Fivebranes from gauge theory

We study theories with 16 supercharges and a discrete energy spectrum. One class of theories has symmetry group $SU(2|4)$. They arise as truncations of N=4 super Yang-Mills theory. They include the plane-wave matrix model, $2+1$ super Yang-Mills theory on R×S² and N=4 super Yang-Mills theory on R×S³/Zₖ. We explain how to obtain their gravity duals in a unified way. We explore the regions of the geometry that are relevant for the study of some $1/2$ BPS and near BPS states. This leads to a class of two dimensional (4,4) supersymmetric sigma models with nonzero H flux, including a massive deformed WZW model. We show how to match some features of the string spectrum with the Yang-Mills theory. The other class of theories are also connected to N=4 super Yang-Mills theory and arise by making some of the transverse scalars compact. Their vacua are characterized by a 2D Yang-Mills theory or 3D Chern-Simons theory. These theories realize peculiar super-Poincar\'e symmetry algebras in $2+1$ or $1+1$ dimensions with ``noncentral'' charges. We finally discuss gravity duals of N=4 super Yang-Mills theory on AdS₃×S¹.