# The spinorial connections

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Let $(M,g)$ be a spin manifold with spinors bundle $\Sigma$, a spinorial connection is an operator:

$$\nabla : \Sigma \rightarrow \Sigma$$

$$\nabla(f \psi)= (df)^* .\psi + f \nabla \psi$$

with $\psi$, a spinor and $f$ a smooth function.

What are the properties of the spinorial connections?

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We have:

$$\nabla ={\cal D}+A$$

Where $\cal D$ is the Dirac operator and $A$ is an endomorphism.

answered Feb 20 by (-35 points)

Is this supposed to be an answer to your question "What are the properties of the spinorial connections?" ?

It is a response for the case where the fiber bundle is trivial, otherwise we have non trivial connections.

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