Airy Function and 4d Quantum Gravity

We study four-dimensional quantum gravity with negative cosmological constant in the minisuperspace approximation and compute the partition function for the $S^3$ boundary geometry. In this approximation scheme the path integrals become dominated by a class of asymptotically AdS "microstate geometries." Despite the fact that the theory is pure Einstein gravity without supersymmetry, the result precisely reproduces, up to higher curvature corrections, the Airy function in the $S^3$ partition function of the maximally supersymmetric Chern-Simons-matter (CSM) theory which sums up all perturbative $1/N$ corrections. We also show that this can be interpreted as a concrete realization of the idea that the CFT partition function is a solution to the Wheeler-DeWitt equation as advocated in the holographic renormalization group. Furthermore, the agreement persists upon the inclusion of a string probe and it reproduces the Airy function in the vev of half-BPS Wilson loops in the CSM theory. These results may suggest that the supergravity path integrals localize to the minisuperspace in certain cases and the use of the minisuperspace approximation in AdS/CFT may be a viable approach to study $1/N$ corrections to large $N$ CFTs.