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 $|\ell|\geq 3$ 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 $|\ell|\geq 3$ 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.