Related papers: De Rham - Witt KZ equations
We present a new expansion of the zeta-function of Riemann. The current formalism -- which combines both the idea of interpolation with constraints and the concept of hypergeometric functions -- can, in a natural way, be generalised within…
Given a Lax system of equations with the spectral parameter on a Riemann surface we construct a projective unitary representation of the Lie algebra of Hamiltonian vector fields by Knizhnik-Zamolodchikov operators. This provides a…
Some aspects of Aomoto's generalized hypergeometric functions on Grassmannian spaces $Gr(k+1,n+1)$ are reviewed. Particularly, their integral representations in terms of twisted homology and cohomology are clarified with an example of the…
We define a system of "dynamical" differential equations compatible with the KZ differential equations. The KZ differential equations are associated to a complex simple Lie algebra $\mathbf{g}$. These are equations on a function of $n$…
We consider the Knizhnik-Zamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of the (gl_k,gl_n) duality. We show that the KZ and dynamical equations naturally exchange under the duality.
We investigate the Knizhnik-Zamolodchikov linear differential system. The coefficients of this system are rational functions. We have proved that the solution of the KZ system is rational when k is equal to two and n is equal to three (see…
By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.
It is shown that from some solutions of generalized Knizhnik-Zamolodchikov equations one can construct eigenfunctions of the Calogero-Sutherland-Moser Hamiltonians with exchange terms, which are characterized by any given permutational…
We construct polynomial solutions of the KZ differential equations over a finite field $F_p$ as analogs of hypergeometric solutions.
We investigate the GKZ $A$-hypergeometric $\mathscr{D}$-modules, introduced by Gel'fand, Kapranov, and Zelevinskii, arising from cyclic covers of toric varieties and find its Riemann--Hilbert partner. This extends our earlier results in…
We consider a rational-trigonometric deformation in context of rational and trigonometric deformations. The simplest examples of these deformations are presented in different fields of mathematics. Rational-trigonometric differential…
Beginning with a relativistic action principle for the irrotational flow of collisionless matter, we compute higher order corrections to the Zel'dovich approximation by deriving a nonlinear Hamilton-Jacobi equation for the velocity…
In this paper we first note a result of birational automorphisms with bounded degree of projective varieties related with the Zariski dense orbit conjecture (ZDO) and the Zariski density of periodic points. Next, we give a reduced result of…
We give a $K$-theoretic account of the basic properties of Witt vectors. Along the way we re-prove basic properties of the little-known Witt vector norm, give a characterization of Witt vectors in terms of algebraic $K$-theory, and a…
In the spirit of the quantum Hamiltonian reduction we establish a relation between the chiral $n$-point functions, as well as the equations governing them, of the $A_1^{(1)}$ WZNW conformal theory and the corresponding Virasoro minimal…
We describe the solutions to a family of rotationally symmetric second order partial differential equations in the complex plane that arises from a four-dimensional complex Lie algebra whose spanning set generates the algebra from which…
The quantized version of a discrete Knizhnik-Zamolodchikov system is solved by an extension of the generalized Bethe Ansatz. The solutions are constructed to be of highest weight which means they fully reflect the internal quantum group…
An elliptic analogue of the $q$ deformed Knizhnik-Zamolodchikov equations is introduced. A solution is given in the form of a Jackson-type integral of Bethe vectors of the XYZ-type spin chains.
We introduce the Z-polynomial of a matroid, which we define in terms of the Kazhdan-Lusztig polynomial. We then exploit a symmetry of the Z-polynomial to derive a new recursion for Kazhdan-Lusztig coefficients. We solve this recursion,…
The first part of this paper is a survey on algebro-geometric aspects of sheaves of logarithmic vector fields of hyperplane arrangements. In the second part we prove that the relative de Rham cohomology (of degree two) of ADE-type adjoint…