On the Scalar Manifold of Exceptional Supergravity

TL;DR

This paper constructs two parametrizations of the exceptional Lie group E7(-25) relevant to supergravity, highlighting its geometric structure and symmetries, with potential generalizations to other E7-type groups.

Contribution

It provides explicit realizations of the scalar manifold of exceptional supergravity using exponential map and Iwasawa decomposition, emphasizing covariance and triality.

Findings

Realization of G/K with E6-invariant tensor showing maximal covariance.

Iwasawa decomposition highlighting SO(8) covariance and triality.

Framework generalizable to broader classes of E7-related groups.

Abstract

We construct two parametrizations of the non compact exceptional Lie group G=E7(-25), based on a fibration which has the maximal compact subgroup K=(E6 x U(1))/Z_3 as a fiber. It is well known that G plays an important role in the N=2 d=4 magic exceptional supergravity, where it describes the U-duality of the theory and where the symmetric space M=G/K gives the vector multiplets' scalar manifold. First, by making use of the exponential map, we compute a realization of G/K, that is based on the E6 invariant d-tensor, and hence exhibits the maximal possible manifest [(E6 x U(1))/Z_3]-covariance. This provides a basis for the corresponding supergravity theory, which is the analogue of the Calabi-Vesentini coordinates. Then we study the Iwasawa decomposition. Its main feature is that it is SO(8)-covariant and therefore it highlights the role of triality. Along the way we analyze the relevant chain of maximal embeddings which leads to SO(8). It is worth noticing that being based on the properties of a "mixed" Freudenthal-Tits magic square, the whole procedure can be generalized to a broader class of groups of type E7.