Holography and Quantum Gravity in 3+1 Dimensions

A holographic theory on N is described by a holographic dual theory on the boundary of N with action S, an integral over the boundary of N . Thus, in the limit of a vanishing boundary, the action of the holographic dual is S = 0, the action of a topological field theory. So, if general relativity is holographic, then when formulated on M a manifold without boundary it must be topological. In this article we prove that on M , a 3 + 1 dimensional manifold without boundary, general relativity with a positive cosmological constant is equivalent to a so(4, 1) Donaldson-Witten theory. For the case of a negative cosmological constant, we find it is equivalent to a so(3, 2) Donaldson-Witten theory. As theories of Donaldson-Witten type are exactly solvable, this proves that exact, non-perturbative calculations can be preformed in 3+ 1 dimensional quantum gravity. We then go on to perform a representative set of such exact calculations. Next we show that the familiar structure of general relativity arises through spontaneous symmetry breaking of the Donaldson-Witten theory’s “topological symmetry.” We then prove that general relativity on N , a 3 + 1 dimensional manifold with boundary, is holographic, i.e. all local degrees of freedom in the bulk are described by a theory residing on the boundary. Next we derive the holographic dual of general relativity and find a so(4, 1) Chern-Simons theory in the case of positive cosmological constant and a so(3, 2) Chern-Simons theory in the case of a negative cosmological constant. Finally, we conclude with some remarks of a general sort on the properties possessed by any holographic theory.