The general pattern of Kač–Moody extensions in supergravity and the issue of cosmic billiards Original Research Article

In this paper we study the systematics of the affine extension of supergravity duality algebras when one steps down from image to image which is instrumental for the study of cosmic billiards. For all image supergravities (with image) there is a universal field theoretical mechanism promoting the extension, which relies on the coexistence of two non-locally related Lagrangian descriptions of the corresponding image degrees of freedom: the Ehlers Lagrangian and the Matzner–Misner one. This and the existence of a generalized Kramer–Neugebauer non-local transformation relating the two models, provide a Chevalley–Serre presentation of the affine Kač–Moody algebra which follows a universal pattern for all supergravities. This is an extension of the mechanism considered by Nicolai for pure image supergravity, but has general distinctive features in extended theories (image) related to the presence of vector fields and to their symplectic description. Moreover the novelty is that in the general case the Matzner–Misner Lagrangian is structurally different from the Ehlers one, since half of the scalars are replaced by gauge 0-forms subject to image electric–magnetic duality rotations representing in image the image rotations of image. The role played by the symplectic bundle of vectors in this context suggests that the mechanism of the affine extension can be studied also for image supergravity, where one deals with geometries rather than with algebras, the scalar manifold being not necessarily a homogeneous manifold image. We also show that the mechanism of the affine extension commutes with the Tits–Satake projection of the relevant duality algebras.