Aspects of Closed Matricial Worlds
We investigate the Hilbert space structure for simple theories of gravity with $\Lambda>0$ on closed spatial sections. Our motivation ties to the presence of gravitational saddles in four-dimensional $\Lambda>0$ Einstein-Maxwell theory with $S^2\times \Sigma_h$ topology, where $\Sigma_h$ is a genus-$h$ Riemann surface. Here, as a concrete starting point, the problem is explored for two-dimensional $\Lambda>0$ quantum gravity. We revisit and elaborate on exact results in the matrix model literature. We study gravitational wavefunctions from both the perspective of the Wheeler-DeWitt equation and the gravitational path integral. Though simple, the setting displays many features of general interest such as large volume effects that disrupt the perturbative expansion, topological corrections to the path-integral wavefunction that offend the exact Wheeler-DeWitt equation, and a sphere path integral ${Z}^{(0)}_{\text{grav}}$ with non-trivial structure in $\Lambda$. We discuss candidate inner products for the infinite-dimensional canonical gravitational Hilbert space uncovered by Lian and Zuckerman. By establishing explicit results up to genus-six, we argue that the dominant contribution to the genus-$h$ gravitational disk path integral at large spatial size mimics the behavior of two-dimensional topological gravity. In passing, we show that for ${Z}^{(0)}_{\text{grav}}$ to give rise to a positive counting problem for discretised Riemann surfaces, it must have a negative pre-factor. We contrast our analysis to the more realistic case of timelike Liouville theory.