Related papers: Virasoro constraints and polynomial recursion for …

We study the BKP hierarchy and its $n$--reduction, for the case that $n$ is odd. This is related to the principal realization of the basic module of the twisted affine Lie algebra $\hat{sl}_n^{(2)}$. We show that the following two…

We study the Virasoro constraints for moduli spaces of representations of quiver with relations by Joyce's vertex algebras. Using the framed Virasoro constraints, we construct a representation of half of the Virasoro algebra on the…

In this paper, we construct the multicomponent modified KP hierarchy and its additional symmetries. The additional symmetries constitute an interesting multi-folds quantum torus type Lie algebra. By a reduction, we also construct the…

We derive a formula for the connected $n$-point functions of a tau-function of the BKP hierarchy in terms of its affine coordinates. This is a BKP-analogue of a formula for KP tau-functions proved by Zhou in [arXiv:1507.01679]. Moreover, we…

The non-isospectral symmetries of a general class of integrable hierarchies are found, generalizing the Galilean and scaling symmetries of the Korteweg--de Vries equation and its hierarchy. The symmetries arise in a very natural way from…

We reformulate the $q$-difference linear system corresponding to the $q$-Painlev\'e equation of type $A_7^{(1)'}$ as a Riemann-Hilbert problem on a circle. Then, we consider the Fredholm determinant built from the jump of this…

This is the first of two articles devoted to a comprehensive exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper is entirely devoted to the study of the tensor-product…

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…

The extended bigraded Toda hierarchy (EBTH) is an integrable system satisfied by the total descendant potential of $\mathbb{CP}^1$ with two orbifold points. We construct additional symmetries of the EBTH and describe explicitly their action…

We construct Virasoro algebra of differential operators for the Jones-Rosso matrix model. These operators generate various relations between Wilson loops. Then we discuss the con- structed operators and corresponding relations in the…

We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials considered by S.Milne \cite{Milne}. For $q=1$ these functions are also known as hypergeometric functions of matrix argument…

In this paper, we discuss the properties of the generating functions of spin Hurwitz numbers. In particular, for spin Hurwitz numbers with arbitrary ramification profiles, we construct the weighed sums which are given by Orlov's…

Modular graph functions arise in the calculation of the low-energy expansion of closed-string scattering amplitudes. For toroidal world-sheets, they are ${\rm SL}(2,\mathbb{Z})$-invariant functions of the torus complex structure that have…

We derive series representations for the tau functions of the $q$-Painlev\'e V, $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations, as degenerations of the tau functions of the $q$-Painlev\'e VI equation in [Jimbo M., Nagoya…

We give an algorithm to compute weighted Ehrhart functions of lattice polytopes for polynomial weights using Lagrange interpolation. We show how to compute generating functions of polynomials using those of unit cubes and Eulerian numbers,…

In this paper I study the functional representation of the Volterra hierarchy (VH). Using the Miwa's shifts I rewrite the infinite set of Volterra equations as one functional equation. This result is used to derive a formal solution of the…

We construct the ($\beta$-deformed) higher order total derivative operators and analyze their remarkable properties. In terms of these operators, we derive the higher order constraints for the ($\beta$-deformed) Hermitian matrix models. We…

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a + b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an…

We construct quasi-periodic solutions of the universal hierarchy which includes the multi-component KP and Toda hierarchies and show how they fit into the bilinear formalism. The tau-function is expressed in terms of the Riemann…

We prove quadratic estimates for complex perturbations of Dirac-type operators, and thereby show that such operators have a bounded functional calculus. As an application we show that spectral projections of the Hodge--Dirac operator on…