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.

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.

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.

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.

FERMIONS IN THE TOPOLOGICAL EXPANSION OF MATRIX MODELS

1991 ◽  
Vol 06 (09) ◽  
pp. 781-787
Author(s):  
G. FERRETTI
Keyword(s):  
Free Energy ◽  
Matrix Models ◽  
Matrix Model ◽  
Hermitian Matrix ◽  
Coupling Constant ◽  
Counting Problem ◽  
Planar Limit ◽  
Quartic Interaction ◽  
Hermitian Matrix Model ◽  
Topological Expansion

The hermitian matrix model with quartic interaction is studied in presence of fermionic variables. We obtain the contribution to the free energy due to the presence of fermions. The first two terms beyond the planar limit are explicitly found for all values of the coupling constant g. These terms represent the solution of the counting problem for vacuum diagrams with one or two fermionic loops.

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 Bulk–boundary correlators in the hermitian matrix model and minimal Liouville gravity

2012 ◽  
Vol 854 (3) ◽  
pp. 853-877 ◽  
Author(s):  
Jean-Emile Bourgine ◽  
Goro Ishiki ◽  
Chaiho Rim
Keyword(s):  
Matrix Model ◽  
Hermitian Matrix ◽  
Liouville Gravity ◽  
Hermitian Matrix Model
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