Communications in Mathematical Physics

, Volume 354, Issue 1, pp 393–458 | Cite as

Conformal Nets II: Conformal Blocks

  • Arthur Bartels
  • Christopher L. Douglas
  • André Henriques
Open Access


Conformal nets provide a mathematical formalism for conformal field theory. Associated to a conformal net with finite index, we give a construction of the ‘bundle of conformal blocks’, a representation of the mapping class groupoid of closed topological surfaces into the category of finite-dimensional projective Hilbert spaces. We also construct infinite-dimensional spaces of conformal blocks for topological surfaces with smooth boundary. We prove that the conformal blocks satisfy a factorization formula for gluing surfaces along circles, and an analogous formula for gluing surfaces along intervals. We use this interval factorization property to give a new proof of the modularity of the category of representations of a conformal net.


  1. 1.
    Andersen J.E., Ueno K.: Abelian conformal field theory and determinant bundles. Int. J. Math. 18(8), 919–993 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Andersen J.E., Ueno K.: Geometric construction of modular functors from conformal field theory. J. Knot Theory Ramifications 16(2), 127–202 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Andersen J.E., Ueno K.: Modular functors are determined by their genus zero data. Quantum Topol. 3(3-4), 255–291 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Andersen J.E., Ueno K.: Construction of the Reshetikhin–Turaev TQFT from conformal field theory. Invent. Math. 201(2), 519–559 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Axelrod S., Della Pietra S., Witten E.: Geometric quantization of Chern-Simons gauge theory. J. Differ. Geom. 33(3), 787–902 (1991)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bakalov, B., Kirillov, A. Jr.: Lectures on Tensor Categories and Modular Functors, University Lecture Series, vol. 21. American Mathematical Society, Providence (2001)Google Scholar
  7. 7.
    Bartels, A., Douglas, C.L., Henriques, A.: Dualizability and index of subfactors. Quantum Topol., 5, 289–345 (2014). arXiv:1110.5671
  8. 8.
    Bartels, A., Douglas, C.L., Henriques, A.: Conformal nets I: coordinate-free nets. Int. Math. Res. Not., 13, 4975–5052 (2015). arXiv:1302.2604v2
  9. 9.
    Bartels, A., Douglas, C.L., Henriques, A.: Conformal nets V: dualizability (2017) (in preparation)Google Scholar
  10. 10.
    Beauville A., Laszlo Y.: Conformal blocks and generalized theta functions. Commun. Math. Phys. 164(2), 385–419 (1994)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Belavin A.A., Polyakov A.M., Zamolodchikov A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B 241(2), 333–380 (1984)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Beliakova A., Blanchet C.: Modular categories of types B, C and D. Comment. Math. Helv. 76(3), 467–500 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Bisch, D.: Bimodules, higher relative commutants and the fusion algebra associated to a subfactor. In: Operator Algebras and Their Applications (Waterloo, ON, 1994/1995), Fields Institute Communications, vol. 13, pp. 13–63. American Mathematical Society, Providence, (1997)Google Scholar
  14. 14.
    Blanchet C.: Hecke algebras, modular categories and 3-manifolds quantum invariants. Topology 39(1), 193–223 (2000)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Brunetti R., Guido D., Longo R.: Modular structure and duality in conformal quantum field theory. Commun. Math. Phys. 156(1), 201–219 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Carpi S., Conti R., Hillier R., Weiner M.: Representations of conformal nets, universal C*-algebras and K-theory. Commun. Math. Phys. 320(1), 275–300 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Carpi, S., Kawahigashi, Y., Longo, R., Weiner, M.: From Vertex Operator Algebras to Conformal Nets and Back. arXiv:1503.01260 (2015)
  18. 18.
    Connes A.: Noncommutative Geometry. Academic Press Inc., San Diego (1994)MATHGoogle Scholar
  19. 19.
    Dong C., Lin X.: Unitary vertex operator algebras. J. Algebra 397, 252–277 (2014)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Faltings G.: A proof for the Verlinde formula. J. Algebr. Geom. 3(2), 347–374 (1994)MathSciNetMATHGoogle Scholar
  21. 21.
    Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves. In: Mathematical Surveys and Monographs, vol. 88, 2nd edn. American Mathematical Society, Providence (2004)Google Scholar
  22. 22.
    Friedan D., Shenker S.: The analytic geometry of two-dimensional conformal field theory. Nuclear Phys. B 281(3-4), 509–545 (1987)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Gabbiani F., Fröhlich J.: Operator algebras and conformal field theory. Commun. Math. Phys. 155(3), 569–640 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Hitchin N.J.: Flat connections and geometric quantization. Commun. Math. Phys. 131(2), 347–380 (1990)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Huang Y.-Z., Lepowsky J.: Tensor categories and the mathematics of rational and logarithmic conformal field theory. J. Phys. A 46, 494009 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kawahigashi Y., Longo R., Müger M.: Multi-interval subfactors and modularity of representations in conformal field theory. Commun. Math. Phys. 219(3), 631–669 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kumar S., Narasimhan M.S., Ramanathan A.: Infinite Grassmannians and moduli spaces of G-bundles. Math. Ann. 300(1), 41–75 (1994)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Laszlo Y.: Hitchin’s and WZW connections are the same. J. Differ. Geom. 49(3), 547–576 (1998)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Longo, R.: Lectures on Conformal Nets II. (2008)
  30. 30.
    Looijenga E.: Unitarity of SL(2)-conformal blocks in genus zero. J. Geom. Phys. 59(5), 654–662 (2009)ADSMathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Moore G., Seiberg N.: Classical and quantum conformal field theory. Commun. Math. Phys. 123(2), 177–254 (1989)ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Moore G., Seiberg N.: Naturality in conformal field theory. Nuclear Phys. B 313(1), 16–40 (1989)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Müger M.: From subfactors to categories and topology. II. The quantum double of tensor categories and subfactors. J. Pure Appl. Algebra 180(1–2), 159–219 (2003)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Posthuma H.: The Heisenberg group and conformal field theory. Q. J. Math. 63(2), 423–465 (2012)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Ramadas T.R.: The “Harder-Narasimhan trace” and unitarity of the KZ/Hitchin connection: genus 0. Ann. Math. (2) 169(1), 1–39 (2009)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Rehren K.-H.: Braid group statistics and their superselection rules. In: The Algebraic Theory of Superselection Sectors (Palermo, 1989), pp. 333–355. World Scientific Publishing, River Edge (1990)Google Scholar
  37. 37.
    Rowell, E.C.: From quantum groups to unitary modular tensor categories. In: Representations of Algebraic Groups, Quantum Groups, and Lie Algebras, Contemporary Mathematics, vol. 413, pp. 215–230. American Mathematical Society, Providence (2006)Google Scholar
  38. 38.
    Sauvageot J.-L.: Sur le produit tensoriel relatif d’espaces de Hilbert. J. Oper. Theory 9(2), 237–252 (1983)MathSciNetMATHGoogle Scholar
  39. 39.
    Segal, G.: The definition of conformal field theory. In: Topology, geometry and quantum field theory, volume 308 of London Mathematical Society Lecture Note Serie, pp. 421–577. Cambridge University Press, Cambridge (2004)Google Scholar
  40. 40.
    Teleman C.: Borel–Weil–Bott theory on the moduli stack of G-bundles over a curve. Invent. Math. 134(1), 1–57 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Tsuchiya, A., Ueno, K., Yamada, Y.: Conformal field theory on universal family of stable curves with gauge symmetries. In: Integrable Systems in Quantum Field Theory and Statistical Mechanics, vol. 19 Advanced Studies in Pure Mathematics, pp. 459–566. Academic Press, Boston (1989)Google Scholar
  42. 42.
    Wassermann A.: Operator algebras and conformal field theory. III. Fusion of positive energy representations of LSU(N) using bounded operators. Invent. Math. 133(3), 467–538 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Wenzl H.: C* tensor categories from quantum groups. J. Am. Math. Soc. 11(2), 261–282 (1998)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Xu F.: Jones-Wassermann subfactors for disconnected intervals. Commun. Contemp. Math. 2(3), 307–347 (2000)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Mathematisches InstitutWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Mathematical InstituteRadcliffe Observatory QuarterOxfordUK
  3. 3.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherlands

Personalised recommendations