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My question is about Calabi-Yau manifolds and G_2's. Let M be a 6 dim'l CY manifold. Then, the claim is $M \times S^1 $ becomes a manifold with G_2 structure. I really want to understand how this works. I need to see concrete calculations. Does anyone know a good reference for this? The article I have been reading says that it's well known, but I'm not convinced. Moreover, the claim is that we have this freedom of changing the holomorphic $(3,0)$ form by a phase factor and preserving the Ricci-flat metric, but how? Thanks everyone.

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    $\begingroup$ Did you check books by Salamon and Joyce? $\endgroup$ Jan 23, 2016 at 19:40
  • $\begingroup$ Which ones especially? I know Joyce's books don't have it, but I didn't check Salamon's. $\endgroup$
    – Sophie
    Jan 23, 2016 at 20:30
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    $\begingroup$ See people.maths.ox.ac.uk/joyce/books.html and amazon.com/Riemannian-Geometry-Holonomy-Research-Mathematics/dp/…. I am not an expert but certainly $G_2$ contains $SU(3)$ which gives what you are asking. $\endgroup$ Jan 23, 2016 at 20:46
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    $\begingroup$ As suggested by Igor Belegradek, this is essentially obvious if one uses the correct definition of $G_2$. To give a useful answer one should know which definition of $G_2$ you are starting with. $\endgroup$
    – user25309
    Jan 23, 2016 at 21:48
  • $\begingroup$ The group $SU(3)$ is the stabilizer of a point of the 6-sphere inside $G_2$, when $G_2$ acts in its unique irreducible representation in dimension 7. So if you have a product of a Riemannian manifold with holonomy in $SU(3)$ by a 1-dimension Riemannian manifold, the holonomy is actually $SU(3)$ acting on $\mathbb{R}^7=\mathbb{R}^6 + \mathbb{R}$, trivially on the $\mathbb{R}$, i.e. as $SU(3)$ sits inside $G_2$. $\endgroup$
    – Ben McKay
    Jan 24, 2016 at 10:45

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