Quantum Cosmology in 2+1 Dimensions

In the search for simple, solvable models of quantum cosmology it is useful to focus on general relativity in 2+1 dimensions, where - due to the absence of local degrees of freedom - there is some hope that we can say something precise.  I will present two results in this direction.  The first involves cosmological solutions with a negative cosmological constant, where we can study universes where the spatial slice is a torus.  Here, cosmological observables can be related to standard observables in a boundary conformal field theory.  In particular, the amplitude for a cosmology is related to a two point correlation function in the CFT density of states, similar to the spectral form factor studied in the quantum chaos literature.  We then turn to cosmology with a positive cosmological constant, where I will describe progress on the description of the Hilbert space of states of de Sitter quantum gravity in the static patch.  I will use path integral methods compute the spectrum of static patch Hamiltonian exactly.  With a certain prescription we find a surprising result: the spectrum of the Hamiltonian is bounded and has (in some cases) an integer degeneracy of states.  The spectrum includes both positive and negative norm states, which we interpret as necessary in order to obtain a finite de Sitter entropy.  Indeed, the thermal entropy agrees with the Bekenstein-Hawking entropy of the de Sitter event horizon, including the precise numerical coefficient.