$E_8$ geometry

TL;DR

This paper explores the geometric structure of $E_8$ symmetry, defining covariant transformations, and constructing a curvature tensor, advancing the understanding of exceptional generalized geometry.

Contribution

It introduces a covariant tensor formalism for $E_8$ generalized diffeomorphisms, including a method to define a spin connection and curvature tensor.

Findings

Defined covariant field-dependent transformations with connections

Constructed a curvature tensor for $E_8$ geometry

Outlined a geometry related to Ehlers SL(n+1) symmetry

Abstract

We investigate exceptional generalised diffeomorphisms based on $E_{8(8)}$ in a geometric setting. The transformations include gauge transformations for the dual gravity field. The surprising key result, which allows for a development of a tensor formalism, is that it is possible to define field-dependent transformations containing connection, which are covariant. We solve for the spin connection and construct a curvature tensor. A geometry for the Ehlers symmetry SL(n+1) is sketched. Some related issues are discussed.