Holographic formulation of quantum general relativity

We show that there is a sector of quantum general relativity, in the Lorentzian signature case, which may be expressed in a completely holographic formulation in terms of states and operators defined on a finite boundary. The space of boundary states is built out of the conformal blocks of ${\mathrm{SU}\left(2\right)\mathrm{}}_L\oplus {\mathrm{SU}\left(2\right)\mathrm{}}_R,$ WZW field theory on the n-punctured sphere, where n is related to the area of the boundary. The Bekenstein bound is explicitly satisfied. These results are based on a new Lagrangian and Hamiltonian formulation of general relativity based on a constrained $\mathrm{Sp}\left(4\right)\mathrm{}$ topological field theory. The Hamiltonian formalism is polynomial, and also left-right symmetric. The quantization uses balanced ${\mathrm{SU}\left(2\right)\mathrm{}}_L\oplus {\mathrm{SU}\left(2\right)\mathrm{}}_R$ spin networks and so justifies the state sum model of Barrett and Crane. By extending the formalism to $\mathrm{Osp}(4/N)\mathrm{}$ a holographic formulation of extended supergravity is obtained, as will be described in detail in a subsequent paper.