In this talk, I will describe a construction of Hilbert spaces and von Neumann algebras from any UV-completion of asymptotically anti-de Sitter quantum gravity with a Euclidean path integral satisfying a simple and familiar set of axioms. We consider a quantum context in which a standard Lorentz-signature classical bulk limit would have Cauchy slices with two asymptotic boundaries (left and right), both of which are compact manifolds without boundary. Our main result is then that the quantum gravity path integral defines (left and right) type I von Neumann algebras of observables acting respectively at the left and right boundaries, such that the two algebras are commutants. The path integral also defines entropies on the von Neumann algebras. The entropies can also be written in terms of standard density matrices and standard Hilbert space traces. Furthermore, in appropriate semiclassical limits our entropies are computed by the RT-formula with quantum corrections. Our work thus provides a Hilbert space interpretation of the Ryu-Takayanagi entropy. Since our axioms do not restrict UV bulk structures, they may be expected to hold equally well for successful formulations of string field theory, spin-foam models, or any other approach to constructing a UV-complete theory of gravity.
