  Recent work on neutrino oscillations suggests that the three generations of
fermions in the standard model are related by representations of the finite
group A(4), the group of symmetries of the tetrahedron. Motivated by this, we
explore models which extend the EPRL model for quantum gravity by coupling it
to a bosonic quantum field of representations of A(4). This coupling is
possible because the representation category of A(4) is a module category over
the representation categories used to construct the EPRL model. The vertex
operators which interchange vacua in the resulting quantum field theory
reproduce the bosons and fermions of the standard model, up to issues of
symmetry breaking which we do not resolve. We are led to the hypothesis that
physical particles in nature represent vacuum changing operators on a sea of
invisible excitations which are only observable in the A(4) representation
labels which govern the horizontal symmetry revealed in neutrino oscillations.
The quantum field theory of the A(4) representations is just the dual model on
the extended lattice of the Lie group $E_6$, as explained by the quantum Mckay
correspondence of Frenkel Jing and Wang. The coupled model can be thought of as
string field theory, but propagating on a discretized quantum spacetime rather
than a classical manifold.
