# nLab Spin^c Dirac operator

Contents

### 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

spin geometry

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

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)

Last revised on March 21, 2014 at 08:32:35. See the history of this page for a list of all contributions to it.