Reconstructing 1/2 BPS Space-Time Metrics from Matrix Models and Spin Chains

Using the AdS/CFT correspondence, we address the question of how to measure complicated space-time metrics using gauge theory probes. In particular, we consider the case of the 1/2 BPS geometries of type IIB supergravity. These geometries are classified by certain "droplets" in a two dimensional space-like hypersurface. We show how to reconstruct the full metric inside these droplets using the one-loop ${\cal N} = 4$ SYM theory dilatation operator. This is done by considering long operators in the SU(2) sector, which are dual to fast rotating strings on the droplets. We develop new powerful techniques for large $N$ complex matrix models that allow us to construct the Hamiltonian for these strings. We find that the Hamiltonian can be mapped to a "dynamical" spin chain. That is, the length of the chain is not fixed. Moreover, all of these spin chains can be explicitly constructed using an interesting algebra which is derived from the matrix model. Our techniques work for general droplet configurations. As an example, we study a single elliptical droplet and the "Hypotrochoid".