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 $${\mathfrak {F}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> 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 $${A_1^{(1)}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> , the algebra $${\mathfrak {F}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> decomposes into an infinite sum of affine representation spaces of $${A_1^{(1)}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> for all levels $$\ell \in \mathbb {Z}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>Z</mml:mi> </mml:mrow> </mml:math> . For $$|\ell | &gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>|</mml:mo> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> there appear in addition coset Virasoro representations for all minimal models of central charge $$c&lt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , but the different level- $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> sectors of $${\mathfrak {F}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> do not form proper representations of these because they are incompletely realized in $${\mathfrak {F}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> . To get around this problem we propose to nevertheless exploit the coset Virasoro algebra for each level by identifying for each level a (for $$|\ell |\ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>|</mml:mo> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> infinite) set of ‘Virasoro ground states’ that are not necessarily elements of $${\mathfrak {F}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> (in which case we refer to them as ‘virtual’), but from which the level- $$\ell $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> sectors of $${\mathfrak {F}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> 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 $$|\ell |\ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>|</mml:mo> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> 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 $$|\ell | &gt; 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">