THE CUBIC MATRIX MODEL WITH IMAGINARY COUPLING AND 2D GRAVITY

1992 ◽  
Vol 07 (03) ◽  
pp. 501-534
Author(s):  
MAHA SAADI ◽  
GUILLERMO ZEMBA
Keyword(s):  
Analytic Continuation ◽  
Matrix Model ◽  
Continuum Limit ◽  
2D Gravity ◽  
Positive Definite ◽  
Model Formulation ◽  
Loop Equation ◽  
Pure Gravity ◽  
The One ◽  
The Continuum

We study the continuum properties of the Hermitian one-matrix model defined by a cubic potential with imaginary coupling, using the orthogonal polynomials and loop-equation approaches. The model is well defined mathematically and has a continuum limit which cannot be naively interpreted as pure gravity since each term of the sum over surfaces is not positive-definite. Nevertheless, the model may be considered as an analytic continuation of the standard matrix-model formulation of gravity. We study in detail the conditions under which the analytic continuation may be performed. In particular, the specific heat is found to obey a Painlevé equation. Although we find a solution that is compatible, as far as global stability is concerned, with the one proposed by David, it is not completely clear that the two solutions are the same.

scholarly journals Effective Action and Measure in Matrix Model of IIB Superstrings

1997 ◽  
Vol 12 (31) ◽  
pp. 2331-2340 ◽  
Author(s):  
L. Chekhov ◽  
K. Zarembo
Keyword(s):  
Matrix Model ◽  
Effective Action ◽  
Continuum Limit ◽  
Auxiliary Field ◽  
Large N ◽  
The Matrix ◽  
Reparametrization Invariance ◽  
The Continuum ◽  
Induced Measure

We calculate an effective action and measure induced by the integration over the auxiliary field in the matrix model recently proposed to describe IIB superstrings. It is shown that the measure of integration over the auxiliary matrix is uniquely determined by locality and reparametrization invariance of the resulting effective action. The large-N limit of the induced measure for string coordinates is discussed in detail. It is found to be ultralocal and, thus, is possibly irrelevant in the continuum limit. The model of the GKM type is considered in relation to the effective action problem.

EXCITATIONS AND INTERACTIONS IN d=1 STRING THEORY

1991 ◽  
Vol 06 (11) ◽  
pp. 1961-1984 ◽  
Author(s):  
ANIRVAN M. SENGUPTA ◽  
SPENTA R. WADIA
Keyword(s):  
Matrix Model ◽  
Periodic Wave ◽  
Density Fluctuations ◽  
Dirac Fermion ◽  
Unitary Matrix ◽  
Additional Parameter ◽  
Continuum Approach ◽  
The Matrix ◽  
The One ◽  
The Continuum

We discuss the singlet sector of the d=1 matrix model in terms of a Dirac fermion formalism. The leading order two- and three-point functions of the density fluctuations are obtained by this method. This allows us to construct the effective action to that order and hence provide the equation of motion. This equation is compared with the one obtained from the continuum approach. We also compare continuum results for correlation functions with the matrix model ones and discuss the nature of gravitational dressing for this regularization. Finally, we address the question of boundary conditions within the framework of the d=1 unitary matrix model, considered as a regularized version of the Hermitian model, and study the implications of a generalized action with an additional parameter (analogous to the θ parameter) which give rise to quasi-periodic wave functions.

scholarly journals INFINITE-TENSION STRINGS AT d>1

1993 ◽  
Vol 08 (06) ◽  
pp. 557-572 ◽  
Author(s):  
D.V. BOULATOV
Keyword(s):  
Matrix Model ◽  
Continuum Limit ◽  
Mean Field ◽  
Bethe Lattice ◽  
Point Of View ◽  
Simple Application ◽  
Field Equations ◽  
Mean Field Equations ◽  
The Mean ◽  
The One

A matrix model describing surfaces embedded in a Bethe lattice is considered. From the mean field point of view, it is equivalent to the Kazakov-Migdal induced gauge theory and therefore, at N=∞ and d>1, the latter can be interpreted as a matrix model for infinite-tension strings. We show that, in the naive continuum limit, it is governed by the one-matrix model saddle point with an upside-down potential. To derive mean field equations, we consider the one-matrix model in external field. As a simple application, its explicit solution in the case of the inverted W potential is given.

scholarly journals THREE-POINT FUNCTIONS OF NON-UNITARY MINIMAL MATTER COUPLED TO GRAVITY

1993 ◽  
Vol 08 (03) ◽  
pp. 197-207 ◽  
Author(s):  
DEBASHIS GHOSHAL ◽  
SWAPNA MAHAPATRA
Keyword(s):  
Matrix Model ◽  
Correlation Functions ◽  
The Other ◽  
Tree Level ◽  
Minimal Models ◽  
Continuum Approach ◽  
Point Correlation ◽  
Local Operators ◽  
The One ◽  
The Continuum

The tree-level three-point correlation functions of local operators in the general (p, q) minimal models coupled to gravity are calculated in the continuum approach. On one hand, the result agrees with the unitary series (q=p+1); and on the other hand, for p=2, q=2k−1, we find agreement with the one-matrix model results.

ON THE DIMERIZATION OF LINEAR POLYMERS

1989 ◽  
Vol 03 (02) ◽  
pp. 125-133 ◽  
Author(s):  
C. ARAGÃO DE CARVALHO
Keyword(s):  
Free Energy ◽  
Effective Potential ◽  
Ground State ◽  
Finite Temperature ◽  
Continuum Limit ◽  
Linear Polymers ◽  
Renormalization Condition ◽  
Loop Approximation ◽  
The One ◽  
The Continuum

We use the continuum limit of the Su-Schrieffer-Heeger model for linear polymers to construct its effective potential (Gibbs free energy) both at zero and finite temperature. We study both trans and cis-polymers. Our results show that, depending on a renormalization condition to be extracted from experiment, there are several possibilities for the minima of the dimerized ground state of cis-polymers. All calculations are done in the one-loop approximation.

scholarly journals CORRELATION FUNCTIONS OF (2k−1, 2) MINIMAL MATTER COUPLED TO 2D QUANTUM GRAVITY

1993 ◽  
Vol 08 (04) ◽  
pp. 327-334 ◽  
Author(s):  
SHUN-ICHI YAMAGUCHI
Keyword(s):  
Quantum Gravity ◽  
Critical Point ◽  
Matrix Model ◽  
Correlation Functions ◽  
Conformal Weight ◽  
Primary Field ◽  
Field Approach ◽  
Point Correlation ◽  
The One ◽  
The Continuum

We compute N-point correlation functions of non-unitary (2k−1, 2) minimal matter coupled to 2D quantum gravity on a sphere using the continuum Liouville field approach. A gravitational dressing of the matter primary field with the minimum conformal weight is used as the cosmological operator. Our results are in agreement with the correlation functions of the one-matrix model at the kth critical point.

scholarly journals LOOP EQUATIONS AND KDV HIERARCHY IN 2-D QUANTUM GRAVITY

1992 ◽  
Vol 07 (10) ◽  
pp. 2285-2293
Author(s):  
F. FUCITO ◽  
M. MARTELLINI
Keyword(s):  
Quantum Gravity ◽  
Matrix Model ◽  
Virasoro Algebra ◽  
Linear Constraints ◽  
String Equation ◽  
Loop Equation ◽  
Infinite Dimensional ◽  
Kdv Hierarchy ◽  
The One ◽  
Operator Construction

A derivation of the loop equation for two-dimensional quantum gravity from the KdV equations and the string equation of the one-matrix model has been recently given. The loop equation was found to be equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the Virasoro algebra. Starting from the equations expressing these constraints, we are able to rederive the equations of the KdV hierarchy using the vertex operator construction of the [Formula: see text] infinite dimensional twisted Kac-Moody algebra. From these considerations it follows that the solutions of the string equation of the one-matrix model are given by a subset of the solutions of the KdV hierarchy.

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.

scholarly journals Symmetry analysis of and first integrals for the continuum Heisenberg spin chain

2002 ◽  
Vol 44 (1) ◽  
pp. 61-72
Author(s):  
M. C. Nucci ◽  
P. G. L. Leach
Keyword(s):  
Differential Equations ◽  
Spin Chain ◽  
Continuum Limit ◽  
Transverse Field ◽  
First Integrals ◽  
Heisenberg Spin ◽  
One Dimensional ◽  
Heisenberg Spin Chain ◽  
The One ◽  
The Continuum

AbstractDaniel et al. [6] analysed the singularity structure of the continuum limit of the one-dimensional anisotropic Heisenberg spin chain in a transverse field and determined the conditions under which the system is nonintegrable and exhibits chaos. We investigate the governing differential equations for symmetries and find the associated first integrals. Our results complement the results of Daniel et al.

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