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.
This post imported from StackExchange MathOverflow at 2014-09-18 10:44 (UCT), posted by SE-user Greg