U-geometry: SL(5)

A bstract Recently Berman and Perry constructed a four-dimensional -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.