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Schlagwörter:
High Energy Physics - Theory, hep-th,Mathematics, Quantum Algebra, math.QA,Mathematics, Representation Theory, math.RT
Zusammenfassung:
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.
