D-Brane Conformal Field Theory and Bundles of Conformal Blocks

  • Christoph Schweigert
  • Jürgen Fuchs
Part of the Progress in Mathematics book series (PM, volume 202)


Conformal blocks form a system of vector bundles over the moduli space of complex curves with marked points. We discuss various aspects of these bundles. In particular, we present conjectures about the dimensions of sub-bundles. They imply a Verlinde formula for non-simple connected groups like PGL(n, C).

We then explain how conformal blocks enter in the construction of conformal field theories on surfaces with boundaries. Such surfaces naturally appear in the conformal field theory description of string propagation in the background of a D-brane. In this context, the sub-bundle structure of the conformal blocks controls the structure of symmetry breaking boundary conditions.


Modulus Space Conformal Block Conformal Field Theory Mapping Class Group Vertex Operator Algebra 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Beauville, in: Hirzebruch 65 Conference on Algebraic Geometry [Israel Math. Conf. Proc. 9], M. Teicher, ed. (Bar-Ilan University, Ramat Gan 1996), p. 75–96.Google Scholar
  2. 2.
    A. Beauville, in: The Mathematical Beauty of Physics, J.M. Drouffe and J.-B. Zuber, eds. (World Scientific, Singapore 1997), p. 141.Google Scholar
  3. 3.
    R. E. Behrend, P. A. Pearce, V. B. Petkova, and J.-B. Zuber, Boundary conditions in rational conformal field theories, Nucl. Phys. B 570 (2000) 525–589.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    A. A. Beilinson and V. G. Drinfeld, Chiral algebras I, preprint.Google Scholar
  5. 5.
    M. Bianchi and A. Sagnotti, Open strings and the relative modular group, Phys. Lett. B 231 (1989) 389–396.MathSciNetCrossRefGoogle Scholar
  6. 6.
    R. E. Borcherds, Quantum vertex algebras, preprint math.QA/9903038.Google Scholar
  7. 7.
    A. Borel, R. Friedman, and J. W. Morgan, Almost commuting elements in compact Lie groups,preprint math.GR/9907007.Google Scholar
  8. 8.
    G. Faltings, A proof for the Verlinde formula, J. Algebraic Geom. 3 (1994) 347–374.MathSciNetzbMATHGoogle Scholar
  9. 9.
    G. Felder, J. Fröhlich, J. Fuchs, and C. Schweigert, The geometry of WZW braves, preprint hep-th/9909030, J. Geom. Phys. 34 (2000) 162–190.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    G. Felder, J. Fröhlich, J. Fuchs, and C. Schweigert, Conformal boundary conditions and three-dimensional topological field theory, Phys. Rev. Lett. 84 (2000) 1659–1662.zbMATHCrossRefGoogle Scholar
  11. 11.
    G. Felder, J. Fröhlich, J. Fuchs, and C. Schweigert, Correlation functions and boundary conditions in RCFT and three-dimensional topology, preprint hep-th/9912239., to appear in Comp. Math.Google Scholar
  12. 12.
    G. Felder, J. Fröhlich, and G. Keller, On the structure of unitary conformal field theory II: Representation theoretic approach, Comm. Math. Phys. 130 (1990) 1–49.zbMATHCrossRefGoogle Scholar
  13. 13.
    E. Frenkel, D. Ben-Zvi, Vertex algebras and algebraic curves, AMS, to appear.Google Scholar
  14. 14.
    J. Fröhlich, B. Pedrini, C. Schweigert, and J. Walcher, Universality in quantum Hall systems: coset construction of incompressible states, J. Stat. Phys. 103 (2001) 527–567.zbMATHCrossRefGoogle Scholar
  15. 15.
    J. Fuchs, Simple WZW currents, Comm. Math. Phys. 136 (1991) 345–356.zbMATHCrossRefGoogle Scholar
  16. 16.
    J. Fuchs, U. Ray, and C. Schweigert, Some automorphisms of Generalized KacMoody algebras, J. Algebra 191 (1997) 518–540.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    J. Fuchs, A. N. Schellekens, and C. Schweigert, The resolution of field identification fixed points in diagonal coset theories, Nucl. Phys. B 461 (1996) 371–404.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    J. Fuchs, A. N. Schellekens, and C. Schweigert, A matrix S for all simple current extensions, Nucl. Phys. B 473 (1996) 323–366.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    J. Fuchs and C. Schweigert, A classifying algebra for boundary conditions, Phys. Lett. B 414 (1997) 251–259.MathSciNetCrossRefGoogle Scholar
  20. 20.
    J. Fuchs and C. Schweigert, The action of outer automorphisms on bundles of chiral blocks, Comm Math. Phys. 206 (1999) 691–736.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    J. Fuchs and C. Schweigert, Orbifold analysis of broken bulk symmetries, Phys. Lett. B 447 (1999) 266–276.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    J. Fuchs and C. Schweigert, Symmetry breaking boundaries I. General theory, Nucl. Phys. B 558 (1999) 419–483.zbMATHCrossRefGoogle Scholar
  23. 23.
    J. Fuchs and C. Schweigert, Symmetry breaking boundaries II. More structures; examples, Nucl. Phys. B 568 (2000) 543–593.zbMATHCrossRefGoogle Scholar
  24. 24.
    V. G. Kac, Vertex Algebras for Beginners (American Mathematical Society, Providence 1996.)Google Scholar
  25. 25.
    F. Malikov, V. Schechtman, and A. Vaintrob, Chiral de Rham complex, Comm. Math. Phys. 204 (1999) 439–473.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    C. Schweigert, On moduli spaces of flat connections with non-simply connected structure group, Nucl. Phys. B 492 (1997) 743–755.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    F. Xu, Algebraic coset conformal field theories I,II, preprint math.OA/9810035, math.OA/9903096.Google Scholar

Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • Christoph Schweigert
    • 1
  • Jürgen Fuchs
    • 2
  1. 1.Lpthe, Université Paris 6Paris Cedex 05France
  2. 2.Institutionen För FysikKarlstads UniversitetKarlstadSweden

Personalised recommendations