INTEGRABLE STRUCTURES OF UNITARY MATRIX MODELS

1992 ◽  
Vol 07 (20) ◽  
pp. 4803-4824 ◽  
Author(s):  
S. KHARCHEV ◽  
A. MIRONOV
Keyword(s):  
Matrix Models ◽  
Matrix Model ◽  
Toda Lattice ◽  
Discrete Analog ◽  
Unitary Matrix ◽  
Component Systems ◽  
Unitary Model ◽  
Two Component Systems ◽  
Hermitian Matrix Model ◽  
Toda Lattice Hierarchy

The unitary matrix model is considered from the viewpoint of integrability. We demonstrate that this is an integrable system embedded into a two-dimensional Toda lattice hierarchy which corresponds to an integrable chain (modified Volterra) under a special reduction. The interrelations between this chain and other chains (like the Toda one) are demonstrated to be given by Bäcklund transformations. The case of the symmetric unitary model is discussed in detail and demonstrated to be connected with the Hermitian matrix model. This connection as a discrete analog of the correspondence between KdV and MKdV systems is investigated more thoroughly. We also demonstrate that unitary matrix models can be considered as two-component systems as well.

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 Multiple phases in a generalized Gross-Witten-Wadia matrix model

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Jorge G. Russo ◽  
Miguel Tierz
Keyword(s):  
Partition Function ◽  
Matrix Models ◽  
Characteristic Polynomial ◽  
Matrix Model ◽  
Phase Structure ◽  
Unitary Matrix ◽  
Two Dimensional ◽  
Toeplitz Determinants ◽  
Dimensional Parameter ◽  
Planar Limit

Abstract We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large N results are obtained by using Szegö theorem with a Fisher-Hartwig singularity. In the large N (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space.

Unitary integrals and related matrix models

2018 ◽  
Author(s):  
Nicolas Orantin
Keyword(s):  
Matrix Models ◽  
Unitary Matrix ◽  
Technical Tool ◽  
Basic Properties ◽  
Matrix Integrals ◽  
Unitary Model ◽  
M Theory ◽  
Pair Correlator

This article examines the basic properties of unitary matrix integrals using three matrix models: the ordinary unitary model, the Brézin-Gross-Witten (BGW) model and the Harish-Chandra-Itzykson-Zuber (HCIZ) model. The tricky sides of the story are given special attention, such as the de Wit-’t Hooft anomaly in unitary integrals and the problem of correlators with Itzykson-Zuber measure. The method of character expansions is also emphasized as a technical tool. The article first provides an overview of the theory of the BGW model, taking into account the de Wit-’t Hooft anomaly and the M-theory of matrix models, before discussing the theory of the HCIZ integral. In particular, it describes the basics of character calculus, character expansion of the HCIZ integral, character expansion for the BGW model and Leutwyler-Smilga integral, and pair correlator in HCIZ theory.

scholarly journals ABJM matrix model and 2D Toda lattice hierarchy

2019 ◽  
Vol 2019 (3) ◽  
Author(s):  
Tomohiro Furukawa ◽  
Sanefumi Moriyama
Keyword(s):  
Matrix Model ◽  
Toda Lattice ◽  
Toda Lattice Hierarchy

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 Faces of Relativistic Toda Chain

1997 ◽  
Vol 12 (15) ◽  
pp. 2675-2724 ◽  
Author(s):  
S. Kharchev ◽  
A. Mironov ◽  
A. Zhedanov
Keyword(s):  
Orthogonal Polynomial ◽  
Toda Lattice ◽  
Toda Chain ◽  
Polynomial Systems ◽  
Function Approach ◽  
Unitary Matrix ◽  
Lax Representation ◽  
Two Dimensional ◽  
Two Component ◽  
Toda Lattice Hierarchy

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 the 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.

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.

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.

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