Geometry of Dual Pairs of Complex Supercurves

  • Jeffrey M. RabinEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2027)


Supercurves are a generalization to supergeometry of Riemann surfaces or algebraic curves. I review the definitions, examples, key results, and open problems in this area.


Modulus Space Riemann Surface Line Bundle Transition Function Theta Function 
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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My thanks to those who have worked with me on the subject of supercurves over the years, notably Maarten Bergvelt, Louis Crane, Fausto Ongay, and Mitchell Rothstein.


  1. 1.
    M.A. Baranov, A.S. Schwarz, Multiloop contribution to string theory. JETP Lett. 42, 419–421 (1985)MathSciNetGoogle Scholar
  2. 2.
    M.J. Bergvelt, J.M. Rabin, Supercurves, their Jacobians, and super KP equations. Duke Math. J. 98(1), 1–57 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    L. Crane, J.M. Rabin, Super Riemann surfaces: Uniformization and Teichmueller theory. Commun. Math. Phys. 113, 601–623 (1988)MathSciNetCrossRefGoogle Scholar
  4. 4.
    P. Deligne, J. Morgan, in Notes on Supersymmetry (Following Joseph Bernstein), ed. by P. Deligne et al. Quantum Fields and Strings, A Course For Mathematicians, vol. I (American Mathematical Society, RI, 1999), pp. 41–97Google Scholar
  5. 5.
    E. D’Hoker, D.H. Phong, The geometry of string perturbation theory. Rev. Mod. Phys. 60, 917–1065 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    J. Dieudonné, Remarks on quasi-Frobenius rings. Ill. J. Math. 2, 346–354 (1958)zbMATHGoogle Scholar
  7. 7.
    S.N. Dolgikh, A.A. Rosly, A.S. Schwarz, Supermoduli spaces. Commun. Math. Phys. 135, 91–100 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    D. Friedan, in Notes on String Theory and Two-Dimensional Conformal Field Theory. Workshop on Unified String Theories (World Scientific, Singapore, 1986), pp. 162–213Google Scholar
  9. 9.
    S.B. Giddings, P. Nelson, Line bundles on super Riemann surfaces. Commun. Math. Phys. 118, 289–302 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    P. Griffiths, J. Harris, Principles of Algebraic Geometry (Wiley, New York, 1978)zbMATHGoogle Scholar
  11. 11.
    C. Haske, R.O. Wells Jr., Serre duality on complex supermanifolds. Duke Math. J. 54, 493–500 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    C. LeBrun, M. Rothstein, Moduli of super Riemann surfaces. Commun. Math. Phys. 117, 159–176 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    A.M. Levin, Supersymmetric elliptic curves. Funct. Anal. Appl. 21, 243–244 (1987)zbMATHGoogle Scholar
  14. 14.
    Y.I. Manin, in Gauge Field Theory and Complex Geometry. Grundlehren Math. Wiss., vol. 289 (Springer, Berlin, 1988)Google Scholar
  15. 15.
    Y.I. Manin, Topics in Noncommutative Geometry (Princeton University Press, NJ, 1991), p. 47zbMATHGoogle Scholar
  16. 16.
    O.V. Ogievetsky, I.B. Penkov, Serre duality for projective supermanifolds. Funct. Anal. Appl. 18, 78–79 (1984)CrossRefGoogle Scholar
  17. 17.
    F. Ongay, J.M. Rabin, On decomposing N = 2 line bundles as tensor products of N = 1 line bundles. Lett. Math. Phys. 61, 101–106 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    J.M. Rabin, Super elliptic curves. J. Geom. Phys. 15, 252–280 (1995)MathSciNetzbMATHGoogle Scholar
  19. 19.
    J.M. Rabin, M.J. Rothstein, D-modules on 1 | 1 supercurves. J. Geom. Phys. 60, 626–636 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    A.A. Rosly, A.S. Schwarz, A.A. Voronov, Geometry of superconformal manifolds. Commun. Math. Phys. 119, 129–152 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Y. Tsuchimoto, On super theta functions. J. Math. Kyoto Univ. 34(3), 641–694 (1994)MathSciNetzbMATHGoogle Scholar
  22. 22.
    A.Yu. Vaintrob, Deformations of complex superspaces and coherent sheaves on them. J. Soviet Math. 51, 2140–2188 (1990)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Department of MathematicsUCSDLa JollaUSA

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