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A. Mironov

Publications and source records attributed to A. Mironov.

266 records · Page 15Linked to original sources
arXiv1994

Generalized Hirota Equations and Representation Theory. I. The case of $SL(2)$ and $SL_q(2)$"

This paper begins investigation of the concept of ``generalized $τ$-function'', defined as a generating function of all the matrix elements of a group element $g \in G$ in a given highest-weight representation of a universal enveloping algebra ${\cal G}$. In the generic situation, the time-variables correspond to the elements of maximal nilpotent subalgebras rather than Cartanian elements. Moreover, in the case of quantum groups such $τ$-``functions'' are not $c$-numbers but take their values in non-commutative algebras (of functions on the quantum group $G$). Despite all these differences from the particular case of conventional $τ$-functions of integrable (KP and Toda lattice) hierarchies (which arise when $G$ is a Kac-Moody (1-loop) algebra of level $k=1$), these generic $τ$-functions also satisfy bilinear Hirota-like equations, which can be deduced from manipulations with intertwining operators. The most important applications of the formalism should be to $k>1$ Kac-Moody and multi-loop algebras, but this paper contains only illustrative calculations for the simplest case of ordinary (0-loop) algebra $SL(2)$ and its quantum counterpart $SL_q(2)$, as well as for the system of fundamental representations of $SL(n)$.

hep-th↗
arXiv1994

On a c-number quantum $τ$-function

We first review the properties of the conventional $τ$-functions of the KP and Toda-lattice hierarchies. A straightforward generalization is then discussed. It corresponds to passing from differential to finite-difference equations; it does not involve however the concept of operator-valued $τ$-function nor the one associated with non-Cartanian (level $k\ne1$) algebras. The present study could be useful to understand better $q$-free fields and their relation to ordinary free fields.

hep-th↗
arXiv1993

$2d$ gravity and matrix models. I. $2d$ gravity

Some approaches to $2d$ gravity developed for the last years are reviewed. They are physical (Liouville) gravity, topological theories and matrix models. A special attention is paid to matrix models and their interrelations with different approaches. Almost all technical details are omitted, but examples are presented.

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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.

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arXiv1992

On the Continuum Limit of the Conformal Matrix Models

The double scaling limit of a new class of the multi-matrix models proposed in \cite{MMM91}, which possess the $W$-symmetry at the discrete level, is investigated in details. These models are demonstrated to fall into the same universality class as the standard multi-matrix models. In particular, the transformation of the W-algebra at the discrete level into the continuum one of the paper \cite{FKN91a} is proposed, the corresponding partition functions being compared. All calculations are demonstrated in full in the first non-trivial case of $W^{(3)}$-constraints.

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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.

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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.

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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.

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arXiv1992

$c = r_g$ Theories of $W_G$-Gravity: The Set of Observables as a Model of Simply Laced $G$

We propose to study a generalization of the Klebanov-Polyakov-Witten (KPW) construction for the algebra of observables in the $c = 1$ string model to theories with $c > 1$. We emphasize the algebraic meaning of the KPW construction for $c = 1$ related to occurrence of a {\it model} of {\it SU}(2) as original structure on the algebra of observables. The attempts to preserve this structure in generalizations naturally leads to consideration of $W$-gravities. As a first step in the study of these generalized KPW constructions we design explicitly the subsector of the space of observables in appropriate $W_G$-string theory, which forms the {\it model} of $G$ for any simply laced {\it G}. The {\it model} structure is confirmed by the fact that corresponding one-loop Kac-Rocha-Caridi $W_G$-characters for $c = r_G$ sum into a chiral (open string) $k=1$ $G$-WZW partition function.

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arXiv1992

On Equivalence of Topological and Quantum 2d Gravity

We demonstrate the equivalence of Virasoro constraints imposed on continuum limit of partition function of Hermitean 1-matrix model and the Ward identities of Kontsevich's model. Since the first model describes ordinary $d = 2$ quantum gravity, while the second one is supposed to coincide with Witten's topological gravity, the result provides a strong implication that the two models are indeed the same.

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arXiv1992

Towards unified theory of $2d$ gravity

We introduce a new 1-matrix model with arbitrary potential and the matrix-valued background field. Its partition function is a $τ$-function of KP-hierarchy, subjected to a kind of ${\cal L}_{-1}$-constraint. Moreover, partition function behaves smoothly in the limit of infinitely large matrices. If the potential is equal to $X^{K+1}$, this partition function becomes a $τ$-function of $K$-reduced KP-hierarchy, obeying a set of ${\cal W} _K$-algebra constraints identical to those conjectured in \cite{FKN91} for double-scaling continuum limit of $(K-1)$-matrix model. In the case of $K=2$ the statement reduces to the early established \cite{MMM91b} relation between Kontsevich model and the ordinary $2d$ quantum gravity . Kontsevich model with generic potential may be considered as interpolation between all the models of $2d$ quantum gravity with $c<1$ preserving the property of integrability and the analogue of string equation.

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arXiv1992

The Asymptotics of the Correlations Functions in $(1+1)d$ Quantum Field Theory From Finite Size Effects in Conformal Theories

Using the finite-size effects the scaling dimensions and correlation functions of the main operators in continuous and lattice models of 1d spinless Bose-gas with pairwise interaction of rather general form are obtained. The long-wave properties of these systems can be described by the Gaussian model with central charge $c=1$. The disorder operators of the extended Gaussian model are found to correspond to some non-local operators in the {\it XXZ} Heisenberg antiferromagnet. Just the same approach is applicable to fermionic systems. Scaling dimensions of operators and correlation functions in the systems of interacting Fermi-particles are obtained. We present a universal treatment for $1d$ systems of different kinds which is independent of the exact integrability and gives universal expressions for critical exponents through the thermodynamic characteristics of the system.

hep-th↗
arXiv1991

Unification of All String Models with $c<1$

A 1-matrix model is proposed, which nicely interpolates between double-scaling continuum limits of all multimatrix models. The interpolating partition function is always a KP $τ$-function and always obeys ${\cal L}_{-1}$-constraint and string equation. Therefore this model can be considered as a natural unification of all models of 2d-gravity (string models) with $c\leq 1.$

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