# nLab Dolbeault-Dirac operator

complex geometry

## Ingredients

• supergeometry

• higher differential geometry

• ## Spin geometry

spin geometry

• spinor bundle

• Dirac operator

• Dirac equation

• Dirac field

• ## String geometry

string geometry

• ## Fivebrane geometry

• fivebrane 6-group

• fivebrane structure

• #### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

# Contents

## Idea

On a Kähler manifold $(X,\omega)$ with a choice of spin structure given by a Theta characteristic $\sqrt{\Omega^{0,n}}$, the sum of the Dolbeault operator $\overline{\partial}$ with its adjoint $\overline{\partial}^\ast$ are identified with the corresponding Dirac operator

$\overline{\partial} + \overline{\partial}^\ast \;\colon\; S_X\simeq \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{0,n}} \to \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{0,n}} \simeq S_X \,.$

Here the action of $\overline{\partial} + \overline{\partial}^\ast$ on $\Omega^{0,\bullet}$ is the canonical one. For the action on $\sqrt{\Omega^{n,0}}$ choose any connection $\nabla$ on this line bundle. Then on a local coordinate patch the action is given by differentiating along a coordinate vector, multiplying with the corresponding Clifford element and projecting on the antiholomorphic part, then summing this over all coordinate vectors (e.g. Friedrich 97, p. 79).

## Properties

### Index and genus

The index is the Todd genus (see there).

## References

Textbook accounts include

section 3.4 (specifically around p. 79) of

• Thomas Friedrich, Dirac operators in Riemannian geometry, Graduate studies in mathematics 25, AMS (1997)

Last revised on March 21, 2014 at 09:17:35. See the history of this page for a list of all contributions to it.