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## How do fermions affect the shape of the physical universe?

On S4(1/h) fermions are sections of the spinor bundle satisfying a Dirac equation:

$(D-m)\psi = \partial_t \psi$.

In order to understand how a fermion can affect the shape of the hypersurface of S4(1/h) that is the physical universe, a natural approach is to modify the Einstein-Hilbert action (which will be over all possibly hypersurfaces) by a Lagrangian term for the fermion.  The Lagrangian is

$L = i c \bar{\psi} (D - mc) \psi - i c \bar{\psi}\partial_t \psi$

This Lagrangian can be defined over spinor fields $\psi$ on the ambient 4-sphere.  Thus spinor field satisfying the Dirac equation on the ambient space can be plugged into the Einstein-Hilbert action so the minimizing hypersurface leads to gravitational field equation with the appropriate stress-energy tensor constructued from the known Dirac field.  In other words, the second fundamental form of the hypersurface will pick up the effect of the Dirac field.

By the results of Bertrand Morel, we have an exact characterization of eigenspinors of the Dirac operator on the ambient manifold with the second fundamental form of an isometric immersion of a hypersurface:  MorelEnergyMomentumTensorSecondFundamentalForm