# nLab Spin^c Dirac operator

### Context

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

spin geometry

string 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 $X$ is a complex manifold, then under the identification

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

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

$D^+ \simeq \overline\partial + \overline \partial^\ast \colon \Gamma(\wedge^{0,even} T^\ast X) \to \Gamma(\wedge^{0,odd} T^\ast X) \,.$

### Genus

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

## References

Original articles include

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

A quick statement of the definition is in

• Ulrich Krämer, $Spin^{\mathbb{C}}$-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 $Spin_{\mathbb{C}}$-Dirac operators, I (pdf)

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