Holographic Cosmology on Closed Slices in 2+1 Dimensions

We apply the framework of Cauchy Slice Holography to the quantization of gravity on closed slices with $\Lambda>0$ (with a focus on $2+1$ dimensions for concreteness). We obtain solutions to the Wheeler-DeWitt equation in a basis of CPT-dual branches. Each branch is a $T^2$-deformed CFT partition function with imaginary central charge. We compute explicit solutions in $2+1$ dimensions in a minisuperspace toy model of pure gravity. This analysis gives us evidence to conjecture a connection between the choice of superposition of branches and the choice of class of geometries to sum over in the gravitational path integral. We further show that, in full quantum gravity on closed slices, bulk unitarity holds, despite the Euclidean holographic field theory not being reflection-positive. We conjecture about the (non)uniqueness of the quantum state of the Universe, in light of this analysis.