Introduction to vertex algebras, Borcherds algebras and the Monster Lie algebras

The theory of vertex algebras constitutes a mathematically rigorous axiomatic formulation of the algebraic origins of conformal field theory. In this context Borcherds algebras arise as certain ''physical'' subspaces of vertex algebras. The aim of this review is to give a pedagogical introduction into this rapidly-developing area of mathematics. Based on the machinery of formal calculus we present the axiomatic definition of vertex algebras. We discuss the connection with conformal field theory by deriving important implications of these axioms. In particular, many explicit calculations are presented to stress the eminent role of the Jacobi identity axiom for vertex algebras. As a class of concrete examples the vertex algebras associated with even lattices are constructed and it is shown in detail how affine Lie algebras and the fake Monster Lie algebra naturally appear. This leads us to the abstract definition of Borcherds algebras as generalized Kac-Moody algebras and their basic properties. Finally, the results about the simplest generic Borcherds algebras are analysed from the point of view of symmetry in quantum theory and the construction of the Monster Lie algebra is sketched. (orig.)Available from TIB Hannover: RA 2999(93-120) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

An affine string vertex operator construction at arbitrary level

An affine vertex operator construction at arbitrary level is presented which is based on a completely compactified chiral bosonic string whose momentum lattice is taken to be the (Minkowskian) affine weight lattice. This construction is manifestly physical in the sense of string theory, i.e., the vertex operators are functions of DDF ''oscillators'' and the Lorentz generators, both of which commute with the Virasoro constraints. We therefore obtain explicit representations of affine highest weight modules in terms of physical (DDF) string states. This opens new perspectives on the representation theory of affine Kac-Moody algebras, especially in view of the simultaneous treatment of infinitely many affine highest weight representations of arbitrary level within a single state space as required for the study of hyperbolic Kac-Moody algebras. A novel interpretation of the affine Weyl group as the ''dimensional null reduction'' of the corresponding hyperbolic Weyl group is given, which follows upon re-expression of the affine Weyl translations as Lorents boosts. (orig.)SIGLEAvailable from TIB Hannover: RA 2999(96-166) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

The Sugawara generators at arbitrary level

We construct an explicit representation of the Sugawara generators for arbitrary level in terms of the homogeneous Heisenberg subalgebra, which generalizes the well-known expression at level 1. This is achieved by employing a physical vertex operator realization of the affine algebra at arbitrary level, in contrast to the Frenkel-Kac-Segal construction which uses unphysical oscillators and is restricted to level 1. At higher level, the new operators are transcendental functions of DDF oscillators unlike the quadratic expressions for the level-1 generators. An essential new feature of our construction is the appearance, beyond level 1, of new types of poles in the operator product expansions in addition to the ones at coincident points, which entail (controllable) non-localities in our formulas. We demonstrate the utility of the new formalism by explicitly working out some higher-level examples. Our results have important implications for the problem of constructing explicit representations for higher-level root spaces of hyperbolic Kac-Moody algebras, and E_1_0 in particular. (orig.)Available from TIB Hannover: RA 2999(96-072) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

The Painleve property, W algebras and Toda field theories associated with hyperbolic Kac-Moody algebras

We show that the Painleve test is useful not only for probing (non-)integrability but also for finding the values of spins of conserved currents (W currents) in Toda field theories (TFTs). In the case of the TFTs based on simple Lie algebras the locations of resonances are shown to give precisely the spins of conserved W currents. We apply this test to TFTs based on strictly hyperbolic Kac-Moody algebras and show that there exist no resonances other than that at n=2, which corresponds to the energy-momentum tensor, indicating their non-integrability. We also check by direct calculation that there are no spin-3 nor -4 conserved currents for all the hyperbolic TFTs in agreement with the result of our Painleve analysis. (orig.)44 refs.Available from TIB Hannover: RA 2999(95-076) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

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, and E_1_0 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 uncompacitified 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_1_0. 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_1_0 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. (orig.)64 refs.Available from TIB Hannover: RA 2999(95-092) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman