We study a quantum corrected SO(6) invariant matrix quantum mechanics obtained from the s-wave modes of the scalars of N = 4 SYM on S 3 . For commuting matrices, this model is believed to describe the 1/8 BPS states of the full SYM theory. In the large N limit the ground state corresponds to a distribution of eigenvalues on a S 5 which we identify with the sphere on the dual geometry AdS 5 × S 5 . We then consider non-BPS excitations by studying matrix perturbations where the off-diagonal modes are treated perturbatively. To a first approximation, these modes can be described by a free theory of "string bits" whose energies depend on the diagonal degrees of freedom. We then consider a state with two string bits and large angular momentum J on the sphere. In the large J limit we use a simple saddle point approximation to show that the energy of these states coincides precisely with the BMN spectrum to all orders in the 't Hooft coupling. We also find some new problems with the all loop Bethe Ansatz conjecture of the N = 4 SYM planar spin chain model.

We prove that in the presence of a maximal giant graviton state in N = 4 SYM, the states dual to open strings attached to the giant graviton give rise to an P SU(2, 2|4) open spin chain model with integrable boundary conditions in the SO(6) sector of the spin chain to one loop order.

We systematically study the spectrum of open strings attached to half BPS giant gravitons in the N = 4 SYM AdS/CFT setup. We find that some null trajectories along the giant graviton are actually null geodesics of AdS 5 × S 5 , so that we can study the problem in a plane wave limit setup. We also find the description of these states at weak 't Hooft coupling in the dual CFT. We show how the dual description is given by an open spin chain with variable number of sites. We analyze this system in detail and find numerical evidence for integrability. We also discover an interesting instability of long open strings in Ramond-Ramond backgrounds that is characterized by having a continuum spectrum of the string, which is separated from the ground state by a gap. This instability arises from accelerating the Dbrane on which the strings end via the Ramond-Ramond field. From the integrable spin chain point of view, this instability prevents us from formulating the integrable structure in terms of a Bethe Ansatz construction.

We present a simple procedure to construct non-local conserved charges for classical open strings on coset spaces. This is done by including suitable reflection matrices on the classical transfer matrix. The reflection matrices must obey certain conditions for the charges to be conserved and in involution. We then study bosonic open strings on AdS 5 ×S 5 . We consider boundary conditions corresponding to Giant Gravitons on S 5 , AdS 4 × S 2 D5-branes and AdS 5 × S 3 D7-branes. We find that we can construct the conserved charges for the full bosonic string on a Maximal Giant Graviton or a D7-brane. For the D5-brane, we find that this is possible only in a SU(2) sub-sector of the open string. Moreover, the charges can not be constructed at all for non-maximal Giant Gravitons. We discuss the interpretation of these results in terms of the dual gauge theory spin chains.

We argue that higher-curvature terms in the gravitational Lagrangian lead, via non-relativistic gauge-gravity duality, to finite renormalization of the dynamical exponent of the dual conformal field theory. Our argument includes a proof of the non-renormalization of the Schrödinger and Lifshitz metrics beyond rescalings of their parameters, directly generalizing the AdS case. We use this effect to construct string-theory duals of non-relativistic critical systems with non-integer dynamical exponents, then use these duals to predict the viscosity/entropy ratios of these systems. The predicted values weakly violate the KSS bound.