# 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

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

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

${D}^{+}\simeq \overline{\partial }+{\overline{\partial }}^{*}:\Gamma \left({\wedge }^{0,\mathrm{even}}{T}^{*}X\right)\to \Gamma \left({\wedge }^{0,\mathrm{odd}}{T}^{*}X\right)\phantom{\rule{thinmathspace}{0ex}}.$D^+ \simeq \overline\partial + \overline \partial^\ast \colon \Gamma(\wedge^{0,even} T^\ast X) \to \Gamma(\wedge^{0,odd} T^\ast X) \,.

## References

A quick statement of the definition is in

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

Revised on August 29, 2013 17:39:12 by Urs Schreiber (89.204.130.26)