Skip to main content
SpringerLink
Log in

Chiral differential operators on supermanifolds

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

The first part of this paper provides a new description of chiral differential operators (CDOs) in terms of global geometric quantities. The main result is a recipe to define all sheaves of CDOs on a smooth cs-manifold; its ingredients consist of an affine connection ∇ and an even 3-form that trivializes p 1(∇). With ∇ fixed, two suitable 3-forms define isomorphic sheaves of CDOs if and only if their difference is exact. Moreover, conformal structures are in one-to-one correspondence with even 1-forms that trivialize c 1(∇). Applying our work in the first part, we then construct what may be called “chiral Dolbeault complexes” of a complex manifold M, and analyze conditions under which these differential vertex superalgebras admit compatible conformal structures or extra gradings (by fermion numbers). When M is compact, their cohomology computes (in various cases) the Witten genus, the two-variable elliptic genus and a spinc version of the Witten genus. This part contains some new results as well as provides a geometric formulation of certain known facts from the study of holomorphic CDOs and σ-models.

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. Ben-Zvi D., Heluani R., Szczesny M.: Supersymmetry of the chiral de Rham complex. Compos. Math. 144(2), 503–521 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Borisov L.A., Libgober A.: Elliptic genera of toric varieties and applications to mirror symmetry. Invent. Math. 140(2), 453–485 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bott R., Taubes C.: On the rigidity theorems of Witten. J. Am. Math. Soc. 2(1), 137–186 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen Q., Han F., Zhang W.: Witten genus and vanishing results on complete intersections. C. R. Acad. Sci. Paris, Série I. 348, 295–298 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheung, P.: Equivariant chiral differential operators and associated modules (in preparation)

  6. Costello, K.J.: A geometric construction of the Witten genus, I. arXiv: 1006.5422

  7. Deligne, P., Morgan, J.W.: Notes on supersymmetry. In: Quantum fields and strings: a course for mathematicians, pp. 41–97, American Mathematical Society, Providence (1999)

  8. Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves, 2nd edn. American Mathematical Society, Providence (2004)

  9. Gorbounov V., Malikov F., Schechtman V.: Gerbes of chiral differential operators. Math. Res. Lett. 7(1), 55–66 (2000)

    MathSciNet  MATH  Google Scholar 

  10. Gorbounov V., Malikov F., Schechtman V.: Gerbes of chiral differential operators II. Vertex algebroids. Invent. Math. 155(3), 605–680 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hirzebruch F., Berger T., Jung R.: Manifolds and Modular Forms. Friedr. Vieweg & Sohn, Braunschweig (1992)

    MATH  Google Scholar 

  12. Kac, V.: Vertex Algebras For Beginners, 2nd edn. American Mathematical Society, Providence (1998)

  13. Kapustin, A.: Chiral de Rham complex and the half-twisted σ-model. arXiv: hep-th/0504074

  14. Malikov F., Schechtman V., Vaintrob A.: Chiral de Rham complex. Comm. Math. Phys. 204(2), 439–473 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ochanine S.: Sur les genres multiplicatifs définis par des intégrales elliptiques. Topology 26(2), 143–151 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tan M.-C.: Two-dimensional twisted σ-models and the theory of chiral differential operators. Adv. Theor. Math. Phys. 10, 759 (2006)

    MathSciNet  MATH  Google Scholar 

  17. Wells R.O.: Differential Analysis On Complex Manifolds, 2nd edn. Springer, New York (1980)

    Google Scholar 

  18. Witten E.: Elliptic genera and quantum field theory. Comm. Math. Phys. 109(4), 525–536 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  19. Witten, E.: The index of the Dirac operator in loop space. In: Elliptic Curves And Modular Forms In Algebraic Topology. Lecture Notes in Mathematics, vol. 1326, pp. 161–181. Springer, Berlin (1988)

  20. Witten E.: Two-dimensional models with (0, 2) supersymmetry: perturbative aspects. Adv. Theor. Math. Phys. 11(1), 1–63 (2007)

    MathSciNet  MATH  Google Scholar 

  21. Zagier, D.: Note on the Landweber-Stong elliptic genus. In: Elliptic Curves And Modular Forms In Algebraic Topology. Lecture Notes in Mathematics, vol. 1326, pp. 216–224. Springer, Berlin (1988)

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, University of Sheffield, Sheffield, UK

    Pokman Cheung

Authors
  1. Pokman Cheung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pokman Cheung.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cheung, P. Chiral differential operators on supermanifolds. Math. Z. 272, 203–237 (2012). https://doi.org/10.1007/s00209-011-0930-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-011-0930-7

Keywords

Search

Navigation