MULTISTRING VERTICES AND HYPERBOLIC KAC-MOODY ALGEBRAS

Multistring vertices and the overlap identities which they satisfy are exploited to understand properties of hyperbolic Kac-Moody algebras, in particular E(10). Since any such algebra can be embedded in the lar ger Lie algebra of physical states of an associated completely compact ified subcritical bosonic string, one can in principle determine the r oot spaces by analyzing which (positive norm) physical states decouple from the N-string vertex. Consequently, the Lie algebra of physical s tates decomposes into a direct sum of the hyperbolic algebra and the s pace of decoupled states. Both these spaces contain transversal and lo ngitudinal states. Longitudinal decoupling holds more generally, and m ay also be valid for uncompactified strings, with possible consequence s for Liouville theory; the identification of the decoupled states sim ply amounts to finding the zeroes of certain ''decoupling polynomials. '' This is not the case for transversal decoupling, which crucially de pends on special properties of the root lattice, as we explicitly demo nstrate for a nontrivial root space of E(10). Because the N-vertices o f the compactified string contain the complete information about decou pling, all the properties of the hyperbolic algebra are encoded into t hem. In view of the integer grading of hyperbolic algebras such as E(1 0) by the level, these algebras can be interpreted as interacting stri ngs moving on the respective group manifolds associated with the under lying finite-dimensional Lie algebras.