Reinhold W. Gebert

OnE10 and the DDF construction

An attempt is made to understand the root spaces of Kac Moody algebras of hyperbolic type, and in particularE10, in terms of a DDF construction appropriate to a subcritical compactified bosonic string. While the level-one root spaces can be completely characterized in terms of transversal DDF states (the level-zero elements just span the affine subalgebra), longitudinal DDF states are shown to appear beyond level one. In contrast to previous treatments of such algebras, we find it necessary to make use of a rational extension of the self-dual root lattice as an auxiliary device, and to admit non-summable operators (in the sense of the vertex algebra formalism). We demonstrate the utility of the method by completely analyzing a non-trivial level-two root space, obtaining an explicit and comparatively simple representation for it. We also emphasize the occurrence of several Virasoro algebras, whose interrelation is expected to be crucial for a better understanding of the complete structure of the Kac Moody algebra.

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 E10. 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 more 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 nontrivial root space of E10· 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 E10 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.

An affine string vertex operator construction at an arbitrary level

An affine vertex operator construction at an 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 Del Giudice–Di Vecchia–Fubini (DFF) “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 give...

TODA FIELD THEORIES ASSOCIATED WITH HYPERBOLIC KAC-MOODY ALGEBRA — PAINLEVÉ PROPERTIES AND W 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 (TFT’s). In the case of TFT’s 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 TFT’s based strictly on hyperbolic Kac-Moody algebras and show that there exist no resonance other than that at n = 2, which corresponds to the energy-momentum tensor, indicating their nonintegrability. We also check by direct calculation that there are no spin-3 or -4 conserved currents for all the hyperbolic TFT’s in agreement with the result of our Painleve analysis.

E(10) for beginners

We present a nontechnical introduction to the hyperbolic Kac Moody algebra E_{10} and summarize our recent attempt to understand the root spaces of Kac Moody algebras of hyperbolic type in terms of a DDF construction appropriate to a subcritical compactified bosonic string.

The Sugawara generators at arbitrary level

Abstract: We construct an explicit representation of the affine 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 level1, 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

On the fundamental representation of borcherds algebras with one imaginary simple root

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INTRODUCTION TO VERTEX ALGEBRAS, BORCHERDS ALGEBRAS AND THE MONSTER LIE ALGEBRA

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E10 for beginners - Contribution to Gürsey Memorial Conference Proceedings '94

Borcherds algebras represent a new class of Lie algebras which have almost all the properties that ordinary Kac-Moody algebras have, but the only major difference is that these generalized Kac-Moody algebras are allowed to have imaginary simple roots. The simplest nontrivial examples one can think of are those where one adds ‘by hand’ one imaginary simple root to an ordinary Kac-Moody algebra. We study the fundamental representation of this class of examples and prove that an irreducible module is given by the full tensor algebra over some integrable highest weight module of the underlying Kac-Moody algebra. We also comment on possible realizations of these Lie algebras in physics as symmetry algebras in quantum field theory.