nLab spinᶜ Dirac operator

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).

Applications

• Freed-Witten anomaly

• geometric quantization by push-forward

Related concepts

• spinᶜ structure, twisted spinᶜ structure

• K-orientation

• spinᶜ quantization

• Dolbeault-Dirac operator, Dirac operator

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)