  Gravity is uniquely situated in between classical topological field theories
and standard local field theories. This can be seen in the the quasi-local
nature of gravitational observables, but is nowhere more apparent than in
gravity's holographic formulation. Holography holds promise for simplifying
computations in quantum gravity. While holographic descriptions of
three-dimensional spacetimes and of spacetimes with a negative cosmological
constant are well-developed, a complete boundary description of zero curvature,
four-dimensional spacetime is not currently available. Building on previous
work in three-dimensions, we provide a new route to four-dimensional holography
and its boundary gravitons. Using Regge calculus linearized around a flat
Euclidean background with the topology of a solid hyper-torus, we obtain the
effective action for a dual boundary theory which describes the dynamics of the
boundary gravitons. Remarkably, in the continuum limit and at large radii this
boundary theory is local and closely analogous to the corresponding result in
three-dimensions. The boundary effective action has a degenerate kinetic term
that leads to singularities in the one-loop partition function that are
independent of the discretization. These results establish a rich boundary
dynamics for four-dimensional flat holography.
