New theories in six dimensions and matrix description of M-theory on T5 and T5Z2

We present four infinite series of new quantum theories with super-Poincare symmetry in six dimensions, which are not local quantum field theories. They have string like excitations but the string is of order one. Compactifying these theories on T5 we find a Matrix theory description of M theory on T5 and on T5Z2, which is well defined and is manifestly U-duality invariant.