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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
Article

Abstract

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.

Keywords

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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References

  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)CrossRefADSMathSciNetzbMATHGoogle 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)CrossRefADSzbMATHMathSciNetGoogle 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)CrossRefADSMathSciNetzbMATHGoogle Scholar
  7. 7.
    Carpi S., Kawahigashi Y., Longo R.: How to add a boundary condition. Commun. Math. Phys. 322, 149–166 (2013)CrossRefADSMathSciNetzbMATHGoogle 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)CrossRefADSMathSciNetzbMATHGoogle Scholar
  11. 11.
    Evans D., Gannon T.: Near-group fusion categories and their doubles. Adv. Math. 255, 586–640 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Evans D., Pinto P.: Subfactor realizations of modular invariants. Commun. Math. Phys. 237, 309–363 (2003)CrossRefADSMathSciNetzbMATHGoogle 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)CrossRefADSMathSciNetzbMATHGoogle Scholar
  14. 14.
    Frieler K., Rehren K.-H.: A non-Abelian square root of Abelian vertex operators. J. Math. Phys. 39, 3073–3090 (1998)CrossRefADSMathSciNetzbMATHGoogle 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)zbMATHMathSciNetGoogle 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)CrossRefADSzbMATHMathSciNetGoogle 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)CrossRefMathSciNetzbMATHGoogle 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)CrossRefADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    Guido D., Longo R.: The conformal spin and statistics theorem. Commun. Math. Phys. 181, 11–35 (1996)CrossRefADSMathSciNetzbMATHGoogle Scholar
  23. 23.
    Izumi M.: The structure of sectors associated with Longo-Rehren inclusions II. Examples Rev. Math. Phys. 13, 603–674 (2001)CrossRefMathSciNetzbMATHGoogle 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)CrossRefADSzbMATHMathSciNetGoogle Scholar
  25. 25.
    Kong L., Runkel I.: Morita classes of algebras in modular tensor categories. Adv. Math. 219, 1548–1576 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Kosaki H.: Extension of Jones’ theory on index to arbitrary factors. J. Funct. Anal. 66, 123–140 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Kosaki H., Longo R.: A remark on the minimal index of subfactors. J. Funct. Anal. 107, 458–470 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Longo R.: Index of subfactors and statistics of quantum fields I. Commun. Math. Phys. 126, 217–247 (1989)CrossRefADSMathSciNetzbMATHGoogle Scholar
  29. 29.
    Longo R.: A duality for Hopf algebras and for subfactors. Commun. Math. Phys. 159, 133–150 (1994)CrossRefADSMathSciNetzbMATHGoogle Scholar
  30. 30.
    Longo R.: Conformal subnets and intermediate subfactors. Commun. Math. Phys. 237, 7–30 (2003)CrossRefADSMathSciNetzbMATHGoogle Scholar
  31. 31.
    Longo R., Rehren K.-H.: Nets of subfactors. Rev. Math. Phys. 7, 567–597 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Longo R., Rehren K.-H.: Local fields in boundary CFT. Rev. Math. Phys. 16, 909–960 (2004)CrossRefMathSciNetzbMATHGoogle 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)CrossRefADSMathSciNetzbMATHGoogle 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)CrossRefADSMathSciNetzbMATHGoogle Scholar
  35. 35.
    Longo R., Roberts J.E.: A theory of dimension. K-Theory 11, 103–159 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Longo R., Xu F.: Topological sectors and a dichotomy in conformal field theory. Commun. Math. Phys. 251, 321–364 (2004)CrossRefADSMathSciNetzbMATHGoogle 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)CrossRefADSMathSciNetzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar

Copyright information

© 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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