  The classical notion of manifold may be augmented to institute the algebras
associated with the generators of the spinor and vector representations of the
Lorentz group on two copies of the cotangent space. In the former case, this
results in the emergence of a Dirac operator out of the vacuum of General
Relativity. In the backdrop of the Higgs VEV, a volume form with such a dual
character endows the augmented manifold with dark matter content as the
simplest conceivable geometric structure that could emerge out of the unified
vacuum.
