Related papers: Hermitian One-Matrix Model and KP Hierarchy
In this paper we establish relations between three enumerative geometry tau-functions, namely the Kontsevich-Witten, Hurwitz and Hodge tau-functions. The relations allow us to describe the tau-functions in terms of matrix integrals,…
We study \tau-functions of the KP hierarchy in terms of abelian group actions on finite dimensional Grassmannians, viewed as subquotients of the Hilbert space Grassmannians of Sato, Segal and Wilson. A determinantal formula of Gekhtman and…
In this work we study the tau-function $Z^{1D}$ of the KP hierarchy specified by the topological 1D gravity. As an application, we present two types of algorithms to compute the orbifold Euler characteristics of $\overline{\mathcal…
An element [\Phi] of the Grassmannian of n-dimensional subspaces of the Hardy space H^2, extended over the field C(x_1,..., x_n), may be associated to any polynomial basis {\phi} for C(x). The Pl\"ucker coordinates…
We study $\hbar$ expansion of the KP hierarchy following Takasaki-Takebe arXiv:hep-th/9405096 considering several examples of matrix model $\tau$-functions with natural genus expansion. Among the examples there are solutions of KP equations…
In arXiv:hep-th/0310113 we started a program of creating a reference-book on matrix-model tau-functions, the new generation of special functions, which are going to play an important role in string theory calculations. The main focus of…
We consider an operator of Bernstein for symmetric functions, and give an explicit formula for its action on an arbitrary Schur function. This formula is given in a remarkably simple form when written in terms of some notation based on the…
We study the hermitean and normal two matrix models in planar approximation for an arbitrary number of eigenvalue supports. Its planar graph interpretation is given. The study reveals a general structure of the underlying analytic complex…
Matrix elements in different representations are connected by quadratic relations. If matrix elements are those of a $\textit{group element}$, i.e. satisfying the property $\Delta(X) = X\otimes X$, then their generating functions obey…
We show that partition functions of various matrix models can be obtained by acting on elementary functions with exponents of W-operators. A number of illustrations is given, including the Gaussian Hermitian matrix model, Hermitian model in…
Generalized convolution symmetries of integrable hierarchies of KP and 2KP-Toda type multiply the Fourier coefficients of the elements of the Hilbert space $\HH= L^2(S^1)$ by a specified sequence of constants. This induces a corresponding…
It is shown that all $\tau$-functions of BKP hierarchy can be written as Pfaffians of skew-symmetric matrices. $\tau$-functions of BKP hierarchy are parameterized by points in the universal orthogonal Grassmannian manifold (UOGM). The UOGM…
For an arbitrary solution to the AKNS hierarchy, the logarithmic derivatives of the tau-function of the solution can be computed by the matrix-resolvent method [14,21]. In this paper, we introduce a pair of wave functions of the solution…
We consider the Itzykson-Zuber-Eynard-Mehta two-matrix model and prove that the partition function is an isomonodromic tau function in a sense that generalizes Jimbo-Miwa-Ueno's. In order to achieve the generalization we need to define a…
The degeneration of the hyperelliptic sigma function is studied. We use the Sato Grassmannian for this purpose. A simple decomposition of a rational function gives a decomposition of Pl\"ucker coordinates of a frame of the Sato…
We use a Grassmannian framework to define multi-component tau functions as expectation values of certain multi-component Fermi operators satisfying simple bilinear commutation relations on Clifford algebra. The tau functions contain both…
Several features of an analytic (infinite-dimensional) Grassmannian of (commensurable) subspaces of a Hilbert space were developed in the context of integrable PDEs (KP hierarchy). We extended some of those features when polarized separable…
We rewrite the loop equations of the hermitian matrix model, in a way which allows to compute all the correlation functions, to all orders in the topological $1/N^2$ expansion, as residues on an hyperelliptical curve. Those residues, can be…
For the eigenvalues of $p$ complex hermitian $n\times n$ matrices coupled in a chain, we give a method of calculating the spacing functions. This is a generalization of the one matrix case which has been known for a long time.
We summarize the recent results about complete solvability of Hermitian and rectangular complex matrix models. Partition functions have very simple character expansions with coefficients made from dimensions of representation of the linear…