Multistring Vertices and Hyperbolic Kac Moody Algebras

TL;DR

This paper explores the connection between multistring vertices, decoupling of physical states, and hyperbolic Kac-Moody algebras like E10, revealing how string theory structures encode algebraic properties and root space decompositions.

Contribution

It introduces a method to analyze hyperbolic Kac-Moody algebras via multistring vertices and decoupling polynomials, linking string theory and algebraic structures in a novel way.

Findings

Decoupling polynomials identify zeroes corresponding to decoupled states.

Transversal decoupling depends on root lattice properties.

Hyperbolic algebras can be viewed as interacting strings on group manifolds.

Abstract

Multistring vertices and the overlap identities which they satisfy are exploited to understand properties of hyperbolic Kac Moody algebras, and $E_{10}$ in particular. Since any such algebra can be embedded in the larger Lie algebra of physical states of an associated completely compactified subcritical bosonic string, one can in principle determine the root spaces by analyzing which (positive norm) physical states decouple from the $N$-string vertex. Consequently, the Lie algebra of physical states decomposes into a direct sum of the hyperbolic algebra and the space of decoupled states. Both these spaces contain transversal and longitudinal states. Longitudinal decoupling holds generally, and may also be valid for uncompactified strings, with possible consequences for Liouville theory; the identification of the decoupled states simply amounts to finding the zeroes of certain ``decoupling polynomials''. This is not the case for transversal decoupling, which crucially depends on special properties of the root lattice, as we explicitly demonstrate for a non-trivial root space of $E_{10}$. Because the $N$-vertices of the compactified string contain the complete information about decoupling, all the properties of the hyperbolic algebra are encoded into them. In view of the integer grading of hyperbolic algebras such as $E_{10}$ by the level, these algebras can be interpreted as interacting strings moving on the respective group manifolds associated with the underlying finite-dimensional Lie algebras.