171 research outputs found
Transition function for the Toda chain model
The method of Lambda-operators developed by S. Derkachov, G. Korchemsky, A.
Manashov is applied to a derivation of eigenfunctions for the open Toda chain.
The Sklyanin measure is reproduced using diagram technique developed for these
Lambda-operators. The properties of the Lambda-operators are studied. This
approach to the open Toda chain eigenfunctions reproduces Gauss-Givental
representation for these eigenfunctions
Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model
Even though matrix model partition functions do not exhaust the entire set of
tau-functions relevant for string theory, they seem to be elementary building
blocks for many others and they seem to properly capture the fundamental
symplicial nature of quantum gravity and string theory. We propose to consider
matrix model partition functions as new special functions. This means they
should be investigated and put into some standard form, with no reference to
particular applications. At the same time, the tables and lists of properties
should be full enough to avoid discoveries of unexpected peculiarities in new
applications. This is a big job, and the present paper is just a step in this
direction. Here we restrict our consideration to the finite-size Hermitean
1-matrix model and concentrate mostly on its phase/branch structure arising
when the partition function is considered as a D-module. We discuss the role of
the CIV-DV prepotential (as generating a possible basis in the linear space of
solutions to the Virasoro constraints, but with a lack of understanding of why
and how this basis is distinguished) and evaluate first few multiloop
correlators, which generalize semicircular distribution to the case of
multitrace and non-planar correlators.Comment: 64 pages, LaTe
Liouville Type Models in Group Theory Framework. I. Finite-Dimensional Algebras
In the series of papers we represent the ``Whittaker'' wave functional of
-dimensional Liouville model as a correlator in -dimensional theory
of the sine-Gordon type (for and ). Asypmtotics of this wave function
is characterized by the Harish-Chandra function, which is shown to be a product
of simple -function factors over all positive roots of the
corresponding algebras (finite-dimensional for and affine for ).
This is in nice correspondence with the recent results on 2- and 3-point
correlators in Liouville model, where emergence of peculiar
double-periodicity is observed. The Whittaker wave functions of
-dimensional non-affine ("conformal") Toda type models are given by simple
averages in the dimensional theories of the affine Toda type. This
phenomenon is in obvious parallel with representation of the free-field wave
functional, which is originally a Gaussian integral over interior of a
-dimensional disk with given boundary conditions, as a (non-local)
quadratic integral over the -dimensional boundary itself. In the present
paper we mostly concentrate on the finite-dimensional case. The results for
finite-dimensional "Iwasawa" Whittaker functions were known, and we present
their survey. We also construct new "Gauss" Whittaker functions.Comment: 47 pages, LaTe
Unitary representations of U_{q}(\mathfrak{sl}(2,\RR)), the modular double, and the multiparticle q-deformed Toda chains
The paper deals with the analytic theory of the quantum q-deformed Toda
chain; the technique used combines the methods of representation theory and the
Quantum Inverse Scattering Method. The key phenomenon which is under scrutiny
is the role of the modular duality concept (first discovered by L.Faddeev) in
the representation theory of noncompact semisimple quantum groups. Explicit
formulae for the Whittaker vectors are presented in terms of the double sine
functions and the wave functions of the N-particle q-deformed open Toda chain
are given as a multiple integral of the Mellin-Barnes type. For the periodic
chain the two dual Baxter equations are derived.Comment: AmsLatex, 41 pages, 3 figure
Matrix Models, Complex Geometry and Integrable Systems. I
We consider the simplest gauge theories given by one- and two- matrix
integrals and concentrate on their stringy and geometric properties. We remind
general integrable structure behind the matrix integrals and turn to the
geometric properties of planar matrix models, demonstrating that they are
universally described in terms of integrable systems directly related to the
theory of complex curves. We study the main ingredients of this geometric
picture, suggesting that it can be generalized beyond one complex dimension,
and formulate them in terms of the quasiclassical integrable systems, solved by
construction of tau-functions or prepotentials. The complex curves and
tau-functions of one- and two- matrix models are discussed in detail.Comment: 52 pages, 19 figures, based on several lecture courses and the talks
at "Complex geometry and string theory" and the Polivanov memorial seminar;
misprints corrected, references adde
On a class of integrable systems connected with GL(N,\RR)
In this paper we define a new class of the quantum integrable systems
associated with the quantization of the cotangent bundle to the
Lie algebra . The construction is based on the Gelfand-Zetlin
maximal commuting subalgebra in . We discuss the connection
with the other known integrable systems based on . The construction
of the spectral tower associated with the proposed integrable theory is given.
This spectral tower appears as a generalization of the standard spectral curve
for integrable system.Comment: LaTeX, 13 page
On differential equation on four-point correlation function in the Conformal Toda Field Theory
The properties of completely degenerate fields in the Conformal Toda Field
Theory are studied. It is shown that a generic four-point correlation function
that contains only one such field does not satisfy ordinary differential
equation in contrast to the Liouville Field Theory. Some additional assumptions
for other fields are required. Under these assumptions we write such a
differential equation and solve it explicitly. We use the fusion properties of
the operator algebra to derive a special set of three-point correlation
function. The result agrees with the semiclassical calculations.Comment: 5 page
Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism
The integral representations for the eigenfunctions of particle quantum
open and periodic Toda chains are constructed in the framework of Quantum
Inverse Scattering Method (QISM). Both periodic and open -particle solutions
have essentially the same structure being written as a generalized Fourier
transform over the eigenfunctions of the particle open Toda chain with
the kernels satisfying to the Baxter equations of the second and first order
respectively. In the latter case this leads to recurrent relations which result
to representation of the Mellin-Barnes type for solutions of an open chain. As
byproduct, we obtain the Gindikin-Karpelevich formula for the Harish-Chandra
function in the case of GL(N,\RR) group.Comment: Latex+amssymb.sty, 14 page
Eigenfunctions of GL(N,\RR) Toda chain: The Mellin-Barnes representation
The recurrent relations between the eigenfunctions for GL(N,\RR) and
GL(N-1,\RR) quantum Toda chains is derived. As a corollary, the Mellin-Barnes
integral representation for the eigenfunctions of a quantum open Toda chain is
constructed for the -particle case.Comment: Latex+amssymb.sty, 7 pages; corrected some typos published in Pis'ma
v ZhETF (2000), vol. 71, 338-34
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