  Quantum gravity is studied nonperturbatively in the case in which space has a
boundary with finite area. A natural set of boundary conditions is studied in
the Euclidean signature theory, in which the pullback of the curvature to the
boundary is self-dual (with a cosmological constant). A Hilbert space which
describes all the information accessible by measuring the metric and connection
induced in the boundary is constructed and is found to be the direct sum of the
state spaces of all $SU(2)$ Chern-Simon theories defined by all choices of
punctures and representations on the spatial boundary $\cal S$. The integer
level $k$ of Chern-Simons theory is found to be given by $k= 6\pi /G^2 \Lambda
+ \alpha$, where $\Lambda$ is the cosmological constant and $\alpha$ is a $CP$
breaking phase. Using these results, expectation values of observables which
are functions of fields on the boundary may be evaluated in closed form. The
Beckenstein bound and 't Hooft-Susskind holographic hypothesis are confirmed,
(in the limit of large area and small cosmological constant) in the sense that
once the two metric of the boundary has been measured, the subspace of the
physical state space that describes the further information that the observer
on the boundary may obtain about the interior has finite dimension equal to the
exponent of the area of the boundary, in Planck units, times a fixed constant.
Finally,the construction of the state space for quantum gravity in a region
from that of all Chern-Simon theories defined on its boundary confirms the
categorical-theoretic ``ladder of dimensions picture" of Crane.
