58 research outputs found
On pq-duality and explicit solutions in gravity models
We study the integral representation for the exact solution to
nonperturbative string theory. A generic solution is determined by two
functions and which behaive at infinity like and
respectively. The integral model for arbitrary models is derived which
explicitely demonstrates duality of minimal models coupled to gravity. We
discuss also the exact solutions to string equation and reduction condition and
present several explicit examples.Comment: NORDITA 93/20, FIAN-TD-04/93, latex, 20 p
Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects
The Kazakov-Migdal model, if considered as a functional of external fields,
can be always represented as an expansion over characters of group. The
integration over "matter fields" can be interpreted as going over the {\it
model} (the space of all highest weight representations) of . In the case
of compact unitary groups the integrals should be substituted by {\it discrete}
sums over weight lattice. The version of the model is the Generalized
Kontsevich integral, which in the above-mentioned unitary (discrete) situation
coincides with partition function of the Yang-Mills theory with the target
space of genus and holes. This particular quantity is always a
bilinear combination of characters and appears to be a Toda-lattice
-function. (This is generalization of the classical statement that
individual characters are always singular KP -functions.) The
corresponding element of the Universal Grassmannian is very simple and somewhat
similar to the one, arising in investigations of the string models.
However, under certain circumstances the formal sum over representations should
be evaluated by steepest descent method and this procedure leads to some more
complicated elements of Grassmannian. This "Kontsevich phase" as opposed to the
simple "character phase" deserves further investigation.Comment: 29 pages, UUITP-10/93, FIAN/TD-07/93, ITEP-M4/9
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
Generalized Kontsevich Model Versus Toda Hierarchy and Discrete Matrix Models
We represent the partition function of the Generalized Kontsevich Model (GKM)
in the form of a Toda lattice -function and discuss various implications
of non-vanishing "negative"- and "zero"-time variables: the appear to modify
the original GKM action by negative-power and logarithmic contributions
respectively. It is shown that so deformed -function satisfies the same
string equation as the original one. In the case of quadratic potential GKM
turns out to describe {\it forced} Toda chain hierarchy and, thus, corresponds
to a {\it discrete} matrix model, with the role of the matrix size played by
the zero-time (at integer positive points). This relation allows one to discuss
the double-scaling continuum limit entirely in terms of GKM, essentially
in terms of {\it finite}-fold integrals.Comment: 46
On two-dimensional quantum gravity and quasiclassical integrable hierarchies
The main results for the two-dimensional quantum gravity, conjectured from
the matrix model or integrable approach, are presented in the form to be
compared with the world-sheet or Liouville approach. In spherical limit the
integrable side for minimal string theories is completely formulated using
simple manipulations with two polynomials, based on residue formulas from
quasiclassical hierarchies. Explicit computations for particular models are
performed and certain delicate issues of nontrivial relations among them are
discussed. They concern the connections between different theories, obtained as
expansions of basically the same stringy solution to dispersionless KP
hierarchy in different backgrounds, characterized by nonvanishing background
values of different times, being the simplest known example of change of the
quantum numbers of physical observables, when moving to a different point in
the moduli space of the theory.Comment: 20 pages, based on talk presented at the conference "Liouville field
theory and statistical models", dedicated to the memory of Alexei
Zamolodchikov, Moscow, June 200
Towards unified theory of gravity
We introduce a new 1-matrix model with arbitrary potential and the
matrix-valued background field. Its partition function is a -function of
KP-hierarchy, subjected to a kind of -constraint. Moreover,
partition function behaves smoothly in the limit of infinitely large matrices.
If the potential is equal to , this partition function becomes a
-function of -reduced KP-hierarchy, obeying a set of -algebra constraints identical to those conjectured in \cite{FKN91} for
double-scaling continuum limit of -matrix model. In the case of
the statement reduces to the early established \cite{MMM91b} relation between
Kontsevich model and the ordinary quantum gravity . Kontsevich model with
generic potential may be considered as interpolation between all the models of
quantum gravity with preserving the property of integrability and
the analogue of string equation.Comment: 67 pages (October 1991
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