  We propose a model of quantum gravity in arbitrary dimensions defined in
terms of the BV quantization of a supersymmetric, infinite dimensional matrix
model. This gives an (AKSZ-type) Chern-Simons theory with gauge algebra the
space of observables of a quantum mechanical Hilbert space H. The model is
motivated by previous attempts to formulate gravity in terms of
non-commutative, phase space, field theories as well as the Fefferman-Graham
curved analog of Dirac spaces for conformally invariant wave equations. The
field equations are flat connection conditions amounting to zero curvature and
parallel conditions on operators acting on H. This matrix-type model may give a
better defined setting for a quantum gravity path integral. We demonstrate that
its underlying physics is a summation over Hamiltonians labeled by a conformal
class of metrics and thus a sum over causal structures. This gives in turn a
model summing over fluctuating metrics plus a tower of additional modes-we
speculate that these could yield improved UV behavior.
