nLab
Spin^c Dirac operator

Context

Index theory

Functional analysis

Higher spin geometry

Contents

Idea

A type of Dirac operator defined on manifolds equipped with Spin^c structure.

Properties

Relation to the Dolbeault operator

Every almost complex manifold carries a canonical spin^c structure (as discussed there). If XX is a complex manifold, then under the identification

S(X) 0,T *X S(X) \simeq \wedge^{0,\bullet} T^\ast X

of the spinor bundle with that of holomorphic differential forms, the corresponding Spin cSpin^c-Dirac operator DD is identified with the Dolbeault-Dirac operator

D +¯+¯ *:Γ( 0,evenT *X)Γ( 0,oddT *X). D^+ \simeq \overline\partial + \overline \partial^\ast \colon \Gamma(\wedge^{0,even} T^\ast X) \to \Gamma(\wedge^{0,odd} T^\ast X) \,.

Genus

The A-hat genus for the Spin cSpin^c-operator is the Todd genus (e.g. Kitada 75).

Applications

References

Original articles include

  • Yasuhiko Kitada, Semi-free circle actions on Spin cSpin^c-manifolds, Publ. RIMS, Kyoto Univ. 10(1975), 601-617 (pdf)

A quick statement of the definition is in

  • Ulrich Krämer, Spin Spin^{\mathbb{C}}-Dirac structures and Dirac operators (2009) (pdf)

Detailed accounts include

  • J. J. Duistermaat, The Spin-c Dirac Operator, in The Heat Kernel Lefschetz Fixed Point Formula for the Spin-c Dirac Operator Modern Birkhäuser Classics, 2011, 41-51, DOI: 10.1007/978-0-8176-8247-7_5

  • Eckhard Meinrenken, Symplectic Surgery and the Spin-C Dirac operator (arXiv:dg-ga/9504002)

  • Charles Epstein, Subelliptic Spin Spin_{\mathbb{C}}-Dirac operators, I (pdf)

Revised on March 21, 2014 08:32:35 by Urs Schreiber (89.204.138.115)