The semiclassical gravitational path integral and random matrices

Abstract

We study the genus expansion on compact Riemann surfaces of the gravitational path integral Z (m) grav in two spacetime dimensions with cosmological constant Λ > 0 coupled to one of the non-unitary minimal models M 2m−1,2 . In the semiclassical limit, corresponding to large m, Z (m) grav admits a Euclidean saddle for genus h ≥ 2. Upon fixing the area of the metric, the path integral admits a round two-sphere saddle for h = 0. We show that the OPE coefficients for the minimal weight operators of M 2m−1,2 grow exponentially in m at large m. Employing the sewing formula, we use these OPE coefficients to obtain the large m limit of the partition function of M 2m−1,2 for genus h ≥ 2. Combing these results we arrive at a semiclassical expression for Z (m) grav . Conjecturally, Z (m) grav admits a completion in terms of an integral over large random Hermitian matrices, known as a multicritical matrix integral. This matrix integral is built from an even polynomial potential of order 2m. We obtain explicit expressions for the large m genus expansion of multicritical matrix integrals in the double scaling limit. We compute invariant quantities involving contributions at different genera, both from a matrix as well as a gravity perspective, and establish a link between the two pictures. Inspired by the proposal of Gibbons and Hawking relating the de Sitter entropy to a gravitational path integral, our setup paves a possible path toward a microscopic picture of a two-dimensional de Sitter universe.