U-geometry : SL(5)

Understand

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.