The semiclassical gravitational path integral and random matrices (toward a microscopic picture of a dS2 universe)

Abstract

We study the genus expansion on compact Riemann surfaces of the gravitational path integral $$ {\mathcal{Z}}_{\mathrm{grav}}^{(m)} $$ Z 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, $$ {\mathcal{Z}}_{\mathrm{grav}}^{(m)} $$ Z 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 $$ {\mathcal{Z}}_{\mathrm{grav}}^{(m)} $$ Z grav m . Conjecturally, $$ {\mathcal{Z}}_{\mathrm{grav}}^{(m)} $$ Z 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.

Quasi-local energy and microcanonical entropy in two-dimensional nearly de Sitter gravity

“…For example, the number of degrees of freedom increases in the IR direction of the microscopic theory, indicating the dual quantum theory description of dS is unlikely to be a local quantum field theory. Additional evidence that the underlying microscopics of de Sitter space is not well characterized by a local quantum field theory has been given via matrix model descriptions of dS 2 [94]. Further, the UV/IR connection for dS appears to be inverted: long distances (IR) in the bulk correspond to low energies (IR) in the microscopic theory.…”

Snowmass White Paper: Cosmology at the Theory Frontier

“…Recent developments in this interesting direction include [291]. Moreover, Euclidean calculations starting from global low-dimensional de Sitter such as [262,292] yield formulas expressible as a sum over microstate contributions. These in turn may connect to lower-dimensional versions of the T T +Λ deformation generalizing [281] as in [273] to yield a Lorentzian microstate count as reviewed for the bulk three-dimensional case above.…”

Snowmass White Paper: Implications of Quantum Gravity for Particle Physics

“…This might account for the curious property that local excitations in de Sitter space lower the total entropy [118] (see also [120,121]), which was given a path integral description in terms of constrained states in [122]. Recent work [123,124] discusses possible matrix model descriptions of dS 2 . Along complementary lines there has also been rapid progress in the construction of holographic dualities for dS 3 , utilizing generalizations of the T T deformation [125][126][127] (see also [128]).…”