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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!

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  • $\begingroup$ This can not be true, as the spin or spinc bundle is not unique. For expamle, if $S$ is a spinc bundle, $S\otimes L$ is still a spinc bundle where $L$ is any line bundle. $\endgroup$ – shu Oct 30 '13 at 11:49
  • $\begingroup$ Yes, but I think we can still use $\sigma_{\mu \nu}^{\alpha\beta}$ to obtain a self-dual 2-form charged under the line bundle. I am reading Liviu's note and hope to understand this. $\endgroup$ – Lelouch Oct 30 '13 at 15:58
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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.

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  • $\begingroup$ Yes, I have the assumption in the second line; maybe I should make it a global announcement. And I have a quick look at your note, which is amazing! Thanks. $\endgroup$ – Lelouch Oct 30 '13 at 15:25
  • $\begingroup$ I read the section you recommend, and I found the quadratic map $q$ gives a self-dual 2-form for each spinor $\psi \in S_+$, which is not exactly what I want. So I modify the $q$, such that for any two spinor $\psi_1$ and $\psi_2$, construct $\left| {{\psi _1}} \right\rangle \left\langle {{\psi _2}} \right| + \left| {{\psi _2}} \right\rangle \left\langle {{\psi _1}} \right| - \frac{1}{2}\left[ {\left\langle {{\psi _1}} \right|\left. {{\psi _2}} \right\rangle + \left\langle {{\psi _2}} \right|\left. {{\psi _1}} \right\rangle } \right]$. $\endgroup$ – Lelouch Oct 31 '13 at 20:19
  • $\begingroup$ I wonder if this is the map from $S_+\otimes S_+$ to $\Lambda^2_+T^*M$? It seems right but I am not sure. $\endgroup$ – Lelouch Oct 31 '13 at 20:21
  • $\begingroup$ Yes, that's the map. $\endgroup$ – Liviu Nicolaescu Nov 1 '13 at 9:06

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