U-Duality and Central Charges in Various Dimensions Revisited

TL;DR

This paper develops a unified geometric framework for extended supergravities across dimensions, making duality symmetries explicit and encoding central and matter charges through the G/H coset structure and flat symplectic bundles. It treats D=4 with N>2 via a Gaillard–Zumino–type embedding into Sp(2n,R), and systematically extends the approach to higher dimensions with and without matter, deriving differential relations among charges from Maurer–Cartan equations and establishing sum rules tied to the kinetic matrices. The work connects the D=4 N>2 theories to Special Geometry generalizations, clarifies the role of U-duality groups in organizing BPS charges, and provides explicit group-theoretical data and charge relations for odd and even dimensional maximally extended theories, including D=6 and D=8. The results have implications for understanding BPS spectra, attractor behavior, and entropy formulas in a dimension-spanning, duality-covariant setting.

Abstract

A geometric formulation which describes extended supergravities in any dimension in presence of electric and magnetic sources is presented. In this framework the underlying duality symmetries of the theories are manifest. Particular emphasis is given to the construction of central and matter charges and to the symplectic structure of all D=4, N-extended theories. The latter may be traced back to the existence, for N>2, of a flat symplectic bundle which is the N>2 generalization of N=2 Special Geometry.