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 UNITARY MATRIX MODELS AND 2D QUANTUM GRAVITY

1992 ◽  
Vol 07 (29) ◽  
pp. 2753-2762 ◽  
Author(s):  
S. DALLEY ◽  
C. V. JOHNSON ◽  
T. R. MORRIS ◽  
A. WÄTTERSTAM
Keyword(s):  
Matrix Models ◽  
Integrable Hierarchies ◽  
Renormalization Group Equation ◽  
Closed String ◽  
Singular Solutions ◽  
Unitary Matrix ◽  
Group Equation ◽  
Gravitational System ◽  
Large N ◽  
Matrix Integrals

The KdV and modified KdV integrable hierarchies are shown to be different descriptions of the same 2D gravitational system — open-closed string theory. Non-perturbative solutions of the multicritical unitary matrix models map to non-singular solutions of the 'renormalization group' equation for the string susceptibility, [Formula: see text]. We also demonstrate that the large-N solutions of unitary matrix integrals in external fields, studied by Gross and Newman, equal the non-singular pure closed-string solutions of [Formula: see text].

scholarly journals Unitary Matrix Integrals, Primitive Factorizations, and Jucys-Murphy Elements

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Sho Matsumoto ◽  
Jonathan Novak
Keyword(s):  
Symmetric Group ◽  
Matrix Models ◽  
Unitary Matrix ◽  
Group Characters ◽  
Matrix Integrals ◽  
International Audience

International audience A factorization of a permutation into transpositions is called "primitive'' if its factors are weakly ordered.We discuss the problem of enumerating primitive factorizations of permutations, and its place in the hierarchy of previously studied factorization problems. Several formulas enumerating minimal primitive and possibly non-minimal primitive factorizations are presented, and interesting connections with Jucys-Murphy elements, symmetric group characters, and matrix models are described. Une factorisation en transpositions d'une permutation est dite "primitive'' si ses facteurs sont ordonnés. Nous discutons du problème de l'énumération des factorisations primitives de permutations, et de sa place dans la hiérarchie des problèmes de factorisation précédemment étudiés. Nous présentons plusieurs formules énumérant certaines classes de factorisations primitives,et nous soulignons des connexions intéressantes avec les éléments Jucys-Murphy, les caractères des groupes symétriques, et les modèles de matrices.

scholarly journals UNITARY MATRIX INTEGRALS IN THE FRAMEWORK OF THE GENERALIZED KONTSEVICH MODEL

1996 ◽  
Vol 11 (28) ◽  
pp. 5031-5080 ◽  
Author(s):  
A. MIRONOV ◽  
A. MOROZOV ◽  
G. W. SEMENOFF
Keyword(s):  
Partition Function ◽  
External Field ◽  
Matrix Models ◽  
Strong Coupling ◽  
Weak Coupling ◽  
Unitary Matrix ◽  
Simple Method ◽  
Generating Functional ◽  
New Approach ◽  
Matrix Integrals

We advocate a new approach to the study of unitary matrix models in external fields which emphasizes their relationship to generalized Kontsevich models (GKM's) with nonpolynomial potentials. For example, we show that the partition function of the Brezin–Gross–Witten model (BGWM), which is defined as an integral over unitary N × N matrices, [Formula: see text], can also be considered as a GKM with potential [Formula: see text]. Moreover, it can be interpreted as the generating functional for correlators in the Penner model. The strong and weak coupling phases of the BGWM are identified with the "character" (weak coupling) and "Kontsevich" (strong coupling) phases of the GKM, respectively. This type of GKM deserves classification as a p = −2 model (i.e. c = 28 or c = −2) when in the Kontsevich phase. This approach allows us to further identify the Harish-Chandra–Itzykson–Zuber integral with a peculiar GKM, which arises in the study of c = 1, theory, and, further, with a conventional two-matrix model which is rewritten in Miwa coordinates. Some further extensions of the GKM treatment which are inspired by the unitary matrix models which we have considered are also developed. In particular, as a by-product, a new, simple method of fixing the Ward identities for matrix models in an external field is presented.

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 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.

scholarly journals The replica limit of unitary matrix integrals

2001 ◽  
Vol 592 (3) ◽  
pp. 419-444 ◽  
Author(s):  
D. Dalmazi ◽  
J.J.M. Verbaarschot
Keyword(s):  
Unitary Matrix ◽  
Matrix Integrals

scholarly journals Complex Langevin dynamics in large N unitary matrix models

2018 ◽  
Vol 98 (3) ◽  
Author(s):  
Pallab Basu ◽  
Kasi Jaswin ◽  
Anosh Joseph
Keyword(s):  
Matrix Models ◽  
Langevin Dynamics ◽  
Unitary Matrix ◽  
Large N

scholarly journals STOKES PHENOMENA AND QUANTUM INTEGRABILITY IN NON-CRITICAL STRING/M THEORY

2013 ◽  
Vol 21 ◽  
pp. 147-148
Author(s):  
CHUAN-TSUNG CHAN ◽  
HIROTAKA IRIE ◽  
CHI-HSIEN YEH
Keyword(s):  
String Theory ◽  
Matrix Models ◽  
Recent Progress ◽  
Solvable Model ◽  
First Principle ◽  
Stokes Phenomenon ◽  
Quantum Integrability ◽  
Key Concepts ◽  
M Theory ◽  
Stokes Phenomena

Non-critical string/M theory is a solvable model which has been studied to reveal various non-perturbative aspects of string theory with providing new key concepts to the next developments of string theory. Here we show some recent progress in study of Stokes phenomenon in non-critical string theory of the multi-cut two-matrix models. In particular, we argue that it is Stokes phenomenon which allows us to know concepts of non-perturbative completion with analytic study of string-theory landscape from the first principle.

scholarly journals AN OPERATOR FORMALISM FOR UNITARY MATRIX MODELS

1991 ◽  
Vol 06 (29) ◽  
pp. 2727-2739 ◽  
Author(s):  
K. N. ANAGNOSTOPOULOS ◽  
M. J. BOWICK ◽  
N. ISHIBASHI
Keyword(s):  
Orthogonal Polynomials ◽  
Matrix Models ◽  
Pseudodifferential Operators ◽  
Scaling Limit ◽  
Unitary Matrix ◽  
String Equation ◽  
Operator Formalism ◽  
Multicritical Point ◽  
Compact Formulation ◽  
Double Scaling Limit

We analyze the double scaling limit of unitary matrix models in terms of trigonometric orthogonal polynomials on the circle. In particular we find a compact formulation of the string equation at the kth multicritical point in terms of pseudodifferential operators and a corresponding action principle. We also relate this approach to the mKdV hierarchy which appears in the analysis in terms of conventional orthogonal polynomials on the circle.

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