scholarly journals FROM ONE-MATRIX MODEL TO KONTSEVICH MODEL

1993 ◽  
Vol 08 (30) ◽  
pp. 2875-2890 ◽  
Author(s):  
J. AMBJØRN ◽  
C. F. KRISTJANSEN
Keyword(s):  
Partition Function ◽  
Matrix Models ◽  
Matrix Model ◽  
Correlation Functions ◽  
Hermitian Matrix ◽  
Scaling Limit ◽  
Genus Zero ◽  
Kdv Hierarchy ◽  
Hermitian Matrix Model ◽  
Double Scaling Limit

Loop equations of matrix models express the invariance of the models under field redefinitions. We use loop equations to prove that it is possible to define continuum times for the generic Hermitian one-matrix model such that all correlation functions in the double scaling limit agree with the corresponding correlation functions of the Kontsevich model expressed in terms of KdV times. In addition the double scaling limit of the partition function of the Hermitian matrix model agrees with the τ-function of the KdV hierarchy corresponding to the Kontsevich model (and not the square of the τ-function) except for some complications at genus zero.

MAPPING BETWEEN TODA AND KDV FLOWS IN THE HERMITIAN MATRIX MODEL

1991 ◽  
Vol 06 (09) ◽  
pp. 811-818 ◽  
Author(s):  
WAICHI OGURA
Keyword(s):  
Orthogonal Polynomial ◽  
Matrix Model ◽  
Polynomial Identity ◽  
Hermitian Matrix ◽  
Scaling Limit ◽  
String Equation ◽  
Singular Potentials ◽  
Hermitian Matrix Model ◽  
Double Scaling Limit

The scaling operators are studied at finite N. We find new singular potentials for which an orthogonal polynomial identity gives the string equation at the double scaling limit. They are free from the degeneracy between even and odd potentials, and provide the mapping between the sl(∞) Toda and the generalized KdV flows. The degeneracy in formal Virasoro conditions are derived explicitly.

GENERALIZED c=1 MATRIX MODELS AND SYMMETRIC SPACES

1992 ◽  
Vol 07 (18) ◽  
pp. 4259-4274 ◽  
Author(s):  
M.A. OLSHANETSKY
Keyword(s):  
Matrix Models ◽  
Matrix Model ◽  
Symmetric Spaces ◽  
Hermitian Matrix ◽  
Scaling Limit ◽  
Natural Generalization ◽  
Target Space ◽  
Approximation Procedure ◽  
Hartree Fock ◽  
Hermitian Matrix Model

Random matrix models with a line as a target space are considered. The models are a natural generalization of the Hermitian matrix model and connected with the classical symmetric spaces of the Euclidean type which were classified by Cartan. Ten different types of these spaces exist. Three models on a line related to these models are reduced to one-dimensional free N-fermion problems which have special symmetric configurations. The solutions in the double scaling limit to all orders of perturbation are the same as for the Hermitian matrix model. In the general case the fermions interact with the Calogero-Mozer integrable potential. Due to this fact only the planar limit can be calculated by applying the Hartree-Fock approximation procedure.

PROPERTIES OF HERMITIAN MATRIX MODELS IN AN EXTERNAL FIELD

1991 ◽  
Vol 06 (37) ◽  
pp. 3455-3466 ◽  
Author(s):  
YU. MAKEENKO ◽  
G. W. SEMENOFF
Keyword(s):  
Integral Equation ◽  
Partition Function ◽  
External Field ◽  
Matrix Models ◽  
Matrix Model ◽  
Hermitian Matrix ◽  
Genus Zero ◽  
Virasoro Constraints ◽  
Large N ◽  
Single Integral

We consider a Hermitian one-matrix model in an (Hermitian) external field. We drive the Schwinger-Dyson equations and show that those can be represented as a set of Virasoro constraints which are imposed on the partition function. We prove that these Virasoro constraints are equivalent (at least at large N) to a single integral equation whose solution can be found. We use this solution to study properties of the Kontsevich model in genus zero.

scholarly journals Genus expansion of matrix models and ћ expansion of KP hierarchy

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
A. Andreev ◽  
A. Popolitov ◽  
A. Sleptsov ◽  
A. Zhabin
Keyword(s):  
Matrix Models ◽  
Matrix Model ◽  
Special Interest ◽  
Hermitian Matrix ◽  
Geometric Meaning ◽  
Kp Hierarchy ◽  
Hurwitz Numbers ◽  
Kp Equations ◽  
Genus Expansion ◽  
Hermitian Matrix Model

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

scholarly journals NON-PERTURBATIVE SOLUTION OF THE SUPER-VIRASORO CONSTRAINTS

1993 ◽  
Vol 08 (13) ◽  
pp. 1205-1214 ◽  
Author(s):  
K. BECKER ◽  
M. BECKER
Keyword(s):  
Partition Function ◽  
Matrix Model ◽  
Scaling Limit ◽  
Virasoro Constraints ◽  
Perturbative Solution ◽  
The One ◽  
Genus Expansion ◽  
Double Scaling Limit

We present the solution of the discrete super-Virasoro constraints to all orders of the genus expansion. Integrating over the fermionic variables we get a representation of the partition function in terms of the one-matrix model. We also obtain the non-perturbative solution of the super-Virasoro constraints in the double scaling limit but do not find agreement between our flows and the known supersymmetric extensions of KdV.

NEW CRITICAL BEHAVIORS FOR ONE-HERMITIAN-MATRIX MODELS

1992 ◽  
Vol 07 (07) ◽  
pp. 1527-1551
Author(s):  
P.M.S. PETROPOULOS
Keyword(s):  
Matrix Models ◽  
Hermitian Matrix ◽  
Scaling Limit ◽  
Field Theories ◽  
Polynomial Coefficients ◽  
Multicritical Points ◽  
Exact Correlation Functions ◽  
Genus One ◽  
Exact Correlation ◽  
Double Scaling Limit

In the general framework of one-Hermitian-matrix models, we study critical behaviors such that δx~δRm/n~δSm; δx, δS and δR are, respectively, the bare cosmological constant and the orthogonal-polynomial coefficients around criticality. On the sphere, we prove the existence of consistent multicriticality conditions such that string equations exhibit the above behavior. We define a double scaling limit and write down exact equations for the specific heat for any (m, n) model. Their solutions are unambiguous and the only corrections come from genus-one topology. We compute exact correlation functions for well-defined scaling operators. These belong to two different sectors. One of them is such that any squared operator vanishes when inserted in any correlation function. We discuss briefly the flows between these multicritical points as well as the nature of the 2D field theories coupled to gravity which they can describe.

scholarly journals MATRIX MODELS AND INTEGRABLE C<1 OPEN STRING THEORIES

1994 ◽  
Vol 03 (01) ◽  
pp. 203-206
Author(s):  
LAURENT HOUART
Keyword(s):  
Ising Model ◽  
Boundary Conditions ◽  
Matrix Models ◽  
Matrix Model ◽  
Scaling Limit ◽  
Open String ◽  
String Equation ◽  
Random Surfaces ◽  
Ising Spins ◽  
Double Scaling Limit

We study in the double scaling limit the two-matrix model which represents the sum over closed and open random surfaces coupled to an Ising model. The boundary conditions are characterized by the fact that the Ising spins sitting at the vertices of the boundaries are all in the same state. We obtain the string equation.

scholarly journals DOUBLE SCALING LIMIT OF THE SUPER-VIRASORO CONSTRAINTS

1993 ◽  
Vol 08 (13) ◽  
pp. 2297-2331 ◽  
Author(s):  
L. ALVAREZ-GAUMÉ ◽  
K. BECKER ◽  
M. BECKER ◽  
R. EMPARAN ◽  
J. MAÑES
Keyword(s):  
Scaling Limit ◽  
Majorana Fermion ◽  
Two Dimensional ◽  
Genus Zero ◽  
Heuristic Argument ◽  
Kdv Hierarchy ◽  
Virasoro Constraints ◽  
Bosonic Part ◽  
Double Scaling Limit ◽  
The Continuum

We obtain the double scaling limit of a set of superloop equations recently proposed to describe the coupling of two-dimensional supergravity to minimal superconformal matter of type (2,4m). The continuum loop equations are described in terms of a [Formula: see text] theory with a Z2-twisted scalar field and a Weyl–Majorana fermion in the Ramond sector. We have computed correlation functions in genus zero, one and partially in genus two. An integrable supersymmetric hierarchy describing our model has not yet been found. We present a heuristic argument showing that the purely bosonic part of our model is described by the KdV hierarchy.

MATRIX MODELS AND GAUGE-INVARIANT FIELD THEORY OF SUBCRITICAL STRINGS

1992 ◽  
Vol 07 (11) ◽  
pp. 2559-2588 ◽  
Author(s):  
ASHOKE SEN
Keyword(s):  
Field Theory ◽  
Partition Function ◽  
Matrix Models ◽  
Matrix Model ◽  
String Field Theory ◽  
Correlation Functions ◽  
Ward Identities ◽  
Gauge Invariant ◽  
The Matrix

A gauge-invariant interacting field theory of subcritical closed strings is constructed. It is shown that for d ≤ 1 this field theory reproduces many of the features of the corresponding matrix model. Among them are the scaling dimensions of the relevant primary fields, identities involving the correlation functions of some of the redundant operators in the matrix model, and the flow between different matrix models under appropriate perturbation. In particular, it is shown that some of the constraints on the partition function derived recently by Dijkgraaf et al. and Fukuma et al. may be interpreted as Ward identities in string field theory.

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