U-geometry : SL(5)

TL;DR

This paper develops an SL(5) compatible differential geometry called U-geometry, generalizing Riemannian geometry to describe the effective action of four-dimensional M-theory with manifest SL(5) U-duality.

Contribution

It introduces a semi-covariant derivative and constructs fully covariant curvatures, providing a geometric framework for SL(5) U-duality in M-theory.

Findings

Defined a semi-covariant derivative compatible with SL(5) symmetry.

Derived covariant scalar and Ricci-like curvatures for the effective action.

Formulated equations of motion within the SL(5) U-geometry framework.

Abstract

Recently Berman and Perry constructed a four-dimensional M-theory effective action which manifests SL(5) U-duality. Here we propose an underlying differential geometry of it, under the name `SL(5) U-geometry' which generalizes the ordinary Riemannian geometry in an SL(5) compatible manner. We introduce a `semi-covariant' derivative that can be converted into fully covariant derivatives after anti-symmetrizing or contracting the SL(5) vector indices appropriately. We also derive fully covariant scalar and Ricci-like curvatures which constitute the effective action as well as the equation of motion.