Kac-Moody Algebras on Soft Group Manifolds

TL;DR

This paper develops a method to construct infinite-dimensional Kac-Moody algebras on deformed, 'soft' group manifolds, extending the algebraic framework in (super)gravity and string theories to include non-invariant geometries.

Contribution

It introduces a novel approach to generate generalized Kac-Moody algebras on 'soft' group manifolds, broadening the algebraic structures applicable in theoretical physics.

Findings

Constructed infinite-dimensional Kac-Moody algebras on soft manifolds.

Found trivial isomorphism for soft circle, non-trivial for soft spheres.

Applied the framework to squashed and Berger three-sphere geometries.

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.