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Supersymmetry of the chiral de Rham complex

Part of: Lie algebras and Lie superalgebras Quantum field theory; related classical field theories Global differential geometry

Published online by Cambridge University Press:  01 March 2008

David Ben-Zvi ,
Reimundo Heluani  and
Matthew Szczesny
David Ben-Zvi
Affiliation:
Department of Mathematics, University of Texas, Austin, TX 78712, USA (email: [email protected])
Reimundo Heluani
Affiliation:
Department of Mathematics, MIT, Cambridge, MA 02139, USA (email: [email protected])
Matthew Szczesny
Affiliation:
Department of Mathematics and Statistics, Boston University, Boston, MA 02215, USA (email: [email protected])
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Abstract

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We present a superfield formulation of the chiral de Rham complex (CDR), as introduced by Malikov, Schechtman and Vaintrob in 1999, in the setting of a general smooth manifold, and use it to endow CDR with superconformal structures of geometric origin. Given a Riemannian metric, we construct an N=1 structure on CDR (action of the N=1 super-Virasoro, or Neveu–Schwarz, algebra). If the metric is Kähler, and the manifold Ricci-flat, this is augmented to an N=2 structure. Finally, if the manifold is hyperkähler, we obtain an N=4 structure. The superconformal structures are constructed directly from the Levi-Civita connection. These structures provide an analog for CDR of the extended supersymmetries of nonlinear σ-models.

Keywords

chiral de Rhamhyperkähler

MSC classification

Secondary: 17B69: Vertex operators; vertex operator algebras and related structures 81T60: Supersymmetric field theories 53C26: Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
Type
Research Article
Information
Compositio Mathematica , Volume 144 , Issue 2 , March 2008 , pp. 503 - 521
Copyright
Copyright © Foundation Compositio Mathematica 2008

References

The first author was supported by a Summer Research Assignment from the University of Texas, and the third author was supported by NSF grant DMS-0401619.