Magic Coset Decompositions

TL;DR

This paper analyzes two types of coset decompositions of a specific non-compact symmetric space related to supergravity, exploring their covariance properties and generalizations within the context of exceptional Lie groups and Jordan algebras.

Contribution

It introduces and compares two coset decomposition methods for a special Kähler symmetric space, including their covariance features and extensions to other non-compact real forms of E7.

Findings

First decomposition has maximal manifest covariance.

Second decomposition is triality-symmetric and of Iwasawa type.

Generalizations connect to Jordan algebras and division algebras.

Abstract

By exploiting a "mixed" non-symmetric Freudenthal-Rozenfeld-Tits magic square, two types of coset decompositions are analyzed for the non-compact special K\"ahler symmetric rank-3 coset E7(-25)/[(E6(-78) x U(1))/Z_3], occurring in supergravity as the vector multiplets' scalar manifold in N=2, D=4 exceptional Maxwell-Einstein theory. The first decomposition exhibits maximal manifest covariance, whereas the second (triality-symmetric) one is of Iwasawa type, with maximal SO(8) covariance. Generalizations to conformal non-compact, real forms of non-degenerate, simple groups "of type E7" are presented for both classes of coset parametrizations, and relations to rank-3 simple Euclidean Jordan algebras and normed trialities over division algebras are also discussed.