Gravity as an ensemble and the moment problem

TL;DR

This paper explores the connection between gravitational path integrals and ensemble averages of boundary theories, using the moment problem to analyze existence and uniqueness, with implications for 2D gravities like JT supergravity.

Contribution

It applies the Stieltjes moment problem criteria to gravitational theories, revealing conditions for ensemble interpretations and the necessity of nonperturbative methods for uniqueness.

Findings

Existence criteria can rule out certain ensemble interpretations.

Perturbation theory fails in JT gravity at large boundary numbers.

Nonperturbative completion is needed for definitive results.

Abstract

If a bulk gravitational path integral can be identified with an average of partition functions over an ensemble of boundary quantum theories, then a corresponding moment problem can be solved. We review existence and uniqueness criteria for the Stieltjes moment problem, which include an infinite set of positivity conditions. The existence criteria are useful to rule out an ensemble interpretation of a theory of gravity, or to indicate incompleteness of the gravitational data. We illustrate this in a particular class of 2D gravities including variants of the CGHS model and JT supergravity. The uniqueness criterium is relevant for an unambiguous determination of quantities such as $\overline{\log Z(\beta)}$ or the quenched free energy. We prove in JT gravity that perturbation theory, both in the coupling which suppresses higher-genus surfaces and in the temperature, fails when the number of boundaries is taken to infinity. Since this asymptotic data is necessary for the uniqueness problem, the question cannot be settled without a nonperturbative completion of the theory.