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.
Similar content being viewed by others
References
Ben-Zvi D., Heluani R., Szczesny M.: Supersymmetry of the chiral de Rham complex. Compos. Math. 144(2), 503–521 (2008)
Borisov L.A., Libgober A.: Elliptic genera of toric varieties and applications to mirror symmetry. Invent. Math. 140(2), 453–485 (2000)
Bott R., Taubes C.: On the rigidity theorems of Witten. J. Am. Math. Soc. 2(1), 137–186 (1989)
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)
Cheung, P.: Equivariant chiral differential operators and associated modules (in preparation)
Costello, K.J.: A geometric construction of the Witten genus, I. arXiv: 1006.5422
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)
Frenkel, E., Ben-Zvi, D.: Vertex algebras and algebraic curves, 2nd edn. American Mathematical Society, Providence (2004)
Gorbounov V., Malikov F., Schechtman V.: Gerbes of chiral differential operators. Math. Res. Lett. 7(1), 55–66 (2000)
Gorbounov V., Malikov F., Schechtman V.: Gerbes of chiral differential operators II. Vertex algebroids. Invent. Math. 155(3), 605–680 (2004)
Hirzebruch F., Berger T., Jung R.: Manifolds and Modular Forms. Friedr. Vieweg & Sohn, Braunschweig (1992)
Kac, V.: Vertex Algebras For Beginners, 2nd edn. American Mathematical Society, Providence (1998)
Kapustin, A.: Chiral de Rham complex and the half-twisted σ-model. arXiv: hep-th/0504074
Malikov F., Schechtman V., Vaintrob A.: Chiral de Rham complex. Comm. Math. Phys. 204(2), 439–473 (1999)
Ochanine S.: Sur les genres multiplicatifs définis par des intégrales elliptiques. Topology 26(2), 143–151 (1987)
Tan M.-C.: Two-dimensional twisted σ-models and the theory of chiral differential operators. Adv. Theor. Math. Phys. 10, 759 (2006)
Wells R.O.: Differential Analysis On Complex Manifolds, 2nd edn. Springer, New York (1980)
Witten E.: Elliptic genera and quantum field theory. Comm. Math. Phys. 109(4), 525–536 (1987)
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)
Witten E.: Two-dimensional models with (0, 2) supersymmetry: perturbative aspects. Adv. Theor. Math. Phys. 11(1), 1–63 (2007)
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)
Author information
Authors and Affiliations
Corresponding author
Rights 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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-011-0930-7