Researchers have investigated the Hilbert space structure for theories of gravity with a positive cosmological constant (Λ>0) on closed spatial sections. This work is motivated by the presence of gravitational saddles in four-dimensional Λ>0 Einstein-Maxwell theory with S²×Σh topology, where Σh is a genus-h Riemann surface. As a concrete starting point, the problem is explored for two-dimensional Λ>0 quantum gravity, revisiting and elaborating on exact results in the matrix model literature.

The study examines gravitational wavefunctions from both the perspective of the Wheeler-DeWitt equation and the gravitational path integral. Although simple, this 2D 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 contradict the exact Wheeler-DeWitt equation, and a sphere path integral Z⁽⁰⁾grav with non-trivial structure in Λ. Candidate inner products for the infinite-dimensional canonical gravitational Hilbert space uncovered by Lian and Zuckerman are discussed.

By establishing explicit results up to genus-six, the authors 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, they show that for Z⁽⁰⁾grav to give rise to a positive counting problem for discretised Riemann surfaces, it must have a negative pre-factor. This analysis is contrasted with the more realistic case of timelike Liouville theory.