I have recently calculated three linearly independent Killing vector fields on a G^2 manifold with a specific metric, which can be found in this paper:


The metric can be found on page 13. In the coordinates used there, the Killing vector fields I found were:

$$\frac{\partial}{\partial \psi}, \hspace{.5cm} \frac{\partial}{\partial \tilde{\psi}}, \hspace{.5cm} \frac{\partial}{\partial\phi} + \frac{\partial}{\partial\tilde{\phi}}$$

I was wondering if anyone could tell me what the physical significance of these might be? For example, in General Relativity, a killing vector may represent the direction of time. I know that they generate flows which are the isometries on the manifold but am wondering if there is any easy way to describe what that would entail physically if physics could work on this manifold.

  • $\begingroup$ By "if physics could work", do you mean using the $G_2$ manifold as compactified dimensions in M-theory on a 4-manifold? It is not clear, as far as I understand, whether compactification could work with a noncompact manifold, because you have to argue that you can perturb the product metric and spinor field into an M-theory background. There are no mathematical proofs that such things exist, but I think that the physical arguments make use of the compactness. (But I am not a physicist, so I don't really know.) $\endgroup$ – Ben McKay Apr 17 '18 at 5:17
  • $\begingroup$ I'm not a physicist either, and unfortunately don't know much about M-theory yet. But yes, I do think what you stated is what I mean. $\endgroup$ – pictorexcrucia Apr 17 '18 at 5:23

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