Math., vol. 83. Amer. Math. Soc., Providence (2011). arXiv:1107.0495
  9. 9.
    Doplicher S., Haag R., Roberts J.E.: Local observables and particle statistics. I. Commun. Math. Phys. 23, 199–230 (1971)CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    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)CrossRefADSMathSciNetMATHGoogle Scholar
  11. 11.
    Evans D., Gannon T.: Near-group fusion categories and their doubles. Adv. Math. 255, 586–640 (2014)CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Evans D., Pinto P.: Subfactor realizations of modular invariants. Commun. Math. Phys. 237, 309–363 (2003)CrossRefADSMathSciNetMATHGoogle Scholar
  13. 13.
    Fredenhagen K., Rehren K.-H., Schroer B.: Superselection sectors with braid group statistics and exchange algebras I. Commun. Math. Phys. 125, 201–226 (1989)CrossRefADSMathSciNetMATHGoogle Scholar
  14. 14.
    Frieler K., Rehren K.-H.: A non-Abelian square root of Abelian vertex operators. J. Math. Phys. 39, 3073–3090 (1998)CrossRefADSMathSciNetMATHGoogle Scholar
  15. 15.
    Fröhlich J., Fuchs J., Runkel I., Schweigert C.: Kramers–Wannier duality from conformal defects. Phys. Rev. Lett. 93, 070601 (2004)CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Fröhlich J., Fuchs J., Runkel I., Schweigert C.: Correspondences of ribbon categories. Ann. Math. 199, 192–329 (2006)MATHMathSciNetGoogle Scholar
  17. 17.
    Fröhlich J., Fuchs J., Runkel I., Schweigert C.: Duality and defects in rational conformal field theory. Nucl. Phys. B 763, 354–430 (2007)CrossRefADSMATHMathSciNetGoogle Scholar
  18. 18.
    Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators I: partition functions. Nucl. Phys. B 646 [PM], 353–497 (2002)Google Scholar
  19. 19.
    Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators II: unoriented worldsheets. Nucl. Phys. B 678 [PM], 511–637 (2004)Google Scholar
  20. 20.
    Fuchs J., Runkel I., Schweigert C.: Boundaries, defects and Frobenius algebras. Fortschr. Phys. 51, 850–855 (2003)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Goddard P., Kent A., Olive D.: Unitary representations of the Virasoro and super-Virasoro algebras. Commun. Math. Phys. 103, 105–119 (1986)CrossRefADSMathSciNetMATHGoogle Scholar
  22. 22.
    Guido D., Longo R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181, 11–35 (1996)CrossRefADSMathSciNetMATHGoogle Scholar
  23. 23.
    Izumi M.: The structure of sectors associated with Longo-Rehren inclusions II. Examples Rev. Math. Phys. 13, 603–674 (2001)CrossRefMathSciNetMATHGoogle Scholar
  24. 24.
    Kawahigashi Y., Longo R., Müger M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219, 631–669 (2001)CrossRefADSMATHMathSciNetGoogle Scholar
  25. 25.
    Kong L., Runkel I.: Morita classes of algebras in modular tensor categories. Adv. Math. 219, 1548–1576 (2008)CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    Kosaki H.: Extension of Jones’ theory on index to arbitrary factors. J. Funct. Anal. 66, 123–140 (1986)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Kosaki H., Longo R.: A remark on the minimal index of subfactors. J. Funct. Anal. 107, 458–470 (1992)CrossRefMathSciNetMATHGoogle Scholar
  28. 28.
    Longo R.: Index of subfactors and statistics of quantum fields I. Commun. Math. Phys. 126, 217–247 (1989)CrossRefADSMathSciNetMATHGoogle Scholar
  29. 29.
    Longo R.: A duality for Hopf algebras and for subfactors. Commun. Math. Phys. 159, 133–150 (1994)CrossRefADSMathSciNetMATHGoogle Scholar
  30. 30.
    Longo R.: Conformal subnets and intermediate subfactors. Commun. Math. Phys. 237, 7–30 (2003)CrossRefADSMathSciNetMATHGoogle Scholar
  31. 31.
    Longo R., Rehren K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567–597 (1995)CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    Longo R., Rehren K.-H.: Local fields in boundary CFT. Rev. Math. Phys. 16, 909–960 (2004)CrossRefMathSciNetMATHGoogle Scholar
  33. 33.
    Longo R., Rehren K.-H.: How to remove the boundary in CFT—an operator algebraic procedure. Commun. Math. Phys. 285, 1165–1182 (2009)CrossRefADSMathSciNetMATHGoogle Scholar
  34. 34.
    Longo R., Rehren K.-H.: Boundary quantum field theory on the interior of the Lorentz hyperboloid. Commun. Math. Phys. 311, 769–785 (2012)CrossRefADSMathSciNetMATHGoogle Scholar
  35. 35.
    Longo R., Roberts J.E.: A theory of dimension. K-Theory 11, 103–159 (1997)CrossRefMathSciNetMATHGoogle Scholar
  36. 36.
    Longo R., Xu F.: Topological sectors and a dichotomy in conformal field theory. Commun. Math. Phys. 251, 321–364 (2004)CrossRefADSMathSciNetMATHGoogle Scholar
  37. 37.
    Rehren, K.-H.: Weak C* Hopf symmetry. In: Doebner, H.-D., et al. (eds.) Quantum Groups Symposium at “Group21”, Goslar 1996 Proceedings, pp. 62–69. Heron Press, Sofia (1997). arXiv:q-alg/9611007
  38. 38.
    Rehren K.-H.: Canonical tensor product subfactors. Commun. Math. Phys. 211, 395–406 (2000)CrossRefADSMathSciNetMATHGoogle Scholar
  39. 39.
    Schellekens A., Warner N.: Conformal subalgebras of Kac-Moody algebras. Phys. Rev. D 34, 3092–3096 (1986)CrossRefADSMathSciNetGoogle Scholar
  40. 40.
    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)CrossRefADSMathSciNetGoogle Scholar
  41. 41.
    Xu F.: Jones–Wassermann subfactors for disconnected intervals. Commun. Contemp. Math. 2, 307–347 (2000)MathSciNetMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität GöttingenGöttingenGermany
  2. 2.Department of MathematicsVanderbilt UniversityNashvilleUSA
  3. 3.Department of Mathematical SciencesThe University of TokyoKomaba, TokyoJapan
  4. 4.Kavli IPMU (WPI)The University of TokyoKashiwaJapan
  5. 5.Dipartimento di MatematicaUniversità di Roma “Tor Vergata”RomaItaly

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