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.

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 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].

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 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.

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 ON EQUIVALENCE OF TWO HURWITZ MATRIX MODELS

2009 ◽  
Vol 24 (33) ◽  
pp. 2659-2666 ◽  
Author(s):  
A. MOROZOV ◽  
SH. SHAKIROV
Keyword(s):  
Partition Function ◽  
External Field ◽  
Laplace Transform ◽  
Matrix Models ◽  
Matrix Model ◽  
Classical Equation ◽  
Equation Of Motion ◽  
Model Representation ◽  
Hurwitz Matrix

In Ref. 1 a matrix model representation was found for the simplest Hurwitz partition function, which has Lambert curve ϕe-ϕ= ψ as a classical equation of motion. We demonstrate that Fourier–Laplace transform in the logarithm of external field Ψ converts it into a more sophisticated form, recently suggested in Ref. 2.

scholarly journals Unitary matrix models and random partitions: Universality and multi-criticality

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Taro Kimura ◽  
Ali Zahabi
Keyword(s):  
Free Energy ◽  
Matrix Models ◽  
Strong Coupling ◽  
Unitary Matrix ◽  
Strong Coupling Regime ◽  
Operator Formalism ◽  
Random Partitions ◽  
Integrable Operator ◽  
Critical Model ◽  
Genus Expansion

Abstract The generating functions for the gauge theory observables are often represented in terms of the unitary matrix integrals. In this work, the perturbative and non-perturbative aspects of the generic multi-critical unitary matrix models are studied by adopting the integrable operator formalism, and the multi-critical generalization of the Tracy-Widom distribution in the context of random partitions. We obtain the universal results for the multi-critical model in the weak and strong coupling phases. The free energy of the instanton sector in the weak coupling regime, and the genus expansion of the free energy in the strong coupling regime are explicitly computed and the universal multi-critical phase structure of the model is explored. Finally, we apply our results in concrete examples of supersymmetric indices of gauge theories in the large N limit.

TOPOLOGY OF THE GRASSMANNIAN σ MODEL ON A LATTICE

1987 ◽  
Vol 02 (08) ◽  
pp. 601-608 ◽  
Author(s):  
T. FUKAI ◽  
M. V. ATRE
Keyword(s):  
Partition Function ◽  
Strong Coupling ◽  
Wilson Loop ◽  
Strong Coupling Limit ◽  
Topological Term

The Grassmannian σ model with a topological term is studied on a lattice. The θ dependence of the partition function and the Wilson loop are evaluated in the strong coupling limit. The latter is shown to be independent of the area at θ = π, as in the CPN−1 model.

scholarly journals Partial deconfinement at strong coupling on the lattice

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Hiromasa Watanabe ◽  
Georg Bergner ◽  
Norbert Bodendorfer ◽  
Shotaro Shiba Funai ◽  
Masanori Hanada ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Matrix Models ◽  
Gauge Group ◽  
Strong Coupling ◽  
Direct Evidence ◽  
Lattice Monte Carlo

Abstract We provide evidence for partial deconfinement — the deconfinement of a SU(M) subgroup of the SU(N) gauge group — by using lattice Monte Carlo simulations. We take matrix models as concrete examples. By appropriately fixing the gauge, we observe that the M × M submatrices deconfine. This gives direct evidence for partial deconfinement at strong coupling. We discuss the applications to QCD and holography.

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