Duality constructions from quantum state manifolds

TL;DR

This paper develops a geometric framework for quantum state manifolds using Riemannian and symplectic structures, connecting quantum symmetries with classical gravity models, especially in the AdS_2/CFT_1 context.

Contribution

It introduces a geometric approach to quantum state spaces that links quantum symmetries with classical gravity solutions, extending to the AdS_2/CFT_1 correspondence.

Findings

State manifolds are asymptotically AdS solutions of dilaton gravity.

The radial coordinate acts as an energy scale in the geometry.

Dilaton fields correspond to expectation values of symmetry generators.

Abstract

The formalism of quantum state space geometry on manifolds of generalised coherent states is proposed as a natural setting for the construction of geometric dual descriptions of non-relativistic quantum systems. These state manifolds are equipped with natural Riemannian and symplectic structures derived from the Hilbert space inner product. This approach allows for the systematic construction of geometries which reflect the dynamical symmetries of the quantum system under consideration. We analyse here in detail the two dimensional case and demonstrate how existing results in the AdS_2/CFT_1 context can be understood within this framework. We show how the radial/bulk coordinate emerges as an energy scale associated with a regularisation procedure and find that, under quite general conditions, these state manifolds are asymptotically anti-de Sitter solutions of a class of classical dilaton gravity models. For the model of conformal quantum mechanics proposed by de Alfaro et. al. the corresponding state manifold is seen to be exactly AdS_2 with a scalar curvature determined by the representation of the symmetry algebra. It is also shown that the dilaton field itself is given by the quantum mechanical expectation values of the dynamical symmetry generators and as a result exhibits dynamics equivalent to that of a conformal mechanical system.