We explain how the string spectrum in flat space and pp-waves arises from the large N limit, at fixed g 2 Y M , of U(N) N = 4 super Yang Mills. We reproduce the spectrum by summing a subset of the planar Feynman diagrams. We give a heuristic argument for why we can neglect other diagrams.We also discuss some other aspects of pp-waves and we present a matrix model associated to the DLCQ description of the maximally supersymmetric eleven dimensional pp-waves.

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The operator product expansion for "small" Wilson loops in N = 4, d = 4 SYM is studied. The OPE coefficients are calculated in the large N and g 2 YM N limit by exploiting the AdS/CFT correspondence. We also consider Wilson surfaces in the (0, 2), d = 6 superconformal theory. In this case, we find that the UV divergent terms include a term proportional to the rigid string action.

We study the large N gauged quantum mechanics for a single Hermitian matrix in the Harmonic oscillator potential well as a toy model for the AdS/CFT correspondence. We argue that the dual geometry should be a string in two dimensions with a curvature of stringy size. Even though the dual geometry is not weakly curved, one can still gain knowledge of the system from a detailed study of the openclosed string duality. We give a mapping between the basis of states made of traces (closed strings) and the eigenvalues of the matrix (D-brane picture) in terms of Schur polynomials. This is interpreted as an exact open-closed duality. We connect this model with a decoupling limit of N = 4 SYM and the study of giant gravitons in AdS 5 × S 5 . We show that the two giant gravitons that expand along AdS 5 and S 5 can be interpreted in the matrix model as taking an eigenvalue from the Fermi sea and exciting it very much, or as making a hole in the Fermi sea respectively. This is similar to recent studies of the c = 1 string. This connection gives new insight on how to perform calculations for giant gravitons.

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 S 5 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 S 3 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 S 5 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.