The semiclassical gravitational path integral and random matrices

Abstract

We study the genus expansion on compact Riemann surfaces of the gravitational path integral ${\mathcal{Z}}_{\text{grav}}^{(m)}$ in two spacetime dimensions with cosmological constant $\Lambda >0$ coupled to one of the non-unitary minimal models ${\mathcal{M}}_{2m-1,2}$. In the semiclassical limit, corresponding to large $m$, ${\mathcal{Z}}_{\text{grav}}^{(m)}$ admits a Euclidean saddle for genus $h\ge 2$. 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 ${\mathcal{M}}_{2m-1,2}$ 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 ${\mathcal{M}}_{2m-1,2}$ for genus $h\ge 2$. Combining these results we arrive at a semiclassical expression for ${\mathcal{Z}}_{\text{grav}}^{(m)}$. Conjecturally, ${\mathcal{Z}}_{\text{grav}}^{(m)}$ 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.

arXiv comment21 pages + appendices, v2 typos corrected + references added, v3 typos corrected

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