The general pattern of Kač–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 one steps down from D = 4 to D = 2 which is instrumental for the study of cosmic billiards. 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 of the corresponding D = 2 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 N = 1 supergravity, but has general distinctive features in extended theories ( N ⩾ 3 ) 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 SO ( 2 n , 2 n ) electric–magnetic duality rotations representing in D = 2 the Sp ( 2 n , R ) rotations of D = 4 . 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 N = 2 supergravity, where one deals with geometries rather than with algebras, the scalar manifold being not necessarily a homogeneous manifold U / H . We also show that the mechanism of the affine extension commutes with the Tits–Satake projection of the relevant duality algebras.