Skip to main content
SpringerLink
Log in

Supertori are algebraic curves

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Super Riemann surfaces of genus 1, with arbitrary spin structures, are shown to be the sets of zeroes of certain polynomial equations in projective superspace. We conjecture that the same is true for arbitrary genus. Properties of superelliptic functions and super theta functions are discussed. The boundary of the genus 1 super moduli space is determined.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D'Hoker, E., Giddings, S.B.: Unitarity of the closed bosonic Polyakov string. Princeton preprint PUPT-1046, February 1987

  2. Nelson, P.: Lectures on strings and moduli space. Harvard preprint HUPT-86/A047

  3. Alvarez-Gaumé, L., Nelson, P.: Riemann surfaces and string theories. In: Supersymmetry, supergravity, and superstrings '86. De Wit, B., Fayet, P., Grisaru, M. (eds.). Singapore: World Scientific 1986

    Google Scholar 

  4. Friedan, D.: Notes on string theory and two-dimensional conformal field theory. In: Proceedings of the workshop on unified string theories. Gross, D., Green, M. (eds.) Singapore: World Press 1986

    Google Scholar 

  5. Martinec, E.: Conformal field theory on a (super-)Riemann surface. Nucl. Phys. B281, 157–210 (1987)

    Google Scholar 

  6. Crane, L., Rabin, J.M.: Super Riemann surfaces: Uniformization and Teichmüller theory. Commun. Math. Phys. (to appear)

  7. Rabin, J.M.: Super Riemann surfaces. University of Chicago preprint EFI 86–63, to appear in the proceedings of the Workshop on Mathematical Aspects of String Theory. U.C. San Diego, 1986

  8. Rabin, J.M.: Teichmüller deformations of super Riemann surfaces. Phys. Lett.190B, 40–46 (1987)

    Google Scholar 

  9. Hamidi, S., Vafa, C.: Interactions on orbifolds. Nucl. Phys. B279, 465–513 (1987)

    Google Scholar 

  10. Segal, G., Wilson, G.: Loop groups and equations of KdV type. IHES Publ.61, 5–65 (1985)

    Google Scholar 

  11. Mulase, M.: Cohomological structure in soliton equations and Jacobian varieties. J. Differ. Geom.19, 403–430 (1984)

    Google Scholar 

  12. Shiota, T.: Characterization of Jacobian varieties in terms of soliton equations. Invent. Math.83, 333–382 (1986)

    Google Scholar 

  13. Ishibashi, N., Matsuo, Y., Ooguri, H.: Soliton equations and free fermions on Riemann surfaces. University of Tokyo preprint UT-499, December 1986

  14. Alvarez-Gaumé, L., Gomez, C., Reina, C.: Loop groups, Grassmannians, and string theory. Phys. Lett.190B, 55–62 (1987)

    Google Scholar 

  15. Friedan, D., Shenker, S.H.: The integrable analytic geometry of quantum string. Phys. Lett.175B, 287–296 (1986)

    Google Scholar 

  16. Koblitz, N.: Introduction to elliptic curves and modular forms. Berlin, Heidelberg, New York: Springer 1984

    Google Scholar 

  17. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. New York: Wiley 1978

    Google Scholar 

  18. Rabin, J.M., Crane, L.: How different are the supermanifolds of Rogers and De Witt? Commun. Math. Phys.102, 123–137 (1985)

    Google Scholar 

  19. Cohn, J.D., Friedan, D.: Geometry of superconformal field theories. University of Chicago preprint EFI 87-07 (in preparation)

  20. Chandrasekharan, K.: Elliptic functions. Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  21. Eguchi, T., Ooguri, H.: Conformal and current algebras on general Riemann surfaces. University of Tokyo preprint UT-491, August 1986

  22. Manin, Yu.I.: New directions in geometry. Russ. Math. Surv.39:6, 51–83 (1984)

    Google Scholar 

  23. Rabin, J.M., Crane, L.: Global properties of supermanifolds. Commun. Math. Phys.100, 141–160 (1985)

    Google Scholar 

  24. Kendig, K.: Elementary algebraic geometry. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  25. Gava, E., Iengo, R., Jayaraman, T., Ramachandran, R.: Multiloop divergences in the closed bosonic string theory. Phys. Lett.168B, 207–211 (1986)

    Google Scholar 

  26. Iengo, R.: Cancellation of leading divergences in multiloop closed superstring amplitudes. Phys. Lett.186B, 52–56 (1987)

    Google Scholar 

  27. Rabin, J.M., Topiwala, P.: Work in progress

  28. Rogers, A.: Graded manifolds, supermanifolds, and infinite-dimensional Grassmann algebras. Commun. Math. Phys.105, 375–384 (1986)

    Google Scholar 

Download references

Author information

Author notes

  1. Jeffrey M. Rabin

    Present address: Department of Mathematics, University of California at San Diego, 92093, La Jolla, CA

Authors and Affiliations

  1. Departments of Mathematics and Physics and the Enrico Fermi Institute, University of Chicago, 60637, Chicago, IL, USA

    Jeffrey M. Rabin & Peter G. O. Freund

Authors
  1. Jeffrey M. Rabin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter G. O. Freund
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by S.-T. Yau

Research partially supported by the DOE (DE-AC02-82-ER-40073) and NSF (PHY-85-21588)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rabin, J.M., Freund, P.G.O. Supertori are algebraic curves. Commun.Math. Phys. 114, 131–145 (1988). https://doi.org/10.1007/BF01218292

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01218292

Keywords

Search

Navigation