The semiclassical gravitational path integral and random matrices (toward a microscopic picture of a dS2 universe)
Dionysios Anninos, Beatrix Mühlmann
Abstract
A bstract We study the genus expansion on compact Riemann surfaces of the gravitational path integral $$ {\mathcal{Z}}_{\mathrm{grav}}^{(m)} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mtext>grav</mml:mtext> <mml:mfenced> <mml:mi>m</mml:mi> </mml:mfenced> </mml:msubsup> </mml:math> in two spacetime dimensions with cosmological constant Λ > 0 coupled to one of the non-unitary minimal models ℳ 2 m − 1, 2 . In the semiclassical limit, corresponding to large m , $$ {\mathcal{Z}}_{\mathrm{grav}}^{(m)} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mtext>grav</mml:mtext> <mml:mfenced> <mml:mi>m</mml:mi> </mml:mfenced> </mml:msubsup> </mml:math> 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 2 m − 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 2 m − 1, 2 for genus h ≥ 2. Combining these results we arrive at a semiclassical expression for $$ {\mathcal{Z}}_{\mathrm{grav}}^{(m)} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mtext>grav</mml:mtext> <mml:mfenced> <mml:mi>m</mml:mi> </mml:mfenced> </mml:msubsup> </mml:math> . Conjecturally, $$ {\mathcal{Z}}_{\mathrm{grav}}^{(m)} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mtext>grav</mml:mtext> <mml:mfenced> <mml:mi>m</mml:mi> </mml:mfenced> </mml:msubsup> </mml:math> 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 2 m . 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.