On Discrete U-duality in M-theory

TL;DR

This paper characterizes the discrete U-duality groups in M-theory compactified on tori, providing explicit generators and extending methods to lower dimensions, revealing differences between toy models and full M-theory.

Contribution

It explicitly determines the generators of discrete U-duality groups in M-theory for various dimensions and extends existing methods to derive these groups from lower-dimensional quantization conditions.

Findings

In d=4, the discrete U-duality group is generated by Chevalley unipotents.

In d=3, the U-duality group matches the one generated by all Chevalley generators for M-theory.

The toy model in d=5 shows a smaller U-duality group than the full Chevalley-generated group.

Abstract

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 group inducing integer shifts on the charge lattice. It is generated by fundamental unipotents, which are constructed by exponentiating the Chevalley generators of the corresponding Lie algebra. 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 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 toy 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 groups agree. We illustrate the compactification to d=3 by an embedding of d=4 particle multiplets into the d=3 theory.