20 research outputs found
Beyond the Frenkel-Kac-Segal construction of affine Lie algebras
This contribution reviews recent progress in constructing affine Lie algebras
at arbitrary level in terms of vertex operators. The string model describes a
completely compactified subcritical chiral bosonic string whose momentum
lattice is taken to be the (Lorentzian) affine weight lattice. The main feature
of the new realization is the replacement of the ordinary string oscillators by
physical DDF operators, whereas the unphysical position operators are
substituted by certain linear combinations of the Lorentz generators. As a side
result we obtain simple expressions for the affine Weyl translations as Lorentz
boosts. Various applications of the construction are discussed.Comment: 6 pages, LaTeX209 with twoside, fleqn, amsmath, amsfonts, amssymb,
amsthm style files; contribution to Proceedings of the 30th Int. Symposium
Ahrenshoop on the Theory of Elementary Particles, Buckow, Germany, August
27-31, 199
Is the classical Bukhvostov-Lipatov model integrable? A Painlev\'e analysis
In this work we apply the Weiss, Tabor and Carnevale integrability criterion
(Painlev\'e analysis) to the classical version of the two dimensional
Bukhvostov-Lipatov model. We are led to the conclusion that the model is not
integrable classically, except at a trivial point where the theory can be
described in terms of two uncoupled sine-Gordon models
Sugawara-type constraints in hyperbolic coset models
In the conjectured correspondence between supergravity and geodesic models on
infinite-dimensional hyperbolic coset spaces, and E10/K(E10) in particular, the
constraints play a central role. We present a Sugawara-type construction in
terms of the E10 Noether charges that extends these constraints infinitely into
the hyperbolic algebra, in contrast to the truncated expressions obtained in
arXiv:0709.2691 that involved only finitely many generators. Our extended
constraints are associated to an infinite set of roots which are all imaginary,
and in fact fill the closed past light-cone of the Lorentzian root lattice. The
construction makes crucial use of the E10 Weyl group and of the fact that the
E10 model contains both D=11 supergravity and D=10 IIB supergravity. Our
extended constraints appear to unite in a remarkable manner the different
canonical constraints of these two theories. This construction may also shed
new light on the issue of `open constraint algebras' in traditional canonical
approaches to gravity.Comment: 49 page
Kac-Moody algebras in perturbative string theory
The conjecture that M-theory has the rank eleven Kac-Moody symmetry e11
implies that Type IIA and Type IIB string theories in ten dimensions possess
certain infinite dimensional perturbative symmetry algebras that we determine.
This prediction is compared with the symmetry algebras that can be constructed
in perturbative string theory, using the closed string analogues of the DDF
operators. Within the limitations of this construction close agreement is
found. We also perform the analogous analysis for the case of the closed
bosonic string.Comment: 31 pages, harvmac (b), 4 eps-figure
Renormalization group flows and continual Lie algebras
We study the renormalization group flows of two-dimensional metrics in sigma
models and demonstrate that they provide a continual analogue of the Toda field
equations based on the infinite dimensional algebra G(d/dt;1). The resulting
Toda field equation is a non-linear generalization of the heat equation, which
is integrable in target space and shares the same dissipative properties in
time. We provide the general solution of the renormalization group flows in
terms of free fields, via Backlund transformations, and present some simple
examples that illustrate the validity of their formal power series expansion in
terms of algebraic data. We study in detail the sausage model that arises as
geometric deformation of the O(3) sigma model, and give a new interpretation to
its ultra-violet limit by gluing together two copies of Witten's
two-dimensional black hole in the asymptotic region. We also provide some new
solutions that describe the renormalization group flow of negatively curved
spaces in different patches, which look like a cane in the infra-red region.
Finally, we revisit the transition of a flat cone C/Z_n to the plane, as
another special solution, and note that tachyon condensation in closed string
theory exhibits a hidden relation to the infinite dimensional algebra G(d/dt;1)
in the regime of gravity. Its exponential growth holds the key for the
construction of conserved currents and their systematic interpretation in
string theory, but they still remain unknown.Comment: latex, 73pp including 14 eps fig
Lectures on conformal field theory and Kac-Moody algebras
This is an introduction to the basic ideas and to a few further selected
topics in conformal quantum field theory and in the theory of Kac-Moody
algebras.Comment: 59 pages, LaTeX2e, extended version of lectures given at the Graduate
Course on Conformal Field Theory and Integrable Models (Budapest, August
1996), to appear in Springer Lecture Notes in Physic
Beyond affine Kac-Moody algebras in string theory
This work is devoted to the study of certain infinite-dimensional Lie algebras arising in string theory. The two investigated models describe a chiral sector of a fully compactified closed bosonic string moving on a subcritical 10-dimensional or a critical 26-dimensional spacetime torus, respectively. To analyze the corresponding Lie algebra of physical states, a discrete version of the DDF construction in the framework of vertex algebras is developed. When applied to the subcritical example, the method yields some new insights into the complicated structure of the hyperbolic Kac-Moody algebra E_1_0 in terms of transversal and longitudinal states. Due to the no-ghost theorem, in 26 dimensions only transversal physical states appear which make up the so-called fake monster Lie algebra. The latter represents an example of a Borcherds algebra, which is a generalized Kac-Moody algebras in the sense that imaginary simple roots are allowed for in the defining relations. It is demonstrated for the example, that this feature can be understood by means of the DDF operators, too. Finally, a new result representation theory of these algebras is proved which is analyzed in view of possible applications to physics. (orig.)SIGLEAvailable from TIB Hannover: RA 2999(94-209) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman
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