Integral group actions on symmetric spaces and discrete duality symmetries of supergravity theories

TL;DR

This paper develops a method to compute Iwasawa coordinates for symmetric spaces associated with supergravity theories and explicitly describes the action of discrete U-duality groups on scalar fields and charge lattices.

Contribution

It introduces a new approach for calculating Iwasawa coordinates using Chevalley generators and Steinberg presentation, and explicitly determines the action of integral U-duality groups on supergravity scalar cosets.

Findings

Explicit formulas for $SL_2( ext{Z})$ and $E_7( ext{Z})$ actions on scalar cosets.

Identification of the spectrum-generating symmetry group for BPS solitons.

Description of discrete symmetries of the dyonic charge lattice.

Abstract

For $G(\mathbb{R})$ a split, simply connected, semisimple Lie group of rank $n$ and $K$ the maximal compact subgroup of $G$, we give a method for computing Iwasawa coordinates of $G/K$ using the Chevalley generators and the Steinberg presentation. When $G/K$ is a scalar coset for a supergravity theory in dimensions $\geq 3$, we determine the action of the integral form $G(\mathbb{Z})$ on $G/K$. We give explicit results for the action of the discrete $U$--duality groups $SL_2(\mathbb{Z})$ and $E_7(\mathbb{Z})$ on the scalar cosets $SL_2(\mathbb{R})/SO_2(\mathbb{R})$ and $E_{7(+7)}(\mathbb{R})/[SU(8,\mathbb{R})/\{\pm Id\}]$ for type IIB supergravity in ten dimensions and 11--dimensional supergravity in $D=4$ dimensions, respectively. For the former, we use this to determine the discrete U--duality transformations on the scalar sector in the Borel gauge and we describe the discrete symmetries of the dyonic charge lattice. We determine the spectrum--generating symmetry group for fundamental BPS solitons of type IIB supergravity in $D=10$ dimensions at the classical level and we propose an analog of this symmetry at the quantum level. We indicate how our methods can be used to study the orbits of discrete U--duality groups in general.