scholarly journals CORRELATORS OF THE KAZAKOV-MIGDAL MODEL

1993 ◽  
Vol 08 (25) ◽  
pp. 2387-2401 ◽  
Author(s):  
M. I. DOBROLIUBOV ◽  
YU. MAKEENKO ◽  
G. W. SEMENOFF
Keyword(s):  
Matrix Model ◽  
Explicit Solution ◽  
Scalar Fields ◽  
Algebraic Equations ◽  
Quadratic Potential ◽  
Large N ◽  
Gaussian Case ◽  
The One ◽  
Non Gaussian

We derive loop equations for the one-link correlators of gauge and scalar fields in the Kazakov-Migdal model. These equations determine the solution of the model in the large-N limit and are similar to analogous equations for the Hermitian two-matrix model. We give an explicit solution of the equations for the case of a Gaussian, quadratic potential. We also show how similar calculations in a non-Gaussian case reduce to purely algebraic equations.

TOPOLOGICAL σ MODELS AND LARGE N MATRIX INTEGRAL

1995 ◽  
Vol 10 (29) ◽  
pp. 4203-4224 ◽  
Author(s):  
TOHRU EGUCHI ◽  
KENTARO HORI ◽  
SUNG-KIL YANG
Keyword(s):  
Matrix Model ◽  
Riemann Surfaces ◽  
Holomorphic Maps ◽  
Integrable Structure ◽  
Toda Hierarchy ◽  
Matrix Integral ◽  
Large N ◽  
The Matrix ◽  
Lax Operator ◽  
The One

In this paper we describe in some detail the representation of the topological CP1 model in terms of a matrix integral which we have introduced in a previous article. We first discuss the integrable structure of the CP1 model and show that it is governed by an extension of the one-dimensional Toda hierarchy. We then introduce a matrix model which reproduces the sum over holomorphic maps from arbitrary Riemann surfaces onto CP1. We compute intersection numbers on the moduli space of curves using a geometrical method and show that the results agree with those predicted by the matrix model. We also develop a Landau-Ginzburg (LG) description of the CP1 model using a superpotential eX + et0,Q e-X given by the Lax operator of the Toda hierarchy (X is the LG field and t0,Q is the coupling constant of the Kähler class). The form of the superpotential indicates the close connection between CP1 and N=2 supersymmetric sine-Gordon theory which was noted sometime ago by several authors. We also discuss possible generalizations of our construction to other manifolds and present an LG formulation of the topological CP2 model.

scholarly journals Fundamental Strings and D-Strings in the IIB Matrix Model

1997 ◽  
Vol 12 (18) ◽  
pp. 1301-1315 ◽  
Author(s):  
B. Sathiapalan
Keyword(s):  
Matrix Model ◽  
Scaling Limit ◽  
Duality Group ◽  
Fundamental String ◽  
Large N ◽  
The Matrix ◽  
Model Derivation ◽  
The One ◽  
The Right ◽  
String Worldsheet

The matrix model for IIB superstring proposed by Ishibashi, Kawai, Kitazawa and Tsuchiya is investigated. Consideration of planar and non-planar diagrams suggests that large-N perturbative expansion is consistent with the double scaling limit proposed by the above authors. We write down a Wilson loop that can be interpreted as a fundamental string vertex operator. The one-point tadpole in the presence of a D-string has the right form and this can be viewed as a matrix model derivation of the boundary conditions that define a D-string. We also argue that if worldsheet coordinates σ and τ are introduced to the fundamental string, then the conjugate variable d/dσ and d/dτ can be interpreted as the D-string worldsheet coordinates. In this way the SL (2Z) duality group of the IIB superstring becomes identified with the symplectic group acting on (p,q).

scholarly journals UNIFIED DESCRIPTION OF CORRELATORS IN NON-GAUSSIAN PHASES OF HERMITIAN MATRIX MODEL

2006 ◽  
Vol 21 (12) ◽  
pp. 2481-2517 ◽  
Author(s):  
A. ALEXANDROV ◽  
A. MIRONOV ◽  
A. MOROZOV
Keyword(s):  
Matrix Model ◽  
Hermitian Matrix ◽  
Partition Functions ◽  
Unified Approach ◽  
Small Space ◽  
Linear Differential ◽  
Gaussian Case ◽  
Non Gaussian ◽  
Hermitian Matrix Model ◽  
Unified Description

Following the program, proposed in hep-th/0310113, of systematizing known properties of matrix model partition functions (defined as solutions to the Virasoro-like sets of linear differential equations), we proceed to consideration of non-Gaussian phases of the Hermitian one-matrix model. A unified approach is proposed for description of "connected correlators" in the form of the phase-independent "check-operators" acting on the small space of T-variables (which parametrize the polynomial W(z)). With appropriate definitions and ordering prescriptions, the multidensity check-operators look very similar to the Gaussian case (however, a reliable proof of suggested explicit expressions in all loops is not yet available, only certain consistency checks are performed).

scholarly journals CRITICAL SCALING AND CONTINUUM LIMITS IN THE D>1 KAZAKOV-MIGDAL MODEL

1995 ◽  
Vol 10 (18) ◽  
pp. 2615-2660 ◽  
Author(s):  
YU. MAKEENKO
Keyword(s):  
Phase Transition ◽  
Matrix Model ◽  
Tricritical Point ◽  
Logarithmic Potential ◽  
Coordination Numbers ◽  
Large N ◽  
Arbitrary Potential ◽  
Local Observables ◽  
The One ◽  
Continuum Theories

I investigate the Kazakov-Migdal (KM) model — the Hermitian gauge-invariant matrix model on a D-dimensional lattice. I use an exact large N solution of the KM model with a logarithmic potential to examine its critical behavior. I find critical lines associated with γstr=−1/2 and γ str =0 as well as a tricritical point associated with a phase transition of the Kosterlitz-Thouless type. The continuum theories are constructed expanding around the critical points. The one associated with γ str =0 coincides with a d=1 string while a phase transition of the Kosterlitz-Thouless type separates it from that with γ str =−1/2, which is indistinguishable from pure 2D gravity for local observables but has a continuum limit for correlators of extended Wilson loops at large distances due to a singular behavior of the Itzykson-Zuber correlator of the gauge fields. I re-examine the KM model with an arbitrary potential in the large D limit and show that it reduces at large N to a one-matrix model whose potential is determined self-consistently. A relation with discretized random surfaces is established via the gauged Potts model, which is equivalent to the KM model at large N providing the coordination numbers coincide.

Enumeration of maps

2018 ◽  
Author(s):  
Grigori Olshanski
Keyword(s):  
Matrix Model ◽  
Matrix Theory ◽  
Polynomial Potential ◽  
Quadratic Potential ◽  
The Matrix ◽  
Formal Expansion ◽  
Enumeration Problem ◽  
The One ◽  
Map Enumeration ◽  
The Relationship

This article discusses the relationship between random matrices and maps, i.e. graphs drawn on surfaces, with particular emphasis on the one-matrix model and how it can be used to solve a map enumeration problem. It first provides an overview of maps and related objects, recalling the basic definitions related to graphs and defining maps as graphs embedded into surfaces before considering a coding of maps by pairs of permutations. It then examines the connection between matrix integrals and maps, focusing on the Hermitian one-matrix model with a polynomial potential and how the formal expansion of its free energy around a Gaussian point (quadratic potential) can be represented by diagrams identifiable with maps. The article also illustrates how the solution of the map enumeration problem can be deduced by means of random matrix theory (RMT). Finally, it explains how the matrix model result can be translated into a bijective proof.

Two-dimensional quantum gravity

2018 ◽  
Author(s):  
Leonid Chekhov
Keyword(s):  
Quantum Gravity ◽  
Matrix Models ◽  
Matrix Model ◽  
Planar Graphs ◽  
2D Gravity ◽  
Loop Model ◽  
World Sheet ◽  
Liouville Gravity ◽  
Large N ◽  
The One

This article discusses the connection between large N matrix models and critical phenomena on lattices with fluctuating geometry, with particular emphasis on the solvable models of 2D lattice quantum gravity and how they are related to matrix models. It first provides an overview of the continuum world sheet theory and the Liouville gravity before deriving the Knizhnik-Polyakov-Zamolodchikov scaling relation. It then describes the simplest model of 2D gravity and the corresponding matrix model, along with the vertex/height integrable models on planar graphs and their mapping to matrix models. It also considers the discretization of the path integral over metrics, the solution of pure lattice gravity using the one-matrix model, the construction of the Ising model coupled to 2D gravity discretized on planar graphs, the O(n) loop model, the six-vertex model, the q-state Potts model, and solid-on-solid and ADE matrix models.

OnPoint: A package for online experiments in motor control and motor learning

2020 ◽  
Author(s):  
Jonathan Sanching Tsay ◽  
Alan S. Lee ◽  
Guy Avraham ◽  
Darius E. Parvin ◽  
Jeremy Ho ◽  
...  
Keyword(s):  
Motor Control ◽  
Motor Learning ◽  
Open Source Software ◽  
Research Progress ◽  
Global Pandemic ◽  
Large N ◽  
Open Source Software Package ◽  
Potential Applications ◽  
Temporal And Spatial ◽  
The One

Motor learning experiments are typically run in-person, exploiting finely calibrated setups (digitizing tablets, robotic manipulandum, full VR displays) that provide high temporal and spatial resolution. However, these experiments come at a cost, not limited to the one-time expense of purchasing equipment but also the substantial time devoted to recruiting participants and administering the experiment. Moreover, exceptional circumstances that limit in-person testing, such as a global pandemic, may halt research progress. These limitations of in-person motor learning research have motivated the design of OnPoint, an open-source software package for motor control and motor learning researchers. As with all online studies, OnPoint offers an opportunity to conduct large-N motor learning studies, with potential applications to do faster pilot testing, replicate previous findings, and conduct longitudinal studies (GitHub repository: https://github.com/alan-s-lee/OnPoint).

scholarly journals Comment on “The phase diagram of the multi-matrix model with ABAB-interaction from functional renormalization”

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
Carlos I. Perez-Sanchez
Keyword(s):  
Phase Diagram ◽  
Matrix Model ◽  
Loop Structure ◽  
The One

Abstract Recently, [JHEP12 131 (2020)] obtained (a similar, scaled version of) the (a, b)-phase diagram derived from the Kazakov-Zinn-Justin solution of the Hermitian two-matrix model with interactions$$ -\mathrm{Tr}\left\{\frac{a}{4}\left({A}^4+{B}^4\right)+\frac{b}{2} ABAB\right\}, $$ − Tr a 4 A 4 + B 4 + b 2 ABAB , starting from Functional Renormalization. We comment on something unexpected: the phase diagram of [JHEP12 131 (2020)] is based on a βb-function that does not have the one-loop structure of the Wetterich-Morris equation. This raises the question of how to reproduce the phase diagram from a set of β-functions that is, in its totality, consistent with Functional Renormalization. A non-minimalist, yet simple truncation that could lead to the phase diagram is provided. Additionally, we identify the ensemble for which the result of op. cit. would be entirely correct.

scholarly journals Weak cosmic censorship with self-interacting scalar and bound on charge to mass ratio

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Yan Song ◽  
Tong-Tong Hu ◽  
Yong-Qiang Wang
Keyword(s):  
Scalar Field ◽  
Scalar Fields ◽  
Cosmic Censorship ◽  
Mass Ratio ◽  
Quartic Coupling ◽  
Scalar Theory ◽  
Free Scalar ◽  
The One ◽  
Nonzero Scalar ◽  
Large Charge

Abstract We study the model of four-dimensional Einstein-Maxwell-Λ theory minimally coupled to a massive charged self-interacting scalar field, parameterized by the quartic and hexic couplings, labelled by λ and β, respectively. In the absence of scalar field, there is a class of counterexamples to cosmic censorship. Moreover, we investigate the full nonlinear solution with nonzero scalar field included, and argue that these counterexamples can be removed by assuming charged self-interacting scalar field with sufficiently large charge not lower than a certain bound. In particular, this bound on charge required to preserve cosmic censorship is no longer precisely the weak gravity bound for the free scalar theory. For the quartic coupling, for λ < 0 the bound is below the one for the free scalar fields, whereas for λ > 0 it is above. Meanwhile, for the hexic coupling the bound is always above the one for the free scalar fields, irrespective of the sign of β.

scholarly journals Divergent ⇒ complex amplitudes in two dimensional string theory

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Ashoke Sen
Keyword(s):  
Scattering Amplitudes ◽  
String Theory ◽  
Matrix Model ◽  
Two Dimensional ◽  
Full Matrix ◽  
Instanton Contribution ◽  
The Matrix ◽  
Theory Result ◽  
The One ◽  
Remarkable Agreement

Abstract In a recent paper, Balthazar, Rodriguez and Yin found remarkable agreement between the one instanton contribution to the scattering amplitudes of two dimensional string theory and those in the matrix model to the first subleading order. The comparison was carried out numerically by analytically continuing the external energies to imaginary values, since for real energies the string theory result diverges. We use insights from string field theory to give finite expressions for the string theory amplitudes for real energies. We also show analytically that the imaginary parts of the string theory amplitudes computed this way reproduce the full matrix model results for general scattering amplitudes involving multiple closed strings.

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