A String-Like Realization of Hyperbolic Kac–Moody Algebras

Abstract

We propose a new approach to studying hyperbolic Kac–Moody algebras, focussing on the rank-3 algebra \({\mathfrak {F}}\) first investigated by Feingold and Frenkel. Our approach is based on the concrete realization of this Lie algebra in terms of a Hilbert space of transverse and longitudinal physical string states, which are expressed in a basis using DDF operators. When decomposed under its affine subalgebra \({A_1^{(1)}}\), the algebra \({\mathfrak {F}}\) decomposes into an infinite sum of affine representation spaces of \({A_1^{(1)}}\) for all levels \(\ell \in \mathbb {Z}\). For \(|\ell | >1\) there appear in addition coset Virasoro representations for all minimal models of central charge \(c<1\), but the different level-\(\ell \) sectors of \({\mathfrak {F}}\) do not form proper representations of these because they are incompletely realized in \({\mathfrak {F}}\). To get around this problem we propose to nevertheless exploit the coset Virasoro algebra for each level by identifying for each level a (for \(|\ell |\ge 3\) infinite) set of ‘Virasoro ground states’ that are not necessarily elements of \({\mathfrak {F}}\) (in which case we refer to them as ‘virtual’), but from which the level-\(\ell \) sectors of \({\mathfrak {F}}\) can be fully generated by the joint action of affine and coset Virasoro raising operators. We conjecture (and present partial evidence) that the Virasoro ground states for \(|\ell |\ge 3\) in turn can be generated from a finite set of ‘maximal ground states’ by the additional action of the ‘spectator’ coset Virasoro raising operators present for all levels \(|\ell | > 2\). Our results hint at an intriguing but so far elusive secret behind Einstein’s theory of gravity, with possibly important implications for quantum cosmology.