The general pattern of Kac Moody extensions in supergravity and the issue of cosmic billiards

Abstract

In this paper we study the systematics of the affine extension of supergravity duality algebras when we step down from D=4 to D=2. For all D=4 supergravities (with N >= 3) there is a universal field theoretical mechanism promoting the extension, which relies on the coexistence of two non locally related lagrangian descriptions. This provides a Chevalley-Serre presentation of the affine Kac Moody algebra which follows a universal pattern for all supergravities and is an extension of the mechanism considered by Nicolai for pure N=1 supergravity. There are new distinctive features in extended theories related to the presence of vector fields and to their symplectic description. The novelty is that in supergravity the so named Matzner-Missner description is structurally different from the Ehlers one with gauge 0--forms subject to SO(2n,2n) electric--magnetic duality rotations representing in D=2 the Sp(2n,R) rotations of D=4. The role played by the symplectic bundle of vectors is emphasized in view of implementing the affine extension also in N=2 supergravity, where the scalar manifold is not necessarily a homogeneous manifold U/H. We show that the mechanism of the affine extension commutes with the Tits Satake projection of the duality algebras. This is very important for the issue of cosmic billiards. We also comment on the general field theoretical mechanism of the further hyperbolic extension obtained in D=1. The possible uses of our results and their relation to outstanding problems are pointed out.