Gravity as an ensemble and the moment problem

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 criterion is relevant for an unambiguous determination of quantities such as <(log Z(beta))over bar> 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.

Automated summary

This paper discusses the moment problem in gravity theories, including criteria for existence and uniqueness, as well as the potential ensemble interpretation and data incompleteness. Through a case study of 2D gravities, the importance of the uniqueness criterion for the unambiguous determination of certain quantities is demonstrated.