Logarithmic Bulk and Boundary Conformal Field Theory and the Full Centre Construction

Conference paper
Part of the Mathematical Lectures from Peking University book series (MLPKU)


We review the definition of bulk and boundary conformal field theory in a way suited for logarithmic conformal field theory. The notion of a maximal bulk theory which can be non-degenerately joined to a boundary theory is defined. The purpose of this construction is to obtain the more complicated bulk theories from simpler boundary theories. We then describe the algebraic counterpart of the maximal bulk theory, namely the so-called full centre of an algebra in an abelian braided monoidal category. Finally, we illustrate the previous discussion in the example of the W 2,3-model with central charge 0.


Full Exchange Conformal field Theory Boundary Theory Bulk Theory Braided Monoidal 
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.



This paper is based on talks given by IR at the conferences Conformal field theories and tensor categories (13–17 June 2011, Beijing) and Logarithmic CFT and representation theory (3–7 October 2011, Paris). IR thanks the Beijing International Center for Mathematical Research at Peking University and the Centre Emile Borel at the Institut Henri Poincare in Paris for hospitality during one month stays around these conferences. MRG and SW thank the Centre Emile Borel for hospitality during the second conference. The authors thank Alexei Davydov, Jürgen Fuchs, Azat Gainutdinov, Jérôme Germoni, Antun Milas, Victor Ostrik, Hubert Saleur, Christoph Schweigert, Alexei Semikhatov and Carl Stigner for helpful discussions. The research of MRG is supported in part by the Swiss National Science Foundation. IR is partially supported by the Collaborative Research Centre 676 ‘Particles, Strings and the Early Universe—the Structure of Matter and Space-Time’. SW is supported by the SNSF scholarship for prospective researchers and by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.


  1. 1.
    Adamović, D., Milas, A.: On W-algebras associated to (2,p) minimal models and their representations. Int. Math. Res. Not., 2010, 3896–3934 (2010). arXiv:0908.4053 zbMATHGoogle Scholar
  2. 2.
    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) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cappelli, A., Itzykson, C., Zuber, J.B.: The ADE classification of minimal and A1(1) conformal invariant theories. Commun. Math. Phys. 113, 1–26 (1987) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cardy, J.L.: Conformal invariance and surface critical behavior. Nucl. Phys. B 240, 514–532 (1984) CrossRefGoogle Scholar
  5. 5.
    Cardy, J.L.: Operator content of two-dimensional conformal invariant theories. Nucl. Phys. B 270, 186–204 (1986) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cardy, J.L., Lewellen, D.C.: Bulk and boundary operators in conformal field theory. Phys. Lett. B 259, 274–278 (1991) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Davydov, A.: Centre of an algebra. Adv. Math. 225, 319–348 (2010). arXiv:0908.1250 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Davydov, A., Müger, M., Nikshych, D., Ostrik, V.: The Witt group of non-degenerate braided fusion categories. J. Reine Angew. Math. (2012). doi: 10.1515/crelle.2012.014. arXiv:1009.2117 Google Scholar
  9. 9.
    Deligne, P.: Catégories tannakiennes. In: Grothendieck Festschrift, vol. II, pp. 111–195. Birkhäuser, Boston (2007) CrossRefGoogle Scholar
  10. 10.
    Eberle, H., Flohr, M.: Virasoro representations and fusion for general augmented minimal models. J. Phys. A 39, 15245 (2006). arXiv:hep-th/0604097 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Etingof, P.I., Ostrik, V.: Finite tensor categories. arXiv:math/0301027
  12. 12.
    Feigin, B.L., Gainutdinov, A.M., Semikhatov, A.M., Tipunin, I.Y.: Logarithmic extensions of minimal models: characters and modular transformations. Nucl. Phys. B 757, 303–343 (2006). arXiv:hep-th/0606196 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fjelstad, J., Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators. V: proof of modular invariance and factorisation. Theory Appl. Categ. 16, 342–433 (2006). arXiv:hep-th/0503194 MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fjelstad, J., Fuchs, J., Runkel, I., Schweigert, C.: Uniqueness of open/closed rational CFT with given algebra of open states. Adv. Theor. Math. Phys. 12, 1283–1375 (2008). arXiv:hep-th/0612306 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Frenkel, I.B., Zhu, Y.: Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math. J. 66, 123–168 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Friedan, D., Shenker, S.H.: The analytic geometry of two-dimensional conformal field theory. Nucl. Phys. B 281, 509–545 (1987) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Fröhlich, J., Fuchs, J., Runkel, I., Schweigert, C.: Correspondences of ribbon categories. Adv. Math. 199, 192–329 (2006). arXiv:math/0309465 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Fuchs, J., Runkel, I., Schweigert, C.: Conformal correlation functions, Frobenius algebras and triangulations. Nucl. Phys. B 624, 452–468 (2002). arXiv:hep-th/0110133 MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fuchs, J., Schweigert, C., Stigner, C.: Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms. J. Algebra 363, 29–72 (2012). arXiv:1106.0210 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gaberdiel, M.R.: An introduction to conformal field theory. Rep. Prog. Phys. 63, 607–667 (2000). arXiv:hep-th/9910156 CrossRefGoogle Scholar
  21. 21.
    Gaberdiel, M.R., Goddard, P.: Axiomatic conformal field theory. Commun. Math. Phys. 209, 549–594 (2000). arXiv:hep-th/9810019 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gaberdiel, M.R., Kausch, H.G.: A local logarithmic conformal field theory. Nucl. Phys. B 538, 631–658 (1999). arXiv:hep-th/9807091 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gaberdiel, M.R., Runkel, I.: The logarithmic triplet theory with boundary. J. Phys. A 39, 14745 (2006). arXiv:hep-th/0608184 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gaberdiel, M.R., Runkel, I.: From boundary to bulk in logarithmic CFT. J. Phys. A 41, 075402 (2008). arXiv:0707.0388 MathSciNetCrossRefGoogle Scholar
  25. 25.
    Gaberdiel, M.R., Runkel, I., Wood, S.: Fusion rules and boundary conditions in the c=0 triplet model. J. Phys. A 42, 325403 (2009). arXiv:0905.0916 MathSciNetCrossRefGoogle Scholar
  26. 26.
    Gaberdiel, M.R., Runkel, I., Wood, S.: A modular invariant bulk theory for the c=0 triplet model. J. Phys. A 44, 015204 (2011). arXiv:1008.0082 MathSciNetCrossRefGoogle Scholar
  27. 27.
    Hu, P., Kriz, I.: Conformal field theory and elliptic cohomology. Adv. Math. 189, 325–412 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Huang, Y.-Z., Kong, L.: Full field algebras. Commun. Math. Phys. 272, 345–396 (2007). arXiv:math/0511328 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Huang, Y.-Z., Lepowsky, J.: Tensor products of modules for a vertex operator algebra and vertex tensor categories. In: Brylinski, R., Brylinski, J.-L., Guillemin, V., Kac, V. (eds.) Lie Theory and Geometry: In Honor of Bertram Kostant, pp. 349–383. Birkhäuser, Boston (1994). arXiv:hep-th/9401119 CrossRefGoogle Scholar
  30. 30.
    Huang, Y.-Z., Lepowsky, J., Zhang, L.: Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I–VIII. arXiv:1012.4193, arXiv:1012.4196, arXiv:1012.4197, arXiv:1012.4198, arXiv:1012.4199, arXiv:1012.4202, arXiv:1110.1929, arXiv:1110.1931
  31. 31.
    Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102, 20–78 (1993) MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kapustin, A., Orlov, D.: Vertex algebras, mirror symmetry, and D-branes: the case of complex tori. Commun. Math. Phys. 233, 79–136 (2003). arXiv:hep-th/0010293 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kong, L.: Open-closed field algebras. Commun. Math. Phys. 280, 207–261 (2008). arXiv:math/0610293 CrossRefzbMATHGoogle Scholar
  34. 34.
    Kong, L., Runkel, I.: Cardy algebras and sewing constraints, I. Commun. Math. Phys. 292, 871–912 (2009). arXiv:0807.3356 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lewellen, D.C.: Sewing constraints for conformal field theories on surfaces with boundaries. Nucl. Phys. B 372, 654–682 (1992) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Longo, R., Rehren, K.H.: Local fields in boundary conformal QFT. Rev. Math. Phys. 16, 909–960 (2004). arXiv:math-ph/0405067 MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Mac Lane, S.: Categories for the Working Mathematician, 2nd edn. Springer, Berlin (1998) zbMATHGoogle Scholar
  38. 38.
    Milas, A.: Weak modules and logarithmic intertwining operators for vertex operator algebras. Contemp. Math. 297, 201–227 (2002). arXiv:math/0101167 MathSciNetCrossRefGoogle Scholar
  39. 39.
    Müger, M.: From subfactors to categories and topology II. The quantum double of tensor categories and subfactors. J. Pure Appl. Algebra 180, 159–219 (2003). arXiv:math/0111205 MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Ostrik, V.: Module categories, weak Hopf algebras and modular invariants. Transform. Groups, 8, 177–206 (2003). arXiv:math/0111139 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Pearce, P.A., Rasmussen, J.: Coset graphs in bulk and boundary logarithmic minimal models. Nucl. Phys. B 846, 616–649 (2011). arXiv:1010.5328 MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Quella, T., Schomerus, V.: Free fermion resolution of supergroup WZNW models. J. High Energy Phys. 0709, 085 (2007). arXiv:0706.0744 MathSciNetCrossRefGoogle Scholar
  43. 43.
    Rasmussen, J., Pearce, P.A.: W-extended fusion algebra of critical percolation. J. Phys. A 41, 295208 (2008). arXiv:0804.4335 MathSciNetCrossRefGoogle Scholar
  44. 44.
    Runkel, I.: Boundary structure constants for the A-series Virasoro minimal models. Nucl. Phys. B 549, 563–578 (1999). arXiv:hep-th/9811178 MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Runkel, I.: Structure constants for the D-series Virasoro minimal models. Nucl. Phys. B 579, 561–589 (2000). arXiv:hep-th/9908046 MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Runkel, I., Fjelstad, J., Fuchs, J., Schweigert, C.: Topological and conformal field theory as Frobenius algebras. Contemp. Math. 431, 225–248 (2007). arXiv:math/0512076 MathSciNetCrossRefGoogle Scholar
  47. 47.
    Saleur, H.: Polymers and percolation in two-dimensions and twisted N=2 supersymmetry. Nucl. Phys. B 382, 486–531 (1992). arXiv:hep-th/9111007 MathSciNetCrossRefGoogle Scholar
  48. 48.
    Segal, G.: The definition of conformal field theory. In: Tillmann, U. (ed.) Topology, Geometry and Quantum Field Theory. London Math. Soc. Lect. Note Ser., vol. 308, pp. 421–577 (2002) Google Scholar
  49. 49.
    Vafa, C.: Conformal theories and punctured surfaces. Phys. Lett. B 199, 195–202 (1987) MathSciNetCrossRefGoogle Scholar
  50. 50.
    Van Oystaeyen, F., Zhang, Y.H.: The Brauer group of a braided monoidal category. J. Algebra 202, 96–128 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Vasseur, R., Gainutdinov, A.M., Jacobsen, J.L., Saleur, H.: The puzzle of bulk conformal field theories at central charge c=0. Phys. Rev. Lett. 108, 161602 (2012). arXiv:1110.1327 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität HamburgHamburgGermany
  2. 2.Institute for Theoretical PhysicsETH ZürichZürichSwitzerland
  3. 3.IPMUThe University of TokyoKashiwaJapan

Personalised recommendations