Hermitian One-matrix Model Research Articles

1/N expansion of the D3-D5 defect CFT at strong coupling

We consider four dimensional U(N) mathcal{N} = 4 SYM theory interacting with a 3d mathcal{N} = 4 theory living on a codimension-one interface and holographically dual to the D3-D5 system without flux. Localization captures several observables in this dCFT, including its free energy, related to the defect expectation value, and single trace frac{1}{2} -BPS composite scalars. These quantities may be computed in a hermitian one-matrix model with non-polynomial single-trace potential. We exploit the integrable Volterra hierarchy underlying the matrix model and systematically study its 1/N expansion at any value of the ’t Hooft coupling. In particular, the strong coupling regime is determined — up to non-perturbative exponentially suppressed corrections — by differential relations that constrain higher order terms in the 1/N expansion. The analysis is extended to the model with SU(N) gauge symmetry by resorting to the more general Toda lattice equations.

Superintegrability for (beta -deformed) partition function hierarchies with W-representations

We construct the (beta -deformed) partition function hierarchies with W-representations. Based on the W-representations, we analyze the superintegrability property and derive their character expansions with respect to the Schur functions and Jack polynomials, respectively. Some well known superintegrable matrix models such as the Gaussian hermitian one-matrix model (in the external field), Ntimes N complex matrix model, beta -deformed Gaussian hermitian and rectangular complex matrix models are contained in the constructed hierarchies.

CFT approach to constraint operators for (β-deformed) hermitian one-matrix models

Since the (β-deformed) hermitian one-matrix models can be represented as the integrated conformal field theory (CFT) expectation values, we construct the operators in terms of the generators of the Heisenberg algebra such that the constraints can be derived by inserting the constructed operators into the integrated expectation values. We also obtain the second order total derivative operators associating with the derived constraint operators and analyze their properties. We explore the intrinsic connection between the derived constraint operators and W-representations of some matrix models. For the Gaussian hermitian one-matrix model in the external field and β-deformed N×N complex matrix model, we investigate the superintegrability and derive the corresponding character expansions from their W-representations. Moreover a conjectured formula for the averages of Jack polynomials in the literature is proved.

On the planar free energy of matrix models

In this work we obtain the planar free energy for the Hermitian one-matrix model with various choices of the potential. We accomplish this by applying an approach that bypasses the usual diagonalization of the matrices and the introduction of the eigenvalue density, to directly zero in the evaluation of the planar free energy. In the first part of the paper, we focus on potentials with finitely many terms. For various choices of potentials, we manage to find closed expressions for the planar free energy, and in some cases determine or bound their radius of convergence as a series in the ’t Hooft coupling. In the second part of the paper we consider specific examples of potentials with infinitely many terms, that arise in the study of mathcal{N} = 2 super Yang-Mills theories on S4, via supersymmetric localization. In particular, we manage to write the planar free energy of two non-conformal examples: SU(N) with Nf < 2N, and mathcal{N} = 2*.

Deformed Cauchy random matrix ensembles and large N phase transitions

We study a new hermitian one-matrix model containing a logarithmic Penner’s type term and another term, which can be obtained as a limit from logarithmic terms. For small coupling, the potential has an absolute minimum at the origin, but beyond a certain value of the coupling the potential develops a double well. For a higher critical value of the coupling, the system undergoes a large N third-order phase transition.

On n-algebraic structures in the Calogero model

We construct two realizations of the [Formula: see text] algebra in the classical Calogero model based on the [Formula: see text] constraints for the Hermitian one-matrix model. We also discuss the quantum Calogero model and construct the new operators which yield the closed [Formula: see text][Formula: see text]-algebra. The super [Formula: see text][Formula: see text]-algebra is derived for the supersymmetric quantum Calogero model.

On Itzykson-Zuber Ansatz

We apply the renormalized coupling constants and Virasoro constraints to derive the Itzykson-Zuber Ansatz on the form of the free energy in 2D topological gravity. We also treat the 1D topological gravity and the Hermitian one-matrix models in the same fashion. Some uniform behaviors are discovered in this approach.

W1+∞ constraints for the hermitian one-matrix model

We construct the multi-variable realizations of the W1+∞ algebra such that they lead to the W1+∞n-algebra. Based on our realizations of the W1+∞ algebra, we derive the W1+∞ constraints for the hermitian one-matrix model. The constraint operators yield not only the W1+∞ algebra but also the closed W1+∞n-algebra.

New algebraic structures from Hermitian one-matrix model

Virasoro constraint is the operator algebra version of one-loop equation for a Hermitian one-matrix model, and it plays an important role in solving the model. We construct the realization of the Virasoro constraint from the Conformal Field Theory (CFT) method. From multi-loop equations of the one-matrix model, we get a more general constraint. It can be expressed in terms of the operator algebras, which is the Virasoro subalgebra with extra parameters. In this sense, we named as generalized Virasoro constraint. We enlarge this algebra with central extension, this is a new kind of algebra, and the usual Virasoro algebra is its subalgebra. And we give a bosonic realization of its subalgebra.

Solution of string equations for asymmetric potentials

We consider the large N expansion of the partition function for the Hermitian one-matrix model. It is well known that the coefficients of this expansion are generating functions F(g) for a certain kind of graph embedded in a Riemann surface. Other authors have made a simplifying assumption that the potential V is an even function. We present a method for computing F(g) in the case that V is not an even function. Our method is based on the string equations, and yields “valence independent” formulas which do not depend explicitly on the potential. We introduce a family of differential operators, the “string polynomials”, which make clear the valence independent nature of the string equations.

The matrix model for dessins d'enfants

We present the matrix models that are the generating functions for branched covers of the complex projective line ramified over 0, 1, and \infty (Grotendieck’s dessins d’enfants) of fixed genus, degree, and the ramification profile at infinity. For general ramifications at other points, the model is the two-logarithm matrix model with the external field studied previously by one of the authors (L.Ch.) and K.Palamarchuk. It lies in the class of the generalised Kontsevich models (GKM) thus being the Kadomtsev–Petviashvili (KP) hierarchy tau function and, upon the shift of times, this model is equivalent to a Hermitian one-matrix model with a general potential whose coefficients are related to the KP times by a Miwa-type transformation. The original model therefore enjoys a topological recursion and can be solved in terms of shifted moments of the standard Hermitian one-matrix model at all genera of the topological expansion. We also derive the matrix model for clean Belyi morphisms, which turns out to be the Kontsevich–Penner model introduced by the authors and Yu. Makeenko. Its partition function is also a KP hierarchy tau function, and this model is in turn equivalent to a Hermitian one-matrix model with a general potential. Finally we prove that the generating function for general two-profile Belyi morphisms is a GKM thus proving that it is also a KP hierarchy tau function in proper times.

Painlevé Kernels in Hermitian Matrix Models

After reviewing the Hermitian one-matrix model, we will give a brief introduction to the Hermitian two-matrix model and present a summary of some recent results on the asymptotic behavior of the two-matrix model with a quartic potential. In particular, we will discuss a limiting kernel in the quartic/quadratic case that is constructed out of a 4×4 Riemann–Hilbert problem related to the Painleve II equation. Also an open problem will be presented.

Higher order analogues of Tracy–Widom distributions via the Lax method

We study the distribution of the largest eigenvalue in Hermitian one-matrix models when the spectral density acquires an extra number of k − 1 zeros at the edge. The distributions are directly expressed through the norms of orthogonal polynomials on a semi-infinite interval, as an alternative to using Fredholm determinants. They satisfy nonlinear recurrence relations which we show form a Lax pair, making contact to the string literature in the early 1990s. The technique of pseudo-differential operators allows us to give compact expressions for the logarithm of the gap probability in terms of the Painlevé XXXIV hierarchy. These are the higher order analogues of the Tracy–Widom distribution. Using known Bäcklund transformations we show how to simplify earlier equivalent results that are derived from Fredholm determinant theory, valid for even k in terms of the Painlevé II hierarchy.

Monte Carlo approach to nonperturbative strings—demonstration in noncritical string theory

We show how Monte Carlo approach can be used to study the double scaling limit in matrix models. As an example, we study a solvable hermitian one-matrix model with the double-well potential, which has been identified recently as a dual description of noncritical string theory with worldsheet supersymmetry. This identification utilizes the nonperturbatively stable vacuum unlike its bosonic counterparts, and therefore it provides a complete constructive formulation of string theory. Our data with the matrix size ranging from 8 to 512 show a clear scaling behavior, which enables us to extract the double scaling limit accurately. The ``specific heat'' obtained in this way agrees nicely with the known result obtained by solving the Painleve-II equation with appropriate boundary conditions.

Matrix models with hard walls: geometry and solutions

We discuss various aspects of most general multisupport solutions to matrix models in the presence of hard walls, i.e., in the case where the eigenvalue support is confined to subdomains of the real axis. The structure of the solution at the leading order is described by semiclassical, or generalized Whitham--Krichever hierarchies as in the unrestricted case. Derivatives of tau-functions for these solutions are associated with families of Riemann surfaces (with possible double points) and satisfy the Witten--Dijkgraaf--Verlinde--Verlinde equations. We then develop the diagrammatic technique for finding free energy of this model in all orders of the 't~Hooft expansion in the reciprocal matrix size generalizing the Feynman diagrammatic technique for the Hermitian one-matrix model due to Eynard.

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

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).

Hermitian matrix model free energy: Feynman graph technique for all genera

We present a diagrammatic technique for calculating the free energy of the Hermitian one-matrix model to all orders of 1/N expansion in the case where the limiting eigenvalue distribution spans arbitrary (but fixed) number of disjoint intervals (curves).

1/N2 Correction to Free Energy in Hermitian Two-Matrix Model

Using the loop equations we find an explicit expression for genus 1 correction in hermitian two-matrix model in terms of holomorphic objects associated to spectral curve arising in large N limit. Our result generalises known expression for $F^1$ in hermitian one-matrix model. We discuss the relationship between $F^1$, Bergmann tau-function on Hurwitz spaces, G-function of Frobenius manifolds and determinant of Laplacian over spectral curve.

Partition functions of matrix models as the first special functions of string theory: Finite Hermitian one-matrix model

Although matrix model partition functions do not exhaust the entire set of τ-functions relevant for string theory, they are elementary blocks for constructing many other τ-functions and seem to capture the fundamental nature of quantum gravity an string theory properly. We propose taking matrix model partition functions as new special functions. This means that they should be investigated and represented in some standard form without reference to particular applications. At the same time, the tables and lists of properties should be sufficiently full to exclude unexpected peculiarities appearing in new applications. Accomplishing this task requires considerable effort, and this paper is only a first step in this direction. We restrict our consideration to the finite Hermitian one-matrix model an concentrate mostly on its phase and branch structure that arises when the partition function is considered as a D-module. We discuss the role of the CIV-DV prepotential (which generates a certain basis in the linear space of solutions of the Virasoro constraints, although an understanding of why and how this basis is distinguished is lacking) an evaluate several first multiloop correlators, which generalize the semicircular distribution to the case of multitrace and nonplanar correlators.

Genus-One Correction to Multicut Matrix Model Solutions

We calculate genus one corrections to Hermitian one-matrix model solution with arbitrary number of cuts directly from the loop equation confirming the answer previously obtained from algebro-geometrical considerations and generalizing it to the case of arbitrary potentials.