Kac-Moody Algebras on Soft Group Manifolds

Abstract

Within the so-called group geometric approach to (super)gravity and (super)string theories, any compact Lie group manifold $G_c$ can be smoothly deformed into a group manifold $G_c^{\mu }$ (locally diffeomorphic to $G_c$ itself), which is `soft', namely, based on a non-left-invariant, intrinsic one-form Vielbein $\mu$, which violates the Maurer-Cartan equations and consequently has a non-vanishing associated curvature two-form. Within the framework based on the above deformation (`softening'), we show how to construct an infinite-dimensional (infinite-rank), generalized Kac-Moody (KM) algebra associated to $G_c^{\mu }$, starting from the generalized KM algebras associated to $G_c$. As an application, we consider KM algebras associated to deformed manifolds such as the `soft' circle, the `soft' two-sphere and the `soft' three-sphere. While the generalized KM algebra associated to the deformed circle is trivially isomorphic to its undeformed analogue, and hence not new, the `softening' of the two- and three- sphere includes squashed manifolds (and in particular, the so-called Berger three-sphere) and yields to non-trivial results.

33 pages, a number of clarifications and further comments are added along the text, to appear in J. Geom. Phys

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