  Recently it has been explicitly shown how a theory with global
$GL(d,\mathbb{R})$ coordinate (affine) invariance which is spontaneously broken
down to its Lorentz subgroup will have as its Goldstone fields enough degrees
of freedom to create a metric and a covariant derivative arXiv:1105.5848. Such
a theory would constitute an effective theory of gravity. So far however, no
explicit theory has been found which exhibits this symmetry breaking pattern,
mainly due to the difficulty of even writing down a $GL(d,\mathbb{R})$
invariant actions in the absence of a metric. In this paper we explicitly
construct an affine generalization of the Dirac action employing infinite
dimensional spinorial representations of the group. This implies that it is
built from an infinite number of spinor Lorentz multiplets. We introduce a
systematic procedure for obtaining $GL(d,\mathbb{R})$ invariant interaction
terms to obtain quite general interacting models. Such models have order
operators whose expectation value can break affine symmetry to Poincar\'{e}
symmetry. We discuss possible interactions and mechanisms for this symmetry
breaking to occur, which would provide a dynamical explanation of the
Lorentzian signature of spacetime.

High Energy Physics - Theory