Large N BPS states and emergent quantum gravity

This paper provides a heuristic derivation of how classical gravitational physics in the AdS/CFT correspondence appears from the strong dynamics of the N = 4 SYM theory in a systematic way. We do this in a minisuperspace approximation by studying 1/8 BPS configurations. We can show that our description matches the semiclassical physics of 1/8 BPS states in supergravity. We also provide a heuristic description of how massive strings appear in the geometry, and how at strong 't Hooft coupling they become local on the S5 suggesting that they can be realized as a sigma model on a weakly curved background. We show that the dynamics of 1/8 BPS dynamics of N = 4 SYM on a round S3 can be reduced to that of a matrix model for commuting matrices. Including measure factors, we show that this effective dynamics is related to bosons living on a six dimensional phase space with repulsive interactions. Because of these interactions, we can argue that on the ground state the bosons assemble themselves on a spherical shell in the shape of a round five sphere. This sphere will be identified with the S5 in the AdS dual geometry. To do this, we first define a precise way to coarse grain the dynamics. We use half BPS configurations as a toy model for this coarse graining, and we can reproduce the droplet picture of these half BPS states systematically. The droplet appears as the saddle point approximation of a statistical ensemble related to the square of the wave function of the eigenvalues of a complex matrix. This procedure is also applied to the set of 1/8 BPS configurations to extract the geometry, giving an analog of the droplet picture of half BPS states for the case of 1/8 BPS configurations. We also have a conjectured realization of some 1/8 BPS giant graviton wave functions in the dynamics, which captures all 1/8 BPS giant gravitons constructed by Mikhailov. This leads to a lot of different topology changes which can be treated heuristically