On discrete U-duality in M-theory

We give a complete set of generators for the discrete exceptional U-duality groups of toroidal compactified type II theory and M-theory in d>2. For this, we use the DSZ quantization in d=4 as originally proposed by Hull and Townsend, and determine the discrete groups inducing integer shifts on the charge lattice. We then extend a method suggested by the above authors and used by Sen for the heterotic string to get the discrete U-duality group in d=3, thereby obtaining a quantized symmetry in d=3 from a d=4 quantization condition. This method is also studied in a toy model corresponding to d=5 simple supergravity, which turns out to give a significantly different result, even though it can be identified with a truncation of M-theory. We illustrate the compactification to d=3 by an embedding of d=4 particle multiplets into the d=3 theory.