M-Theory of Matrix Models

Small M-theories unify various models of a given family in the same way as the M-theory unifies a variety of superstring models. We consider this idea in application to the family of eigenvalue matrix models: their M-theory unifies various branches of Hermitean matrix model (including Dijkgraaf-Vafa partition functions) with Kontsevich tau-function. Moreover, the corresponding duality relations look like direct analogues of instanton and meron decompositions, familiar from Yang-Mills theory.Comment: 12 pages, contribution to the Proceedings of the Workshop "Classical and Quantum Integrable Systems", Protvino, Russia, January, 200

Fermionic construction of partition function for multi-matrix models and multi-component TL hierarchy

We use $p$-component fermions $(p=2,3,...)$ to present $(2p-2)N$-fold integrals as a fermionic expectation value. This yields fermionic representation for various $(2p-2)$-matrix models. Links with the $p$-component KP hierarchy and also with the $p$-component TL hierarchy are discussed. We show that the set of all (but two) flows of $p$-component TL changes standard matrix models to new ones.Comment: 16 pages, submitted to a special issue of Theoretical and Mathematical Physic

Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model

Even though matrix model partition functions do not exhaust the entire set of tau-functions relevant for string theory, they seem to be elementary building blocks for many others and they seem to properly capture the fundamental symplicial nature of quantum gravity and string theory. We propose to consider matrix model partition functions as new special functions. This means they should be investigated and put into some standard form, with no reference to particular applications. At the same time, the tables and lists of properties should be full enough to avoid discoveries of unexpected peculiarities in new applications. This is a big job, and the present paper is just a step in this direction. Here we restrict our consideration to the finite-size Hermitean 1-matrix model and concentrate mostly on its phase/branch structure arising when the partition function is considered as a D-module. We discuss the role of the CIV-DV prepotential (as generating a possible basis in the linear space of solutions to the Virasoro constraints, but with a lack of understanding of why and how this basis is distinguished) and evaluate first few multiloop correlators, which generalize semicircular distribution to the case of multitrace and non-planar correlators.Comment: 64 pages, LaTe

Fermionic approach to the evaluation of integrals of rational symmetric functions

We use the fermionic construction of two-matrix model partition functions to evaluate integrals over rational symmetric functions. This approach is complementary to the one used in the paper ``Integrals of Rational Symmetric Functions, Two-Matrix Models and Biorthogonal Polynomials'' \cite{paper2}, where these integrals were evaluated by a direct method.Comment: 34 page

Effect of Various Infusion Solutions on Microrheology

Objective: to evaluate the in vitro and in vivo effects of various infusion solutions on red blood cell rheology in the early posttraumatic period. Material and methods. The in vitro study assessed crystalloids, albumin, dextrans, modified gelatin, and different generations of hydroxyethyl starches (HES). The preparations were added to blood in a 1:10 dilution; before and after their addition, the values of erythrocyte aggregation and erythrocyte deformability were estimated. The in vivo study covered 59 patients with severe concomitant injury, who were divided into 3 groups: 1) those who received crystalloids only; 2) those who had crystalloids + 6% HES 130/0.42; 3) those who had crystalloids + gelofusine. The same parameters of red blood cell rheology were estimated as in the in vitro study. Results. Albumin, repolyglycan, and HES 130/0.42 were found to have the most pronounced disaggregatory effect in vitro. At the same time, polyglycan, gelofusine, and HES 450/0.7 in particular, enhanced erythrocyte aggregation. In vitro, albumin, HES 130/0.42, and HES 200/0.5 exerted the most beneficial effect on erythrocyte deformability whereas dextrans made the latter worse and HES 450/0.7 and gelofusine failed to have a considerable effect on it. The early posttraumatic period was marked by progressive erythrocyte hyperaggregation and phasic deformability changes. Significant microrheological disorders persisted in the patients on infusion therapy with crystalloid solutions only. Addition of HES 130/0.42 to infusion therapy improved the deformability of erythrocytes and lowered their aggregation. The use of gelofusine as a component of infusion therapy caused a moderate increase in erythrocyte aggregation. Key words: infusion therapy, erythrocyte deformability, erythrocyte aggregation

Challenges of Matrix Models

Brief review of concepts and unsolved problems in the theory of matrix models.Comment: Contribution to Proceedings of Cargese 200

Classical A_n--W-Geometry

This is a detailed development for the $A_n$ case, of our previous article entitled "W-Geometries" to be published in Phys. Lett. It is shown that the $A_n$--W-geometry corresponds to chiral surfaces in $CP^n$. This is comes out by discussing 1) the extrinsic geometries of chiral surfaces (Frenet-Serret and Gauss-Codazzi equations) 2) the KP coordinates (W-parametrizations) of the target-manifold, and their fermionic (tau-function) description, 3) the intrinsic geometries of the associated chiral surfaces in the Grassmannians, and the associated higher instanton- numbers of W-surfaces. For regular points, the Frenet-Serret equations for $CP^n$--W-surfaces are shown to give the geometrical meaning of the $A_n$-Toda Lax pair, and of the conformally-reduced WZNW models, and Drinfeld-Sokolov equations. KP coordinates are used to show that W-transformations may be extended as particular diffeomorphisms of the target-space. This leads to higher-dimensional generalizations of the WZNW and DS equations. These are related with the Zakharov- Shabat equations. For singular points, global Pl\"ucker formulae are derived by combining the $A_n$-Toda equations with the Gauss-Bonnet theorem written for each of the associated surfaces.Comment: (60 pages