Related papers: Integrable structures of specialized hypergeometri…

We propose a new bilinear Hirota equation for $\tau$-functions associated with the $E_8$ root lattice, that provides a "lens" generalisation of the $\tau$-functions for the elliptic discrete Painlev\'e equation. Our equations are…

Parametric families in the centre ${\bf Z}({\bf C}[S_n])$ of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements.…

We show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2D Toda integrable hierarchy. The maps are determined by a particular solution to the hierarchy…

Multivariate orthogonal polynomials in $D$ real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials,…

The modified Toda (mToda) hierarchy is a two-component generalization of the 1-st modified KP (mKP) hierarchy, which connects the Toda hierarchy via Miura links and has two tau functions. Based on the fact that the mToda and 1-st mKP…

In this paper, the authors investigate the case of discrete multiple orthogonal polynomials with two weights on the step line, which satisfy Pearson equations. The discrete multiple orthogonal polynomials in question are expressed in terms…

In this work, we employ the Tannaka-Krein reconstruction to compute the quantum double $\mathcal D(\mathcal G)$ of a finite 2-group $\mathcal G$ as a Hopf monoidal category. We also construct a 3+1D lattice model from the Dijkgraaf-Witten…

The Brezin-Gross-Witten tau function is a tau function of the KdV hierarchy which arises in the weak coupling phase of the Brezin-Gross-Witten model. It falls within the family of generalized Kontsevich matrix integrals, and its…

We define two types of Witten's zeta functions according to Cartan's classification of compact symmetric spaces. The type II is the original Witten zeta function constructed by means of irreducible representations of the simple compact Lie…

We study the Pfaff lattice, introduced by us in the context of a Lie algebra splitting of gl(infinity) into sp(infinity) and lower-triangular matrices. We establish a set of bilinear identities, which we show to be equivalent to the Pfaff…

We present a family of matrix models such that their partition functions are tau functions of the universal character (UC) hierarchy. This develops one of the topics of our previous paper arXiv:2410.14823. We found new matrix models…

We extend the notion of pseudo-differential operators that are used to represent the Gelfand-Dickey hierarchies, and obtain a similar representation for the full Drinfeld-Sokolov hierarchies of $D_n$ type. By using such pseudo-differential…

We consider the topological string partition function, including the Nekrasov deformation, for type IIB geometries with an A_{n-1} singularity over a Riemann surface. These models realize the N=2 SU(n) superconformal gauge systems recently…

We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one can reduce the matrix integral to the…

We give a self-contained introduction to the relations between Integrable Systems and the Geometry of Riemann Surfaces. We start from a historical introduction to the topic of integrable systems. Afterwards, we study the polynomial…

We investigate the special K\"ahler geometry of the base of the Hitchin integrable system in terms of spectral curves and topological recursion. The Taylor expansion of the special K\"ahler metric about any point in the base may be computed…

The reduction by restricting the spectral parameters $k$ and $k'$ on a generic algebraic curve of degree $\mathcal{N}$ is performed for the discrete AKP, BKP and CKP equations, respectively. A variety of two-dimensional discrete integrable…

The exact solution of $N=2$ supersymmetric $SU(N)$ Yang-Mills theory is studied in the framework of the Whitham hierarchies. The solution is identified with a homogeneous solution of a Whitham hierarchy. This integrable hierarchy…

Bloch and Okounkov introduced an n-point correlation function on the infinite wedge space and found an elegant closed formula in terms of theta functions. This function has connections to Gromov-Witten theory, Hilbert schemes, symmetric…

We show that any polynomial tau-function of the s-component KP and the BKP hierarchies can be interpreted as a zero mode of an appropriate combinatorial generating function. As an application, we obtain explicit formulas for all polynomial…