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I do not understand, how I get a connection on the spinor bundle from some connection on the $Spin^{\mathbb{C}}$-structure: Let $M$ be an oriented, Riemannian manifold with a $Spin^{\mathbb{C}}$-structure $P_M$. Then from lifting the Levi-Civita connection, we can get a connection on $P_M$. I then want to know: How can we (out of this connection on $P_M$) obtain a connection on the spinor bundle, which is given by $P_M \times_{Spin^{\mathbb{C}}} V$ where $V$ is the unique irreducible representation $\mathbb{C}l_n(\mathbb{C})$?

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It is explained in the first few pages of Chapter 3 in Dirac Operators in Riemannian Geometry by Thomas Friedrich.

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I'll also throw 1 into the mix here. A personal favourite for this type of thing.

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