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The geometry of super Riemann surfaces

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Abstract

We define super Riemann surfaces as smooth 2|2-dimensional supermanifolds equipped with a reduction of their structure group to the group of invertible upper triangular 2×2 complex matrices. The integrability conditions for such a reduction turn out to be (most of) the torsion constraints of 2d supergravity. We show that they are both necessary and sufficient for a frame to admit local superconformal coordinates. The other torsion constraints are merely conditions to fix some of the gauge freedom in this description, or to specify a particular connection on such a manifold, analogous to the Levi-Civita connection in Riemannian geometry. Unlike ordinary Riemann surfaces, a super Riemann surface cannot be regarded as having only one complex dimension. Nevertheless, in certain important aspects super Riemann surfaces behave as nicely as if they had only one dimension. In particular they posses an analog\(\widehat\partial \) of the Cauchy-Riemann operator on ordinary Riemann surfaces, a differential operator taking values in the bundle of half-volume forms. This operator furnishes a short resolution of the structure sheaf, making possible a Quillen theory of determinant line bundles. Finally we show that the moduli space of super Riemann surfaces is embedded in the larger space of complex curves of dimension 1|1.

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References

  1. Wells, R. O.: Differential analysis on complex manifolds. Berlin, Heidelberg, New York: Springer 1980

    Google Scholar 

  2. Alvarez-Gaumé, L., Bost, J. B., Moore, G., Nelson, P., Vafa, C.: Bosonization on higher genus Riemann surfaces. Commun. Math. Phys.112, 503 (1987)

    Google Scholar 

  3. Baranov, M., Shvarts, A.: Multiloop contribution in string theory. Pisma ZETF42, 340 (1985) [= JETP Lett.42, 419 (1986)]; Baranov, M., Manin, Yu., Frolov, I., Shvarts, A.: Multiloop contribution to fermion string. Yad. Phys.43, 1053 (1986) [= Sov. J. Nucl. Phys.43, 670 (1986)]

    Google Scholar 

  4. Friedan, D.: In Green, M., Gross, D. (eds.). Unified string theories. Singapore World Scientific 1986

  5. Rabin, J., Crane, L.: Super Riemann Surfaces. Commun. Math. Phys.113, 601 (1988)

    Google Scholar 

  6. Le Brun, C., Rothstein, M.: Moduli of super Riemann surfaces. Commun. Math. Phys. (in press)

  7. Batchelor, M., Bryant, P.: Graded Riemann surfaces. Commun. Math. Phys.114, 243–255 (1988)

    Google Scholar 

  8. Gates, S. J.: Harvard Preprint HUTP-78/A028 (1978) (unpublished); Brown, M. and Gates, S. J.: Superspace Bianchi identities and the supercovariant derivative. Ann. Phys.122, 443 (1979)

  9. Howe, P.: Super Weyl transformations in two dimensions. J. Phys.A12, 393 (1979)

    Google Scholar 

  10. Martinec, E.: Superspace geometry of fermionic strings. Phys. Rev.D28, 2604 (1983)

    Google Scholar 

  11. Brooks, R., Muhammad, F., Gates, S.: UnidexterousD=2 supersymmetry in superspace. Nucl. Phys.B268, 47 (1986); Moore, G., Nelson, P., Polchinski, J.: Strings and supermoduli. Phys. Lett.B169 47 (1986); Evans, M., Ovrut, B.: The world sheet supergravity of the heterotic string. Phys. Lett.171B, 177 (1986)

    Google Scholar 

  12. D'Hoker, E., Phong, D.: Loop amplitudes for the fermionic string. Nucl. Phys.B278, 225 (1986)

    Google Scholar 

  13. Nelson, P., Moore, G.: Heterotic geometry. Nucl. Phys.B274 509 (1986)

    Google Scholar 

  14. Giddings, S. B., Nelson, P.: Torsion constraints and super Riemann surfaces. Phys. Rev. Lett.59, 2619 (1987)

    Google Scholar 

  15. Baranov, M., Frolov, I., Schwarz, A.: Geometry of two-dimensional superconformal field theories. Teor. Mat. Fiz.70, 92 (1987) [ = Theor. Math. Phys.70, 64 (1987)]; Baranov, M., Schwarz, A. S.: On the multiloop contribution to string theory. Int. J. Mod. Phys.A2, 1773 (1987)

    Google Scholar 

  16. Nelson, P.: Holomorphic coordinates for supermoduli space. Commun. Math. Phys.115, 167–175 (1988)

    Google Scholar 

  17. Leites, D.: Introduction to the theory of supermanifolds. Usp. Mat. Nauk.35:I, 3 (1980) [= Russ. Math. Surv.35:I, 1 (1980)]

    Google Scholar 

  18. Sternberg, S.: Differential geometry, 2d ed. New York: Chelsea 1983

    Google Scholar 

  19. Martin, J. L.: Proc. Roy. Soc. Lond.A251, 536 (1959); 543

    Google Scholar 

  20. Nelson, P.: Introduction to supermanifolds. Int. J. Mod. Phys.A3, 585 (1988)

    Google Scholar 

  21. Rothstein, M.: Deformations of complex supermanifolds. Proc. AMS95, 255 (1985)

    Google Scholar 

  22. Gawedzki, K.: Supersymmetries. Ann. Inst. Poincaré Sect. A (NS)27, 335 (1977)

    Google Scholar 

  23. Batchelor, M.: The structure of supermanifolds. Trans. Am. Math. Soc.253, 329 (1979)

    Google Scholar 

  24. Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. New York: Academic 1978

    Google Scholar 

  25. Chern, S. S.: Complex manifolds without potential theory. 2d ed. Berlin, Heidelberg, New York: Springer 1979

    Google Scholar 

  26. Manin, Yu.: Critical dimensions of string theories. Funk. Anal.20, 88 (1986) [= Funct. Anal. Appl.20, 244 (1987)]

    Google Scholar 

  27. D'Hoker, E., Phong, D.: Superholomorphic anomalies and supermoduli space. Nucl. Phys.B292, 317 (1987)

    Google Scholar 

  28. See, e.g., Gates, S. J., Grisaru, M., Rocek, M., Siegel, W.: Superspace. Reading, MA: Benjamin/Cummings 1983

    Google Scholar 

  29. Rocek, M., van Nieuwenhuizen, P., Zhang, S.: Ann. Phys.172, 348 (1986)

    Google Scholar 

  30. Spivak, M.: A comprehensive introduction to differential geometry, Vol. 1. Berkely CA: Publish or Perish 1977

    Google Scholar 

  31. Manin, Yu.: Gauge field theory and supersymmetry. Berlin, Heidelberg, New York: Springer 1987

    Google Scholar 

  32. Rothstein, M.: Integration on noncompact supermanifolds. Trans. AMS299, 387 (1987)

    Google Scholar 

  33. Giddings, S. B., Nelson, P.: Line bundles on super Riemann surfaces. BUHEP-87-48 = HUTP-87/A080

  34. See e.g., Nelson, P.: Lectures on strings and moduli space. Phys. Rep.149, 304 (1987)

    Google Scholar 

  35. Manin, Yu.: In: Arbeitstagung Bonn 1984. Lecture Notes on Mathematics, Berlin, Heidelberg, New York: Springer 1985

    Google Scholar 

  36. Deligne, P.: unpublished (1987)

  37. Cohn, J., Friedan, D.: Modular geometry of superconformal field theory. Chicago EFI 87-07

  38. Atick, J. J., Rabin, J. M., Sen, A.: An ambiguity in fermionic string perturbation theory. Nucl. Phys.B299, 279 (1988)

    Google Scholar 

  39. Moore, G., Morozov, A.: Some remarks on two-loop superstring calculations. IASSNS-HEP-87/47

  40. Verlinde, H.: A note on the integral over the fermionic supermoduli. Utrecht preprint THU-87/26

  41. Atick, J. J., Moore, G., Sen, A.: Some global issues in string perturbation theory. SLAC-PUB-4463 = IASSNS-HEP-87/61

  42. Rosly, A., Schwarz, A. S., Voronov, A. A.: Geometry of superconformal manifolds ITEP preprint 1987

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Authors and Affiliations

  1. Lyman Laboratory of Physics, Harvard University, 02138, Cambridge, MA, USA

    Steven B. Giddings

  2. Department of Physics, Boston University, 02215, Boston, MA, USA

    Philip Nelson

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  1. Steven B. Giddings
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  2. Philip Nelson
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Communicated by L. Alvarez-Gaumé

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Giddings, S.B., Nelson, P. The geometry of super Riemann surfaces. Commun.Math. Phys. 116, 607–634 (1988). https://doi.org/10.1007/BF01224903

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