I am confused by two examples of spinor bundles over 4-manifolds, which I saw in various places:


(1) The spinor bundle $S = S\_+ \oplus S\_-$ associated to a spin or spinc structure of Riemannian four-manifold $M$


(2) If the manifold $M$ has almost complex structure, then the bundle $\tilde S \equiv \Lambda^{0,\*}T^\*M$ is also a spin bundle, which can also be split into $\tilde S\_+ = \Lambda^{0,0}T^\*M \oplus \Lambda^{0,2}T^\*M$ and $\tilde S\_- = \Lambda^{0,1}T^\*M$.


My question is: are the two examples the same? In particular, I know that $S\_+ \otimes S\_+ = \Lambda\_+^2 T^\*M \oplus C^\infty(M)$ where $\Lambda\_+^2T^\*M$ denotes the bundle of self-dual two forms (corresponding to symmetric product of two $S\_+$), I wonder if the same is true for $\tilde S\_+$? I think it is true, since I can multiply by $(\sigma\_{\mu \nu})^{\alpha \beta}$ to go from a bi-spinor to a self-dual 2-form. But I want to see if there is a more geometrical way (playing with the $(0,{\rm even})$-forms) to see it.


Thanks!



How to understand two examples of spin bundle
First let me point out that the bundle $\Lambda^2\_+ T^\*M$ makes sense only when $\dim M=4$. Maybe you should add this assumption.


As for your question, I think that you can find the answer in Example 1.3.3 page 30 of [these notes.](http://www3.nd.edu/~lnicolae/swnotes.pdf)


