Abstract
We study the genus expansion on compact Riemann surfaces of the gravitational path integral \( {\mathcal{Z}}_{\mathrm{grav}}^{(m)} \) in two spacetime dimensions with cosmological constant Λ > 0 coupled to one of the non-unitary minimal models ℳ2m − 1, 2. In the semiclassical limit, corresponding to large m, \( {\mathcal{Z}}_{\mathrm{grav}}^{(m)} \) 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 ℳ2m − 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 ℳ2m − 1, 2 for genus h ≥ 2. Combining these results we arrive at a semiclassical expression for \( {\mathcal{Z}}_{\mathrm{grav}}^{(m)} \). Conjecturally, \( {\mathcal{Z}}_{\mathrm{grav}}^{(m)} \) 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.
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Anninos, D., Mühlmann, B. The semiclassical gravitational path integral and random matrices (toward a microscopic picture of a dS2 universe). J. High Energ. Phys. 2021, 206 (2021). https://doi.org/10.1007/JHEP12(2021)206
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DOI: https://doi.org/10.1007/JHEP12(2021)206