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I have previously asked about Weitzenböck identities and received some great answers on MathOverflow.

One question which has arisen from the post is the following:

Let $E$ be a hermitian holomorphic vector bundle on a hermitian manifold. Is $\bar{\partial}_E + \bar{\partial}_E^*$ a Dirac operator?

Equivalently:

Is the operator $\Delta_{\bar{\partial}_E} = \left(\bar{\partial}_E + \bar{\partial}_E^*\right)^2 = \bar{\partial}_E\bar{\partial}_E^* + \bar{\partial}_E^*\bar{\partial}_E = \left[\bar{\partial}_E, \bar{\partial}_E^*\right]$ a generalised Laplacian?

In Nicolaescu's Lectures on the Geometry of Manifolds he shows that (up to a constant) $\bar{\partial} + \bar{\partial}^*$ is a Dirac operator. As I haven't completely understood the concept of Clifford multiplication, his proof of this fact isn't clear to me yet so I'm not sure whether it carries over to the vector bundle case. One thing I do know is that for an $E$-valued $(p,q)$-form written locally as $f^i\otimes e_i$, where $f^i$ is a $(p,q)$-form and $\{e_i\}$ is a local basis of holomorphic sections of $E$, then $\bar{\partial}_E(f^i\otimes e_i) = \bar{\partial}f^i\otimes e_i$ but $(\bar{\partial}_E + \bar{\partial}_E^*)(f^i\otimes e_i) \neq (\bar{\partial} + \bar{\partial}^*)(f^i)\otimes e_i$.

I would also like to know if $\partial_E + \partial_E^*$ is a Dirac operator. In this case, I don't even know if the non-vector bundle counterpart, namely $\partial + \partial^*$, is a Dirac operator.

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I wonder a bit whether this question better fits on MathOverflow, for the reason that Liviu Nicolaescu is active there and not here. –  Willie Wong Jan 18 '13 at 9:10

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