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S. Kharchev

Publications and source records attributed to S. Kharchev.

At least 37 records · Page 2Linked to original sources
arXiv2003

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 $T^*(GL(N))$ to the Lie algebra $\frak{gl}_N$. The construction is based on the Gelfand-Zetlin maximal commuting subalgebra in $U(\frak{gl}_N)$. We discuss the connection with the other known integrable systems based on $T^*GL(N)$. 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.

math.QA↗
arXiv2001

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.

hep-th↗
arXiv2000

Integral representations for the eigenfunctions of quantum open and periodic Toda chains from QISM formalism

The integral representations for the eigenfunctions of $N$ particle quantum open and periodic Toda chains are constructed in the framework of Quantum Inverse Scattering Method (QISM). Both periodic and open $N$-particle solutions have essentially the same structure being written as a generalized Fourier transform over the eigenfunctions of the $N-1$ 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.

hep-th↗
arXiv1999

Integral representation for the eigenfunctions of quantum periodic Toda chain

Integral representation for the eigenfunctions of quantum periodic Toda chain is constructed for N-particle case. The multiple integral is calculated using the Cauchy residue formula. This gives the representation which reproduces the particular results obtained by Gutzwiller for N=2,3 and 4-particle chain. Our method to solve the problem combines the ideas of Gutzwiller and R-matrix approach of Sklyanin with the classical results in the theory of the Whittaker functions. In particular, we calculate Sklyanin's invariant scalar product from the Plancherel formula for the Whittaker functions derived by Semenov-Tian-Shansky thus obtaining the natural interpretation of the Sklyanin measure in terms of the Harish-Chandra function.

hep-th↗
arXiv1998

Kadomtsev-Petviashvili Hierarchy and Generalized Kontsevich Model

The review is devoted to the integrable properties of the Generalized Kontsevich Model which is supposed to be an universal matrix model to describe the conformal field theories with $c<1$. It is shown that the deformations of the "monomial" phase to "polynomial" one have the natural interpretation in context of so-called equivalent hierarchies. The dynamical transition between equivalent integrable systems is exactly along the flows of the dispersionless Kadomtsev-Petviashvili hierarchy; the coefficients of the potential are shown to be directly related with the flat (quasiclassical) times arising in N=2 Landau-Ginzburg topological model. The Virasoro constraint for solution with an arbitrary potential is shown to be a standard $Ł_{-p}$-constraint of the (equivalent) $p$-reduced hierarchy with the times additively corrected by the flat coordinates.

hep-th↗
arXiv1996

Different Aspects of Relativistic Toda Chain

We demonstrate that the generalization of the relativistic Toda chain (RTC) is a special reduction of two-dimensional Toda Lattice hierarchy (2DTL). We also show that the RTC is gauge equivalent to the discrete AKNS hierarchy and the unitary matrix model. Relativistic Toda molecule hierarchy is also considered, along with the forced RTC. The simple approach to the discrete RTC hierarchy based on Darboux-Bäcklund transformation is proposed.

hep-th↗
arXiv1996

Faces of Relativistic Toda Chain

We demonstrate that the generalization of the relativistic Toda chain (RTC) is a special reduction of two-dimensional Toda Lattice hierarchy (2DTL). This reduction implies that the RTC is gauge equivalent to the discrete AKNS hierarchy and, which is the same, to the two-component Volterra hierarchy while its forced (semi-infinite) variant is described by the unitary matrix integral. The integrable properties of the RTC hierarchy are revealed in different frameworks of: Lax representation, orthogonal polynomial systems, and $τ$-function approach. Relativistic Toda molecule hierarchy is also considered, along with the forced RTC. Some applications to biorthogonal polynomial systems are discussed.

hep-th↗
arXiv1996

Liouville Type Models in Group Theory Framework. I. Finite-Dimensional Algebras

In the series of papers we represent the ``Whittaker'' wave functional of $d+1$-dimensional Liouville model as a correlator in $d+0$-dimensional theory of the sine-Gordon type (for $d=0$ and $1$). Asypmtotics of this wave function is characterized by the Harish-Chandra function, which is shown to be a product of simple $Γ$-function factors over all positive roots of the corresponding algebras (finite-dimensional for $d=0$ and affine for $d=1$). This is in nice correspondence with the recent results on 2- and 3-point correlators in $1+1$ Liouville model, where emergence of peculiar double-periodicity is observed. The Whittaker wave functions of $d+1$-dimensional non-affine ("conformal") Toda type models are given by simple averages in the $d+0$ 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 $d+1$-dimensional disk with given boundary conditions, as a (non-local) quadratic integral over the $d$-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.

hep-th↗
arXiv1995

Non-Standard KP Evolution and Quantum $τ$-function

One possible way to fix partly a ``canonical definition'' of $τ$-functions beyond the conventional KP/Toda framework could be to postulate that evolution operators are {\it always} group elements. We discuss implications of this postulate for the first non-trivial case: fundamental representations of quantum groups $SL_q(N)$. It appears that the most suited (simple) for quantum deformation framework is some non-standard formulation of KP/Toda systems. It turns out that the postulate needs to be slightly modified to take into account that no ``nilpotent subgroups'' exist in $SL_q(N)$ for $q\neq 1$. This has some definite and simple implications for $q$-determinant-like representations of quantum $τ$-functions.

q-alg↗
arXiv1994

Quantization of string theory for $c \leq 1$

We consider the canonical quantization scheme for $c \leq 1$ ($(p,q)$ -) string theories and compare it with what is known from matrix model approach. We derive explicitly a trivial ($\equiv $ topological) solution. We discuss a ``dressing" operator which in principle allows one to obtain a non-trivial solution, but an explicit computation runs into a problem of analytic continuation of the formal expressions for $τ$-functions. We discuss also the application of proposed scheme to the case of discrete matrix model and consider some parallels with mirror symmetry and background independence in string theory.

hep-th↗
arXiv1993

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 $GL$ group. The integration over "matter fields" can be interpreted as going over the {\it model} (the space of all highest weight representations) of $GL$. In the case of compact unitary groups the integrals should be substituted by {\it discrete} sums over weight lattice. The $D=0$ version of the model is the Generalized Kontsevich integral, which in the above-mentioned unitary (discrete) situation coincides with partition function of the $2d$ Yang-Mills theory with the target space of genus $g=0$ and $m=0,1,2$ holes. This particular quantity is always a bilinear combination of characters and appears to be a Toda-lattice $τ$-function. (This is generalization of the classical statement that individual $GL$ characters are always singular KP $τ$-functions.) The corresponding element of the Universal Grassmannian is very simple and somewhat similar to the one, arising in investigations of the $c=1$ 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.

hep-th↗
arXiv1993

On pq-duality and explicit solutions in $c \le 1$ $2d$ gravity models

We study the integral representation for the exact solution to nonperturbative $c le 1$ string theory. A generic solution is determined by two functions $W(x)$ and $Q(x)$ which behaive at infinity like $x^p$ and $x^q$ respectively. The integral model for arbitrary $(p,q)$ models is derived which explicitely demonstrates $p-q$ duality of minimal models coupled to gravity. We discuss also the exact solutions to string equation and reduction condition and present several explicit examples.

hep-th↗
arXiv1992

Topological versus Non--Topological Theories and $p-q$ Duality in $c \le 1$ 2d Gravity Models

We discuss the non--perturbative formulation for $c \leq 1$ string theory. The field theory like formulation of topological and non--topological models is presented. The integral representation for arbitrary $(p,q)$ solutions is derived which explicitly obeys $p-q$ duality of these theories. The exact solutions to string equation and various examples are also discussed.

hep-th↗
arXiv1992

Landau-Ginzburg Topological Theories in the Framework of GKM and Equivalent Hierarchies

We consider the deformations of ``monomial solutions'' to Generalized Kontsevich Model \cite{KMMMZ91a,KMMMZ91b} and establish the relation between the flows generated by these deformations with those of $N=2$ Landau-Ginzburg topological theories. We prove that the partition function of a generic Generalized Kontsevich Model can be presented as a product of some ``quasiclassical'' factor and non-deformed partition function which depends only on the sum of Miwa transformed and flat times. This result is important for the restoration of explicit $p-q$ symmetry in the interpolation pattern between all the $(p,q)$-minimal string models with $c<1$ and for revealing its integrable structure in $p$-direction, determined by deformations of the potential. It also implies the way in which supersymmetric Landau-Ginzburg models are embedded into the general context of GKM. From the point of view of integrable theory these deformations present a particular case of what is called equivalent hierarchies.

hep-th↗
arXiv1992

Conformal Matrix Models as an Alternative to Conventional Multi-Matrix Models

We introduce {\it conformal multi-matrix models} (CMM) as an alternative to conventional multi-matrix model description of two-dimensional gravity interacting with $c < 1$ matter. We define CMM as solutions to (discrete) extended Virasoro constraints. We argue that the so defined alternatives of multi-matrix models represent the same universality classes in continuum limit, while at the discrete level they provide explicit solutions to the multi-component KP hierarchy and by definition satisfy the discrete $W$-constraints. We prove that discrete CMM coincide with the $(p,q)$-series of 2d gravity models in a {\it well}-{\it defined} continuum limit, thus demonstrating that they provide a proper generalization of Hermitian one-matrix model.

hep-th↗
arXiv1992

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 $τ$-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 $τ$-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, $i.e.$ essentially in terms of {\it finite}-fold integrals.

hep-th↗