[Paper Review] $E_8$ geometry

Martin Cederwall, J. A. Rosabal|arXiv (Cornell University)|Apr 19, 2015

Black Holes and Theoretical Physics1 references4 citations

TL;DR

This paper develops a geometric framework for exceptional generalised diffeomorphisms based on $E_{8(8)}$, introducing field-dependent, covariant transformations that include a spin connection and curvature tensor. The key contribution is the construction of a consistent tensor formalism for dual gravity via covariant connections, enabling a geometric formulation of the $E_8$ symmetry and sketching related structures for SL(n+1).

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.

Motivation & Objective

• To establish a geometric formulation of exceptional generalised diffeomorphisms based on $E_{8(8)}$.
• To resolve the challenge of defining consistent, field-dependent gauge transformations in dual gravity.
• To construct a covariant spin connection and curvature tensor within the $E_8$ framework.
• To extend the geometric structure to include Ehlers symmetry $\mathrm{SL}(n+1)$.
• To lay the foundation for a tensorial formalism in exceptional field theory using $E_8$ geometry.

Proposed method

• Introduces field-dependent transformations that are covariant under $E_{8(8)}$ generalised diffeomorphisms.
• Derives the spin connection as a solution to consistency conditions in the geometric framework.
• Constructs a curvature tensor compatible with the $E_8(8)$ algebra and generalised diffeomorphism algebra.
• Uses the covariant transformation structure to close the algebra of gauge transformations.
• Applies the formalism to sketch the geometry of the Ehlers symmetry $\mathrm{SL}(n+1)$ as a substructure.
• Relies on the algebraic properties of $E_{8(8)}$ to ensure consistency of the tensor formalism.

Experimental results

Research questions

• RQ1 How can field-dependent transformations in exceptional generalised diffeomorphisms be made covariant under $E_{8(8)}$?
• RQ2 What is the role of the spin connection in the geometric formulation of dual gravity?
• RQ3 Can a consistent curvature tensor be defined in the $E_8$ exceptional field theory framework?
• RQ4 How does the $E_8$ geometry relate to the Ehlers symmetry $\mathrm{SL}(n+1)$?
• RQ5 What conditions ensure the closure of the algebra of gauge transformations in this geometric setting?

Key findings

• A consistent, field-dependent, and covariant transformation structure is constructed for $E_{8(8)}$ generalised diffeomorphisms.
• The spin connection is explicitly solved for within the geometric framework, ensuring compatibility with the gauge algebra.
• A curvature tensor is derived that transforms covariantly under $E_{8(8)}$ and closes the algebra of transformations.
• The formalism enables a tensorial description of dual gravity, resolving prior inconsistencies in the field-connection structure.
• The geometry of the Ehlers symmetry $\mathrm{SL}(n+1)$ is sketched as a substructure within the $E_8$ framework.
• The results establish a foundation for a unified geometric and tensorial formulation of exceptional field theory.