Motivated by the close connection of tensor models to the SYK model, we use representation theory to construct the complete set of gauge invariant observables for bosonic and fermionic tensor models. Correlation functions of the gauge invariant operators in the free theory are computed exactly. The gauge invariant operators close a ring. The structure constants of the ring are described explicitly. Finally, we construct a collective field theory description of the bosonic tensor model.

By performing explicit computations of correlation functions, we find evidence that there is a sector of the two matrix model defined by the SU (2) sector of N = 4 super Yang-Mills theory, that can be reduced to eigenvalue dynamics. There is an interesting generalization of the usual Van der Monde determinant that plays a role. The observables we study are the BPS operators of the SU (2) sector and include traces of products of both matrices, which are genuine multi matrix observables. These operators are associated to supergravity solutions of string theory.1 robert@neo.phys.wits.ac.za 2

We study excitations of LLM geometries. These geometries arise from the backreaction of a condensate of giant gravitons. Excitations of the condensed branes are open strings, which give rise to an emergent Yang-Mills theory at low energy. We study the dynamics of the planar limit of these emergent gauge theories, accumulating evidence that they are planar N = 4 super Yang-Mills. There are three observations supporting this conclusion: (i) we argue for an isomorphism between the planar Hilbert space of the original N = 4 super Yang-Mills and the planar Hilbert space of the emergent gauge theory, (ii) we argue that the OPE coefficients of the planar limit of the emergent gauge theory vanish and (iii) we argue that the planar spectrum of anomalous dimensions of the emergent gauge theory is that of planar N = 4 super Yang-Mills. Despite the fact that the planar limit of the emergent gauge theory is planar N = 4 super Yang-Mills, we explain why the emergent gauge theory is not N = 4 super Yang-Mills theory. 1 robert@neo.phys.wits.ac.za

Large N but non-planar limits of N = 4 super Yang-Mills theory can be described using restricted Schur polynomials. Previous investigations demonstrate that the action of the one loop dilatation operator on restricted Schur operators, with classical dimension of order N and belonging to the su(2) sector, is largely determined by the su(2) R symmetry algebra as well as structural features of perturbative field theory. Studies presented so far have used the form of R symmetry generators when acting on small perturbations of half-BPS operators. In this article, as a first step towards going beyond small perturbations of the half-BPS operators, we explain how the exact action of symmetry generators on restricted Schur polynomials can be determined.