Kac-Moody Algebras on Soft Group Manifolds

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 $μ$, 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.