U-duality and BPS spectrum of super Yang-Mills theory and M-theory

It is shown that the BPS spectrum of Super-Yang-Mills theory on Td × , which fits into representations of the U-duality group for M-theory compactified on Td, in accordance with the matrix-theory conjecture, in fact fits into representations of the U-duality group for M-theory on Td+1, with the extra dualities realised as generalised Nahm transformations. The spectrum of BPS M-branes is analysed, new branes are discussed and matrix theory applications described.