209 research outputs found
Elements of a Global Operator Approach to Wess-Zumino-Novikov-Witten Models
Elements of a global operator approach to the WZNW models for compact Riemann
surfaces of arbitrary genus g with N marked points were given by Schlichenmaier
and Sheinman. This contribution reports on the results. The approach is based
on the multi-point Krichever-Novikov algebras of global meromorphic functions
and vector fields, and the global algebras of affine type and their
representations. Using the global Sugawara construction and the identification
of a certain subspace of the vector field algebra with the tangent space to the
moduli space of the geometric data, Knizhnik-Zamalodchikov equations are
defined. Some steps of the approach of Tsuchia, Ueno and Yamada to WZNW models
are presented to compare it with our approach.Comment: 17 pages, Amslatex, Invited talk presented at the 3rd International
Workshop on "Lie Theory and Its Applications in Physics - Lie III", 11 - 14
July 1999, Clausthal, German
Some Concepts of Modern Algebraic Geometry: Point, Ideal and Homomorphism
Starting from classical algebraic geometry over the complex numbers (as it
can be found for example in Griffiths and Harris it was the goal of these
lectures to introduce some concepts of the modern point of view in algebraic
geometry. Of course, it was quite impossible even to give an introduction to
the whole subject in such a limited time. For this reason the lectures and now
the write-up concentrate on the substitution of the concept of classical points
by the notion of ideals and homomorphisms of algebras.Comment: 36 pages. This is a write-up of lectures given at the ``Kleine
Herbstschule 93'' of the Graduiertenkolleg ``Mathematik im Bereich Ihrer
Wechselwirkungen mit der Physik'' at the Ludwig-Maximilians-Universitaet
Muenche
Higher genus affine algebras of Krichever - Novikov type
For higher genus multi-point current algebras of Krichever-Novikov type
associated to a finite-dimensional Lie algebra, local Lie algebra two-cocycles
are studied. They yield as central extensions almost-graded higher genus affine
Lie algebras. In case that the Lie algebra is reductive a complete
classification is given. For a simple Lie algebra, like in the classical
situation, there is up to equivalence and rescaling only one non-trivial
almost-graded central extension. The classification is extended to the algebras
of meromorphic differential operators of order less or equal one on the
currents algebra.Comment: 35 page
Berezin-Toeplitz Quantization of compact Kaehler manifolds
Invited lecture at the XIV-th workshop on geometric methods in physics,
Bialowieza, Poland, July 9-15, 1995. In this lecture results are reviewed
obtained by the author together with Martin Bordemann and Eckhard Meinrenken on
the Berezin-Toeplitz quantization of compact Kaehler manifolds. Using global
Toeplitz operators, approximation results for the quantum operators are shown.
From them it follows that the quantum operators have the correct classical
limit. A star product deformation of the Poisson algebra is constructed.Comment: Amstex 2.1, 15 pages, minor changes, some annoying typos removed and
2 references adde
Differential Operator Algebras on compact Riemann Surfaces
Invited talk at the International Symposium on Generalized Symmetries in
Physics at the Arnold-Sommerfeld-Institute, Clausthal, Germany, July 26 -- July
29, 1993. This talk reviews results on the structure of algebras consisting of
meromorphic differential operators which are holomorphic outside a finite set
of points on compact Riemann surfaces. For each partition into two disjoint
subsets of the set of points where poles are allowed, a grading of the algebra
and of the modules of lambda - forms is introduced. With respect to this
grading the Lie structure of the algebra and of the modules are almost graded
ones. Central extensions and semi-infinite wedge representations are studied.
If one considers only differential operators of degree 1 then these algebras
are generalizations of the Virasoro algebra in genus zero, resp. of Krichever
Novikov algebras in higher genus.Comment: 11 pages, AmsTeX 2.1 and psbox macro
Local cocycles and central extensions for multi-point algebras of Krichever-Novikov type
Multi-point algebras of Krichever Novikov type for higher genus Riemann
surfaces are generalisations of the Virasoro algebra and its related algebras.
Complete existence and uniqueness results for local 2-cocycles defining
almost-graded central extensions of the functions algebra, the vector field
algebra, and the differential operator algebra (of degree \le 1) are shown.
This is applied to the higher genus, multi-point affine algebras to obtain
uniqueness for almost-graded central extensions of the current algebra of a
simple finite-dimensional Lie algebra. An earlier conjecture of the author
concerning the central extension of the differential operator algebra induced
by the semi-infinite wedge representations is proved.Comment: 38 pages, Amslatex, some minor changes in Section
Higher Genus Affine Lie Algebras of Krichever -- Novikov Type
Classical affine Lie algebras appear e.g. as symmetries of infinite
dimensional integrable systems and are related to certain differential
equations. They are central extensions of current algebras associated to
finite-dimensional Lie algebras g. In geometric terms these current algebras
might be described as Lie algebra valued meromorphic functions on the Riemann
sphere with two possible poles. They carry a natural grading. In this talk the
generalization to higher genus compact Riemann surfaces and more poles is
reviewed. In case that the Lie algebra g is reductive (e.g. g is simple,
semi-simple, abelian, ...) a complete classification of (almost-) graded
central extensions is given. In particular, for g simple there exists a unique
non-trivial (almost-)graded extension class. The considered algebras are
related to difference equations, special functions and play a role in Conformal
Field Theory.Comment: 9 pages, Talk presented at the International Conference on Difference
Equations, Special Functions, and Applications, Munich, July 200
-point Virasoro algebras are multi-point Krichever--Novikov type algebras
We show how the recently again discussed -point Witt, Virasoro, and affine
Lie algebras are genus zero examples of the multi-point versions of
Krichever--Novikov type algebras as introduced and studied by Schlichenmaier.
Using this more general point of view, useful structural insights and an easier
access to calculations can be obtained. The concept of almost-grading will
yield information about triangular decompositions which are of importance in
the theory of representations. As examples the algebra of functions, vector
fields, differential operators, current algebras, affine Lie algebras, Lie
superalgebras and their central extensions are studied. Very detailed
calculations for the three-point case are given.Comment: 46 page
Berezin-Toeplitz Quantization and Star Products for Compact Kaehler Manifolds
For compact quantizable K\"ahler manifolds certain naturally defined star
products and their constructions are reviewed. The presentation centers around
the Berezin-Toeplitz quantization scheme which is explained. As star products
the Berezin-Toeplitz, Berezin, and star product of geometric quantization are
treated in detail. It is shown that all three are equivalent. A prominent role
is played by the Berezin transform and its asymptotic expansion. A few ideas on
two general constructions of star products of separation of variables type by
Karabegov and by Bordemann--Waldmann respectively are given. Some of the
results presented is work of the author partly joint with Martin Bordemann,
Eckhard Meinrenken and Alexander Karabegov. At the end some works which make
use of graphs in the construction and calculation of these star productsComment: 39 pages, Based on a talk presented in the frame of the Thematic
Program on Quantization, Spring 2011, at the University of Notre Dame, USA.
In the revised version some additional references are given in relation to
the role of the metaplectic correction and quotients. Also now there is an
additional section about applications and related reference
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