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Physicists said that for a given Calabi-Yau 3-fold with the topological Euler number $e$, $|e|/2$ corresponds to the number of generations of the elementary particles.

My question is: what invariant of $G_2$ manifolds corresponds to the number of particle generations?

If a topological invariant correspond to a physical constant, let us call it 'physical'. So $|e|$ is physical. Are there other physical invariants of Calabi-Yau 3-folds? The cubic form on the integral cohomology $H^2(X; \Bbb Z)$ and the linear form by the second Chern class $c_2$ on it almost determine the topology of the Calabi-Yau 3-fold $X$. So I expect that this cubic form and $c_2$ may give some physical invariants.

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Chiral matter and non-abelian gauge groups are associated with singularities in a G2 manifold. I didn't pay too much attention to this area, but Acharya and Gukov's hep-th/0409191 is probably a good place to start. –  Aaron Bergman Jun 20 '13 at 15:07

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