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 greater t han or equal to 3. For this, we use the DSZ quantization in d = 4 as origin ally proposed by Hull and Townsend, and determine the discrete group induci ng integer shifts on the charge lattice. It is generated by fundamental uni potents, which are constructed by exponentiating the Chevalley generators o f the corresponding Lie algebra. We then extend a method suggested by the a bove authors and used by Sen for the heterotic string to obtain the discret e U-duality group in d = 3, thereby obtaining a quantized symmetry in d = 3 from a d = 4 quantization condition. This is studied first in a toy model, corresponding to d = 5 simple supergravity, and then applied to M-theory. It turns out that, in the: top model, the resulting u-duality group in d = 3 is strictly smaller than the one generated by the fundamental unipotents corresponding to all Chevalley generators. However, for M-theory, both grou ps agree. We illustrate the compactification to d = 3 by an embedding of d = 4 particle multiplets into the d = 3 theory.