Heisenberg Action and Verlinde Formulas

  • Bert van Geemen
  • Emma Previato
Part of the Progress in Mathematics book series (PM, volume 115)

Abstract

In this paper we review some of the recent work on ‘nonabelian theta functions’. We discuss various links between abelian and nonabelian theta functions as well as links with the Schottky problem and open questions.

Keywords

Manifold 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ABI]
    D. Altschüler, M. Bauer and C. Itzykson, The branching rules of conformai embeddings, Comm. Math. Phys. 132 (1990), 349–364.MathSciNetCrossRefGoogle Scholar
  2. [ADPW]
    S. Axelrod, S. Delia Pietra and E. Witten, Geometric quantization of Chern-Simons gauge theory, J. Differential Geom. 33 (1991), 787–902.MathSciNetMATHGoogle Scholar
  3. [Bl]
    A. Beauville, Fibrés de rang 2 sur une courbe, fibre déterminant et fonctions thêta, Bull. Soc. Math. France 116 (1988), 431–448.MathSciNetMATHGoogle Scholar
  4. [B2]
    A. Beauville, Fibrés de rang 2 sur une courbe, fibré déterminant et fonctions thêta, II, Bull. Soc. Math. France 119 (1991), 259–291.MathSciNetMATHGoogle Scholar
  5. [BD]
    A. Beauville and O. Debarre, Sur les fonctions thêta du second ordre, Arithmetic of complex manifolds, 27–39, Springer-Verlag, Berlin 1989.CrossRefGoogle Scholar
  6. [BNR]
    A. Beauville, M.S. Narasimhan and S. Ramanan, Spectral curves and the generalized theta divisor, J. Reine Angew. Math. 398 (1989), 169–179.MathSciNetMATHGoogle Scholar
  7. [Bel]
    A. Bertram, A partial verification of the Verlinde formulae for vector bundles of rank 2, Preprint 1991.Google Scholar
  8. [Be2]
    A. Bertram, Moduli of rank 2 vector bundles, theta divisors, and the geometry of curves in projective space, Preprint 1991.Google Scholar
  9. [BeSz]
    A. Bertram and A. Szenes, Hilbert polynomials of moduli spaces of rank 2 vector bundles II, Preprint 1991.Google Scholar
  10. [Bo]
    R. Bott, Stable bundles revisited, J. Differential Geom. Supplement 1 (1991), 1–18.MathSciNetCrossRefGoogle Scholar
  11. [Br]
    J.-L. Brylinski, Propriétés de ramification à l’infini du groupe modulaire de Teichmüller, Ann. Sci. École Norm. Sup. 12 (1979) 295–333.MathSciNetMATHGoogle Scholar
  12. [CIZ]
    A. Cappelli, C. Itzykson and J.-B. Zuber, The A-D-E classification of minimal and A 1(1) conformai invariant theories, Comm. Math. Phys. 113 (1987), 1–26.MathSciNetMATHCrossRefGoogle Scholar
  13. [C]
    P. Cartier, Quantum mechanical commutation relations and theta functions, Proc. Sympos. Pure Math. 9, eds. A. Borel and G. Mostow, pp. 361–383.Google Scholar
  14. [DR]
    U.V. Desale and S. Ramanan, Classification of vector bundles of rank 2 on hyperelliptic curves, Invent. Math. 38 (1976), 161–185.MathSciNetMATHCrossRefGoogle Scholar
  15. [D]
    R. Donagi, Non-Jacobians in the Schottky loci, Ann. of Math. 126 (1987), 193–217.MathSciNetMATHCrossRefGoogle Scholar
  16. [DN]
    J.-M. Drezet and M.S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), 53–94.MathSciNetMATHCrossRefGoogle Scholar
  17. [vG]
    B. van Geemen, Schottky-Jung relations and vector bundles on hyperelliptic curves, Math. Ann. 281 (1988), 431–449.MathSciNetMATHCrossRefGoogle Scholar
  18. [vGvdG]
    B. van Geemen and G. van der Geer, Kummer varieties and the moduli spaces of abelian varieties, Amer. J. Math. 108 (1986), 615–642.MathSciNetMATHCrossRefGoogle Scholar
  19. [vGP]
    B. van Geemen and E. Previato, Prym varieties and the Verlinde formula, MSRI Preprint 04829-91.Google Scholar
  20. [H1]
    N. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91–114.MathSciNetMATHCrossRefGoogle Scholar
  21. [H2]
    N. Hitchin, Flat connections and geometric quantization, Comm. Math. Phys. 131 (1990), 347–380.MathSciNetMATHCrossRefGoogle Scholar
  22. [I]
    J.-L Igusa, Theta functions, Springer-Verlag, Berlin 1972.MATHCrossRefGoogle Scholar
  23. [L1]
    Y. Laszlo, Dimension de l’espace des sections du diviseur thêta généralisé, Bull. Soc. Math. France 119 (1991), 293–306.MathSciNetMATHGoogle Scholar
  24. [L2]
    Y. Laszlo, Un théorème de Riemann pour les diviseurs thêta sur les espaces des modules de fibrés stables sur une courbe, Duke Math. J. 64 (1991), 333–347.MathSciNetMATHCrossRefGoogle Scholar
  25. [Ml]
    D. Mumford, Prym varieties I, in Contributions to Analysis, 325–350, Acad. Pren. New York, 1974.Google Scholar
  26. [M2]
    D. Mumford, Tata lectures on theta I, Birkhäuser, Boston 1983.MATHGoogle Scholar
  27. [NR1]
    M.S. Narasimhan and S. Ramanan, Moduli of vector bundles on a compact Riemann surface, Ann. of Math. 89 (1969), 19–51.MathSciNetCrossRefGoogle Scholar
  28. [NR2]
    M.S. Narasimhan and S. Ramanan, 2Θ-linear systems on abelian varieties, in Vector bundles on algebraic varieties, p. 415–427, Oxford University Press, 1987.Google Scholar
  29. [S]
    C.S. Seshadri, Space of unitary vector bundles on a compact Riemann surface, Ann. of Math. 85 (1967), 303–336.MathSciNetMATHCrossRefGoogle Scholar
  30. [Sz]
    A. Szenes, Hilbert polynomials of moduli spaces of rank 2 vector bundles I, Preprint 1991.Google Scholar
  31. [V]
    E. Verlinde, Fusion rules and modular transformations in 2d conformai field theory, Nuclear Phys. B 300 (1988), 360–376.MathSciNetMATHCrossRefGoogle Scholar
  32. [W]
    G. Welters, Polarized abelian varieties and the heat equation, Compositio Math. 49 (1983), 173–194.MathSciNetMATHGoogle Scholar
  33. [Z]
    D. Zagier, The cohomology ring of the moduli space of rank 2 vector bundles, in preparation.Google Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Bert van Geemen
    • 1
  • Emma Previato
    • 2
  1. 1.Department of MathematicsUniversity of UtrechtUtrechtThe Netherlands
  2. 2.Department of MathematicsBoston UniversityBostonUSA

Personalised recommendations