396 research outputs found
Introduction to Vertex Algebras, Borcherds Algebras, and the Monster Lie Algebra
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 mathemat% ics. 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.Comment: 55 pages, (two minor changes thanks to comment by R. Borcherds
Explicit determination of a 727-dimensional root space of the hyperbolic Lie algebra
The 727-dimensional root space associated with the level-2 root \bLambda_1
of the hyperbolic Kac--Moody algebra is determined using a recently
developed string theoretic approach to hyperbolic algebras. The explicit form
of the basis reveals a complicated structure with transversal as well as
longitudinal string states present.Comment: 12 pages, LaTeX 2
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
Lorentz boosts.Comment: 15 pages, LaTeX2e, packages amsfonts, amssymb, xspace; final version
to appear in J. Math. Phy
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 in particular.Comment: 17 pages, 1 figure, LaTeX2e, amsfonts, amssymb, xspace, PiCTe
Missing Modules, the Gnome Lie Algebra, and
We study the embedding of Kac-Moody algebras into Borcherds (or generalized
Kac-Moody) algebras which can be explicitly realized as Lie algebras of
physical states of some completely compactified bosonic string. The extra
``missing states'' can be decomposed into irreducible highest or lowest weight
``missing modules'' w.r.t. the relevant Kac-Moody subalgebra; the corresponding
lowest weights are associated with imaginary simple roots whose multiplicities
can be simply understood in terms of certain polarization states of the
associated string. We analyse in detail two examples where the momentum lattice
of the string is given by the unique even unimodular Lorentzian lattice
or , respectively. The former leads to the Borcherds
algebra , which we call ``gnome Lie algebra", with maximal Kac-Moody
subalgebra . By the use of the denominator formula a complete set of
imaginary simple roots can be exhibited, whereas the DDF construction provides
an explicit Lie algebra basis in terms of purely longitudinal states of the
compactified string in two dimensions. The second example is the Borcherds
algebra , whose maximal Kac-Moody subalgebra is the hyperbolic algebra
. The imaginary simple roots at level 1, which give rise to irreducible
lowest weight modules for , can be completely characterized;
furthermore, our explicit analysis of two non-trivial level-2 root spaces leads
us to conjecture that these are in fact the only imaginary simple roots for
.Comment: 31 pages, LaTeX2e, AMS packages, PSTRICK
On the fundamental representation of Borcherds algebras with one imaginary simple root
Borcherds algebras represent a new class of Lie algebras which have almost
all the properties that ordinary Kac-Moody algebras have, and 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.Comment: 8 page
BPS Saturation from Null Reduction
We show that any -dimensional strictly stationary, asymptotically
Minkowskian solution of a null reduction of -dimensional pure
gravity must saturate the BPS bound provided that the KK vector field can be
identified appropriately. We also argue that it is consistent with the field
equations.Comment: 10 page
Small grid embeddings of 3-polytopes
We introduce an algorithm that embeds a given 3-connected planar graph as a
convex 3-polytope with integer coordinates. The size of the coordinates is
bounded by . If the graph contains a triangle we can
bound the integer coordinates by . If the graph contains a
quadrilateral we can bound the integer coordinates by . The
crucial part of the algorithm is to find a convex plane embedding whose edges
can be weighted such that the sum of the weighted edges, seen as vectors,
cancel at every point. It is well known that this can be guaranteed for the
interior vertices by applying a technique of Tutte. We show how to extend
Tutte's ideas to construct a plane embedding where the weighted vector sums
cancel also on the vertices of the boundary face
Polytopality and Cartesian products of graphs
We study the question of polytopality of graphs: when is a given graph the
graph of a polytope? We first review the known necessary conditions for a graph
to be polytopal, and we provide several families of graphs which satisfy all
these conditions, but which nonetheless are not graphs of polytopes. Our main
contribution concerns the polytopality of Cartesian products of non-polytopal
graphs. On the one hand, we show that products of simple polytopes are the only
simple polytopes whose graph is a product. On the other hand, we provide a
general method to construct (non-simple) polytopal products whose factors are
not polytopal.Comment: 21 pages, 10 figure
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