Branes and Six Dimensional Supersymmetric Theories

We consider configurations of six-branes, five-branes and eight-branes in various superstring backgrounds. These configurations give rise to $(0,1)$ supersymmetric theories in six dimensions. The condition for RR charge conservation of a brane configuration translates to the condition that the corresponding field theory is anomaly-free. Sets of infinitely many models with non-trivial RG fixed points at strong coupling are demonstrated. Some of them reproduce and generalise the world-volume theories of SO(32) and $E_8\times E_8$ small instantons. All the models are shown to be connected by smooth transitions. In particular, the small instanton transition for which a tensor multiplet is traded for 29 hypermultiplets is explicitly demonstrated. The particular limit in which these theories can be considered as six-dimensional string theories without gravity are discussed. New fixed points (string theories) associated with $E_n$ global symmetries are discovered by taking the strong string coupling limit.