Skip to main content
SpringerLink
Log in

The supermoduli of SUSY curves with Ramond punctures

  • Original Paper
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We construct local and global moduli spaces of supersymmetric curves with Ramond-Ramond punctures. We assume that the underlying ordinary algebraic curves have a level n structure and build these supermoduli spaces as algebraic superspaces, i.e., quotients of étale equivalence relations between superschemes.

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

Notes

  1. There is a notational inconsistency here, as \(\Theta _{{{\mathscr {X}}},x}\) is not the stalk \({{{\mathcal {D}}}er}({{\mathcal {O}}}_{{{\mathscr {X}}}})_x= {{{\mathcal {D}}}er}({{\mathcal {O}}}_{{{\mathscr {X}}},x})\) but rather its fibre \({{{\mathcal {D}}}er}({{\mathcal {O}}}_{{{\mathscr {X}}},x})\otimes _{{{\mathcal {O}}}_{X,x}}k(x)\). We keep this inconsistency for historical reasons.

  2. A level n structure on an ordinary curve \(p:X \rightarrow S\) is an isomorphism between the n-torsion of the relative Jacobian of the curve and the group \(\Gamma (S,R^1p_*{{\mathbb {Z}}}_n)\).

References

  1. Artin, M.: Algebraic spaces, Yale University Press, New Haven, Conn., 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969, Yale Mathematical Monographs, 3

  2. Baranov, M.A., Schwarz, A.S.: On the multiloop contribution to the string theory. Int. J. Modern Phys. A 2, 1773–1796 (1987)

    Article  MathSciNet  Google Scholar 

  3. Bartocci, C., Bruzzo, U., Hernández Ruipérez, D.: The geometry of supermanifolds, vol. 71 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1991)

  4. Berezin, F.A., Leĭtes, D.A.: Supermanifolds. Dokl. Akad. Nauk SSSR 224, 505–508 (1975)

    MathSciNet  MATH  Google Scholar 

  5. Codogni, G., Viviani, F.: Moduli and periods of supersymmetric curves. arXiv:1706.04910v4 [math.AG]

  6. Crane, L., Rabin, J.M.: Super Riemann surfaces: uniformization and Teichmüller theory. Commun. Math. Phys. 113, 601–623 (1988)

    Article  Google Scholar 

  7. Deligne, P.: Letter to Y.I. Manin. https://publications.ias.edu/sites/default/files/lettre-a-manin-1987-09-25.pdf (1987)

  8. Diroff, D.J.: On the super Mumford form in the presence of Ramond and Neveu-Schwarz punctures. J. Geom. Phys. 144, 273–293 (2019)

    Article  MathSciNet  Google Scholar 

  9. Domínguez Pérez, J.A., Hernández Ruipérez, D., Sancho de Salas, C.: Supersymmetric products of SUSY-curves, in Differential geometric methods in theoretical physics (Rapallo, 1990), vol. 375 of Lecture Notes in Phys., Springer, Berlin, pp. 271–285 (1991)

  10. Domínguez Pérez, J.A., Hernández Ruipérez, D., Sancho de Salas, C.: The variety of positive superdivisors of a supercurve (supervortices). J. Geom. Phys. 12, 183–203 (1993)

    Article  MathSciNet  Google Scholar 

  11. Domínguez Pérez, J.A., Hernández Ruipérez, D., Sancho de Salas, C.: Global structures for the moduli of (punctured) super Riemann surfaces. J. Geom. Phys. 21, 199–217 (1997)

    Article  MathSciNet  Google Scholar 

  12. Donagi, R., Witten, E.: Super Atiyah classes and obstructions to splitting of supermoduli space. Pure Appl. Math. Q. 9, 739–788 (2013)

    Article  MathSciNet  Google Scholar 

  13. Donagi, R., Witten, E.: Supermoduli space is not projected In: String-Math 2012, vol. 90 of Proc. Sympos. Pure Math. Am. Math. Soc., Providence, RI, pp. 19–71 (2015)

  14. Falqui, G., Reina, C.: A note on the global structure of supermoduli spaces. Commun. Math. Phys. 128, 247–261 (1990)

    Article  MathSciNet  Google Scholar 

  15. Felder, G., Kazhdan, D., Polishchuk, A.: Regularity of the superstring supermeasure and the superperiod map. arXiv:1905.12805v1 [math.AG], (2019)

  16. Friedan, D.: Notes on string theory and two-dimensional conformal field theory. In: Workshop on unified string theories (Santa Barbara, Calif., 1985), World Sci. Publishing, Singapore, pp. 162–213 (1986)

  17. Grothendieck, A.: Technique de descente et théorèmes d’existence en géométrie algébrique. V. Les schémas de Picard: théorèmes d’existence. In: Séminaire Bourbaki, Vol. 7, Soc. Math. France, Paris, pp. Exp. No. 232, 143–161 (1995)

  18. Hartshorne, R.: Algebraic geometry, Graduate Texts in Mathematics, vol. 52. Springer-Verlag, New York (1977)

    Book  Google Scholar 

  19. Keel, S., Mori, S.: Quotients by groupoids. Ann. Math. (2) 145, 193–213 (1997)

    Article  MathSciNet  Google Scholar 

  20. Knutson, D.: Algebraic spaces, Lecture Notes in Mathematics, Vol. 203, Springer-Verlag, Berlin (1971)

  21. Kostant, B.: Graded manifolds, graded Lie theory, and prequantization. In: Differential geometrical methods in mathematical physics (Proc. Sympos., Univ. Bonn, Bonn, 1975), Springer, Berlin, pp. 177–306. Lecture Notes in Math., Vol. 570 (1977)

  22. LeBrun, C., Rothstein, M.: Moduli of super Riemann surfaces. Commun. Math. Phys. 117, 159–176 (1988)

    Article  MathSciNet  Google Scholar 

  23. Manin, Y.I.: Critical dimensions of string theories and a dualizing sheaf on the moduli space of (super) curves. Funktsional. Anal. i Prilozhen. 20, 88–89 (1986)

    MathSciNet  Google Scholar 

  24. Manin, Y.I.: Quantum strings and algebraic curves. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Providence, RI, Amer. Math. Soc., pp. 1286–1295 (1987)

  25. Manin, Y.I.: Gauge field theory and complex geometry, vol. 289 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1988. Translated from the Russian by N. Koblitz and J. R. King, second edition 1997, with an appendix by S. Merkulov

  26. Manin, Y.I.: Neveu-Schwarz sheaves and differential equations for Mumford superforms. J. Geom. Phys. 5, 161–181 (1988)

    Article  MathSciNet  Google Scholar 

  27. Moĭšezon, B.G.: An algebraic analogue of compact complex spaces with a sufficiently large field of meromorphic functions. I. Izv. Akad. Nauk SSSR Ser. Mat. 33, 174–238 (1969)

    MathSciNet  Google Scholar 

  28. Moĭšezon, B.G.: An algebraic analogue of compact complex spaces with a sufficiently large field of meromorphic functions. II. Izv. Akad. Nauk SSSR Ser. Mat. 33,(1969)

  29. Moĭšezon, B.G.: An algebraic analogue of compact complex spaces with a sufficiently large field of meromorphic functions. III. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 33, 506–548 (1969)

    MathSciNet  Google Scholar 

  30. Ott, N., Voronov, A.: The supermoduli space of genus zero super Riemann surfaces with Ramond punctures. arXiv:1910.05655 [math.AG]

  31. Polyakov, A.M.: Quantum geometry of bosonic strings. Phys. Lett. B 103, 207–210 (1981)

    Article  MathSciNet  Google Scholar 

  32. Salam, A., Strathdee, J.: Super-gauge transformations. Nuclear Phys. B 76, 477–482 (1974)

    Article  MathSciNet  Google Scholar 

  33. Sernesi, E.: Deformations of algebraic schemes, vol. 334 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (2006)

  34. Vaĭntrob, A.Y.: Deformations of complex superspaces and of the coherent sheaves on them. In: Current problems in mathematics. Newest results, Vol. 32, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 125–211. J. Soviet Math. 51 (1990), no. 1, 2140–2188

  35. Witten, E.: Notes on super Riemann surfaces and their moduli. Pure Appl. Math. Q. 15, 57–211 (2019)

    Article  MathSciNet  Google Scholar 

  36. Zariski, O.: On the purity of the branch locus of algebraic functions. Proc. Nat. Acad. Sci. USA 44, 791–796 (1958)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the participants in the “Workshop on Supermoduli” which took place at the Institute for Geometry and Physics in Trieste on September 23 to 26, 2019, for the nice atmosphere and the fruitful interchanges which occurred. We also thank Ron Donagi for stimulating discussions and for sharing ideas, and Alexander Polishchuk for pointing out a small mistake in a previous version. We thank the referee for comments that allowed us to give a more precise statement of Proposition 3.5.

Author information

Authors and Affiliations

  1. SISSA (Scuola Internazionale Superiore di Studi Avanzati), Via Bonomea 265, 34136, Trieste, Italia

    Ugo Bruzzo

  2. Departamento de Matemática, Universidad Federal da Paraíba, Campus I, João Pessoa, PB, Brazil

    Ugo Bruzzo

  3. INFN (Istituto Nazionale di Fisica Nucleare), Sezione di Trieste, Trieste, Italy

    Ugo Bruzzo

  4. IGAP (Institute for Geometry and Physics), Trieste, Italy

    Ugo Bruzzo

  5. Arnold-Regge Center for Algebra, Geometry and Theoretical Physics, Torino, Italy

    Ugo Bruzzo

  6. Departamento de Matemáticas and IUFFYM (Instituto Universitario de Física Fundamental y Matemáticas), Universidad de Salamanca, Plaza de la Merced 1-4, 37008, Salamanca, Spain

    Daniel Hernández Ruipérez

  7. Real Academia de Ciencias Exactas, Físicas y Naturales, Madrid, Spain

    Daniel Hernández Ruipérez

Authors
  1. Ugo Bruzzo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Hernández Ruipérez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Hernández Ruipérez.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

U. Bruzzo: Research partly supported by GNSAGA-INdAM and by the PRIN project “Geometria delle varietà algebriche.”

D. H. Ruipérez: Research partly supported by research projects “Espacios finitos y functores integrales”, MTM2017-86042-P, (Ministerio of Economía, Industria y Competitividad) and “STAMGAD, SA106G19” (Junta de Castilla y León).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bruzzo, U., Ruipérez, D.H. The supermoduli of SUSY curves with Ramond punctures. RACSAM 115, 144 (2021). https://doi.org/10.1007/s13398-021-01078-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13398-021-01078-4

Keywords

Mathematics Subject Classification

Search

Navigation