A. Varchenko

Identities between q-hypergeometric and hypergeometric integrals of different dimensions

V. Tarasov, A. Varchenko

Given complex numbers $m_{1}, l_{1}$ and nonnegative integers $m_{2}, l_{2}$ , such that $m_{1} + m_{2} = l_{1} + l_{2}$ , for any $a, b = 0, ..., min (m_{2}, l_{2})$ we define an $l_{2}$ -dimensional Barnes type q-hy…

Bispectral and (gl_N, gl_M) Dualities, Discrete Versus Differential

E. Mukhin, V. Tarasov, A. Varchenko

Let $V =< x^{I^»_{i}} p_{ij} (x), i = 1, ..., n, j = 1, ..., N_{i} >$ be a space of quasi-polynomials in $x$ of dimension $N = N_{1} + ... + N_{n}$ . The regularized fundamental differential operator of…

A Selberg Integral Type Formula for an sl_2 One-Dimensional Space of Conformal Blocks

A. Varchenko

For distinct complex numbers $z_{1}, ..., z_{2 N}$ , we give a polynomial $P (y_{1}, ..., y_{2 N})$ in the variables $y_{1}, ..., y_{2 N}$ , which is homogeneous of degree $N$ , linear with respect t…

math.AG2006

The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz

E. Mukhin, V. Tarasov, A. Varchenko

We prove the B. and M. Shapiro conjecture that says that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis…

hep-th1994

Solutions to the Quantized Knizhnik-Zamolodchikov Equation and the Bethe Ansatz

V. Tarasov, A. Varchenko

We give an integral representation for solutions to the quantized Knizhnik- Zamolodchikov equation (qKZ) associated with the Lie algebra $g l_{N + 1}$ . Asymptotic solutions to qKZ are…

math-ph2011

Conformal blocks and equivariant cohomology

R. RimÃ¡nyi, V. Schechtman, A. Varchenko

We show that the conformal blocks constructed in the previous article by the first and the third author may be described as certain integrals in equivariant cohomology. When the bu…

math-ph2020

Dynamical $sl_{2}$ Bethe algebra and functions on pairs of quasi-polynomials

A. Slinkin, D. Thompson, A. Varchenko

We consider the space $Fun_{sl_{2}} V [0]$ of functions on the Cartan subalgebra of $sl_{2}$ with values in the zero weight subspace $V [0]$ of a tensor product of ir…

math.AG2011

Intersection cohomology of a rank one local system on the complement of a hyperplane-like divisor

D. Arinkin, A. Varchenko

Under a certain condition A we give a construction to calculate the intersection cohomology of a rank one local system on the complement to a hyperplane-like divisor

math.AG2018

Elliptic and K-theoretic stable envelopes and Newton polytopes

R. RimÃ¡nyi, V. Tarasov, A. Varchenko

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $K$ -theoretic stable envelopes for them and also define a version of the elliptic stable…

math.AG2011

Cohomology of the complement to an elliptic arrangement

A. Levin, A. Varchenko

We consider the complement to an arrangement of hyperplanes in a cartesian power of an elliptic curve and describe its cohomology with coefficients in a nontrivial rank one local s…

Bethe algebra of the gl_{N+1} Gaudin model and algebra of functions on the critical set of the master function

E. Mukhin, V. Tarasov, A. Varchenko

Consider a tensor product of finite-dimensional irreducible gl_{N+1}-modules and its decomposition into irreducible modules. The gl_{N+1} Gaudin model assigns to each multiplicity…

Populations of solutions of the XXX Bethe equations associated to Kac-Moody algebras

E. Mukhin, A. Varchenko

We consider the XXX Bethe equation associated with integral dominant weights of a Kac-Moody algebra and introduce a generating procedure constructing new solutions starting from a…

alg-geom1997

The Determinant of a Hypergeometric Period Matrix

Y. Markov, V. Tarasov, A. Varchenko

We consider a function $U = e^{- f_{0}} \prod_{j}^{N} f_{j}^{I^\pm_{j}}$ on a real affine space, here $f_{0}, .., f_{N}$ are linear functions, $I^\pm_{1}, ..., I^\pm_{N}$ complex numbers. The zeros of the functions…

math-ph2023

Hypergeometric integrals, hook formulas and Whittaker vectors

G. Felder, A. Smirnov, V. Tarasov +1

We determine the coefficient of proportionality between two multidimensional hypergeometric integrals. One of them is a solution of the dynamical difference equations associated wi…

Quiver $D$ -Modules and Homology of Local Systems over an Arrangement of Hyperplanes

S. Khoroshkin, A. Varchenko

Let $C$ be an arrangement of affine hyperplanes in a complex affine space $X$ , $D$ the ring of algebraic differential operators on $X$ . We define a category of quivers associated w…

math.AG2015

Equivariant Chern-Schwartz-MacPherson classes in partial flag varieties: interpolation and formulae

R. Rimanyi, A. Varchenko

Consider the natural torus action on a partial flag manifold $F l$ . Let $I^c ◯_{I} \subset F l$ be an open Schubert variety, and let $c^{s m} (I^c ◯_{I}) \in H_{T}^{*} (F l)$ be its torus equivariant Ch…

math.QA2001

Solutions of Trigonometric KZ Equations satisfy Dynamical Difference Equations

Y. Markov, A. Varchenko

The trigonometric KZ equations associated to a Lie algebra \g depend on a parameter Î»in \h where \h is a Cartan subalgebra of \g. A system of dynamical difference equations with r…

math.CO2011

Path count asymptotics and Stirling numbers

K. Petersen, A. Varchenko

We obtain formulas for the growth rate of the numbers of certain paths in infinite graphs built on the two-dimensional Eulerian graph. Corollaries are identities relating Stirling…

Gaudin Hamiltonians generate the Bethe algebra of a tensor power of vector representation of gl_N

E. Mukhin, V. Tarasov, A. Varchenko

We show that the Gaudin Hamiltonians H_1,...,H_n generate the Bethe algebra of the n-fold tensor power of the vector representation of gl_N. Surprisingly the formula for the genera…

Critical points of master functions and flag varieties

E. Mukhin, A. Varchenko

We consider critical points of master functions associated with integral dominant weights of Kac-Moody algebras and introduce a generating procedure constructing new critical point…

On reality property of Wronski maps

E. Mukhin, V. Tarasov, A. Varchenko

We prove that if all roots of the discrete Wronskian with step 1 of a set of quasi-exponentials with real bases are real, simple and differ by at least 1, then the complex span of…

Solutions of the qKZB Equation in Tensor Products of finite dimensional modules over the elliptic quantum group $E_{I ¨ , I^\cdot} s l_{2}$

E. Mukhin, A. Varchenko

We consider the quantized Knizhnik-Zamolodchikov-Bernard difference equation (qKZB) with step $p$ and values in a tensor product of finite dimensional evaluation modules over the e…

BGG resolutions via configuration spaces

M. Falk, V. Schechtman, A. Varchenko

We study the blow-ups of configuration spaces. These spaces have a structure of what we call an Orlik-Solomon manifold; it allows us to compute the intersection cohomology of certa…

On separation of variables and completeness of the Bethe ansatz for quantum gl_N Gaudin model

E. Mukhin, V. Tarasov, A. Varchenko

In this note, we discuss implications of the results obtained in [MTV4]. It was shown there that eigenvectors of the Bethe algebra of the quantum gl_N Gaudin model are in a one-to-…

math.QA2016

On the Gaudin model associated to Lie algebras of classical types

Kang Lu, E. Mukhin, A. Varchenko

We derive explicit formulas for solutions of the Bethe Ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module…

hep-th1994

Asymptotic Solutions to the Knizhnik-Zamolodchikov Equation and Crystal Base

A. Varchenko

The Knizhnik-Zamolodchikov equation associated with $s ℓ_{2}$ is considered. The transition functions between asymptotic solutions to the Knizhnik-Zamolodchikov equation are descri…

Differential equation for Jacobi-Pineiro polynomials

E. Mukhin, A. Varchenko

For $r \in Z_{\geq 0}$ , we present a linear differential operator %$(\di)^{r+1}+ a_1(x)(\di)^{r}+...+a_{r+1}(x)$ of order $r + 1$ with rational coefficients and depending on paramete…

The gl_2 Bethe algebra associated with a nilpotent element

E. Mukhin, V. Tarasov, A. Varchenko

To any 2x2-matrix K one assigns a commutative subalgebra B^{K}\subset U(gl_2[t]) called a Bethe algebra. We describe relations between the Bethe algebras, associated with the zero…

math.AG2016

Dynamical Gelfand-Zetlin algebra and equivariant cohomology of Grassmannians

R. Rimanyi, A. Varchenko

We consider the rational dynamical quantum group $E_{y} (g l_{2})$ and introduce an $E_{y} (g l_{2})$ -module structure on $\oplus_{k=0}^n H^*_{GL_n\times\C^\times}(T^*Gr(k,n))'$, where $H^*_{G…

The Quantized Knizhnik-Zamolodchikov Equation in Tensor Products of Irreducible sl(2)-Modules

E. Mukhin, A. Varchenko

We consider the quantized Knizhnik-Zamolodchikov difference equation (qKZ) with values in a tensor product of irreducible sl(2) modules, the equation defined in terms of rational R…

Identities for hypergeometric integrals of different dimensions

V. Tarasov, A. Varchenko

Given complex numbers $m_{1}, l_{1}$ and positive integers $m_{2}, l_{2}$ , such that $m_{1} + m_{2} = l_{1} + l_{2}$ , we define $l_{2}$ -dimensional hypergeometric integrals $I_{a, b} (z; m_{1}, m_{2}, l_{1}, l_{2})$ , $a,…

math.CA2006

Higher Lame Equations and Critical Points of Master Functions

E. Mukhin, V. Tarasov, A. Varchenko

Under certain conditions, we give an estimate from above on the number of differential equations of order $r + 1$ with prescribed regular singular points, prescribed exponents at sin…

Conformal blocks in the tensor product of vector representations and localization formulas

R. Rimanyi, A. Varchenko

Using equivariant localization formulas we give a formula for conformal blocks at level one on the sphere as suitable polynomials. Using this presentation we give a generating set…

Modular transformations of the elliptic hypergeometric functions, Macdonald polynomials, and the shift operator

G. Felder, L. Stevens, A. Varchenko

We consider the space of elliptic hypergeometric functions of the sl_2 type associated with elliptic curves with one marked point. This space represents conformal blocks in the sl_…

math.QA2005

Multiple orthogonal polynomials and a counterexample to Gaudin Bethe Ansatz Conjecture

E. Mukhin, A. Varchenko

Jacobi polynomials are polynomials whose zeros form the unique solution of the Bethe Ansatz equation associated with two sl_2 irreducible modules. We study sequences of r polynomia…

How to regularize singular vectors and kill the dynamical Weyl group

K. Styrkas, V. Tarasov, A. Varchenko

We study the meromorphic family of intertwining operators between Verma modules and their products with finite-dimensional ones. A regularizing operator, acting in a finite dimensi…

math.QA2013

Cohomology of a flag variety as a Bethe algebra

R. Rimanyi, V. Schechtman, V. Tarasov +1

We interpret the GL_n equivariant cohomology of a partial flag variety of flags of length N in \C^n as the Bethe algebra of a suitable gl_N[t] module associated with the tensor pow…

math.AG2006

Determinants of the hypergeometric period matrices of an arrangement and its dual

D. Mukherjee, A. Varchenko

We fix three natural numbers $k, n, N$ , such that $n + k + 1 = N$ , and introduce the notion of two dual arrangements of hyperplanes. One of the arrangements is an arrangement of $N$ hype…

math.AT2001

A remark on sl_2 approximation of the Kontsevich integral of the unknot

S. Tyurina, A. Varchenko

We give a formula for an sl_2 approximation of the Kontsevich integral of the unknot.

Solutions to the XXX type Bethe ansatz equations and flag varieties

E. Mukhin, A. Varchenko

We consider a version of the A_N Bethe equation of XXX type and introduce a reproduction procedure constructing new solutions of this equation from a given one. The set of all solu…

Bethe algebra of Gaudin model, Calogero-Moser space and Cherednik algebra

E. Mukhin, V. Tarasov, A. Varchenko

We identify the Bethe algebra of the Gaudin model associated to gl(N) acting on a suitable representation with the center of the rational Cherednik algebra and with the algebra of…

math.QA2018

Self-dual Grassmannian, Wronski map, and representations of $gl_{N}$ , $sp_{2 r}$ , $so_{2 r + 1}$

Kang Lu, E. Mukhin, A. Varchenko

We define a $gl_{N}$ -stratification of the Grassmannian of $N$ planes $Gr (N, d)$ . The $gl_{N}$ -stratification consists of strata $I^c ◯_{I^}$ labeled…

Three sides of the geometric Langlands correspondence for gl_N Gaudin model and Bethe vector averaging maps

E. Mukhin, V. Tarasov, A. Varchenko

We consider the gl_N Gaudin model of a tensor power of the standard vector representation. The geometric Langlands correspondence in the Gaudin model relates the Bethe algebra of t…

math.QA2001

Small Elliptic Quantum Group $e_{t a u, I^^{3}} (s l_{N})$

V. Tarasov, A. Varchenko

The small elliptic quantum group $e_{I ¨ , I^^{3}} (s l_{N})$ , introduced in the paper, is an elliptic dynamical analogue of the universal enveloping algebra $U (s l_{n})$ . We define highest weig…

Bethe algebra and algebra of functions on the space of differential operators of order two with polynomial solutions

E. Mukhin, V. Tarasov, A. Varchenko

We show that the following two algebras are isomorphic. The first is the algebra $A_{P}$ of functions on the scheme of monic linear second-order differential operators on $\C$ with p…

math.QA2001

Special functions, conformal blocks, Bethe ansatz, and SL(3,Z)

G. Felder, A. Varchenko

This is the talk of the second author at the meeting "Topological Methods in Physical Sciences", London, November 2000. We review our work on KZB equations.

Resonance relations, holomorphic trace functions and hypergeometric solutions to qKZB and Macdonald-Ruijsenaars equations

K. Styrkas, A. Varchenko

The resonance relations are identities between coordinates of functions with values in tensor products of representations of the quantum group Uq(sl2). We show that the space of hy…

Schubert calculus and representations of general linear group

E. Mukhin, V. Tarasov, A. Varchenko

We construct a canonical isomorphism between the Bethe algebra acting on a multiplicity space of a tensor product of evaluation gl_N[t]-modules and the scheme-theoretic intersectio…

math.QA2005

Bispectral and $(\glN,\glM)$ Dualities

E. Mukhin, V. Tarasov, A. Varchenko

Let $V = < p_{ij}(x)e^{\la_ix}, i=1,...,n, j=1, ..., N_i >$ be a space of quasi-polynomials of dimension $N = N_{1} + ... + N_{n}$ . Define the regularized fundamental operator of $V$ as the…

Multiplication Formulas for the Elliptic Gamma Function

G. Felder, A. Varchenko

The elliptic gamma function is a generalization of the Euler gamma function. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma funct…

math.DS2009

The Euler adic dynamical system and path counts in the Euler graph

K. Petersen, A. Varchenko

We give a formula for generalized Eulerian numbers, prove monotonicity of sequences of certain ratios of the Eulerian numbers, and apply these results to obtain a new proof that th…

Dynamical differential equations compatible with rational qKZ equations

V. Tarasov, A. Varchenko

For the Lie algebra $g l_{N}$ we introduce a system of differential operators called the dynamical operators. We prove that the dynamical differential operators commute with the $gl_N…

On the new form of Bethe ansatz equations and separation of variables in the $s l_{3}$ Gaudin model

E. Mukhin, V. Schechtman, V. Tarasov +1

A new form of Bethe ansatz equations is introduced. A version of a separation of variables for the quantum $s l_{3}$ Gaudin model is presented.

math.QA2011

Reality property of discrete Wronski map with imaginary step

E. Mukhin, V. Tarasov, A. Varchenko

For a set of quasi-exponentials with real exponents, we consider the discrete Wronskian (also known as Casorati determinant) with pure imaginary step 2h. We prove that if the coeff…

Cohomology classes of conormal bundles of Schubert varieties and Yangian weight functions

R. Rimanyi, V. Tarasov, A. Varchenko

We consider the conormal bundle of a Schubert variety $S_{I}$ in the cotangent bundle $T^{*} G r$ of the Grassmannian $G r$ of $k$ -planes in $C^{n}$ . This conormal bundle has a fundamental…

Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra

V. Gorbounov, R. Rimanyi, V. Tarasov +1

We interpret the equivariant cohomology algebra H^*_{GL_n\times\C^*}(T^*F_Î»;\C) of the cotangent bundle of a partial flag variety F_Î»parametrizing chains of subspaces 0=F_0\subse…

Partial flag varieties, stable envelopes and weight functions

R. Rimanyi, V. Tarasov, A. Varchenko

We consider the cotangent bundle T^*F_Î»of a GL_n partial flag variety, Î»= (Î»_1,...,Î»_N), |Î»|=\sum_iÎ»_i=n, and the torus T=(C^*)^{n+1} equivariant cohomology H^*_T(T^*F_Î»). I…

math.QA2011

Norms of eigenfunctions to trigonometric KZB operators

E. Jensen, A. Varchenko

Let $g$ be a simple Lie algebra and $V [0] = V_{1} \otimes ... \otimes V_{n} [0]$ the zero weight subspace of a tensor product of $g$ -modules. The trigonometric KZB operators are commuting di…

Unitarity of SL(2)-conformal blocks in genus zero

E. Looijenga, A. Varchenko

This submission has been withdrawn by arXiv administrators due to an unresolved conflict between the authors. This article was submitted without consent of E. Looijenga.

math.QA2000

Difference Equations Compatible with Trigonometric KZ Differential Equations

V. Tarasov, A. Varchenko

The trigonometric KZ equations associated with a Lie algebra $\g$ depend on a parameter $Î»\in\h$ where $\h\subset\g$ is the Cartan subalgebra. We suggest a system of dynamical dif…

math.QA2003

Critical points of functions, sl_2 representations, and Fuchsian differential equations with only univalued solutions

I. Scherbak, A. Varchenko

Let a second order Fuchsian differential equation with only univalued solutions have finite singular points at z_1, ..., z_n with exponents (a_1,b_1), ..., (a_n,b_n). Let the expon…

Duality for Knizhnik-Zamolodchikov and Dynamical Equations

V. Tarasov, A. Varchenko

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…

Spaces of quasi-exponentials and representations of gl_N

E. Mukhin, V. Tarasov, A. Varchenko

We consider the action of the Bethe algebra B_K on (\otimes_{s=1}^k L_{Î»^{(s)}})_Î», the weight subspace of weight $I^»$ of the tensor product of k polynomial irreducible gl_N-modu…

Quantization of the space of conformal blocks

E. Mukhin, A. Varchenko

We consider the discrete Knizhnik-Zamolodchikov connection (qKZ) associated to $g l (N)$ , defined in terms of rational R-matrices. We prove that under certain resonance conditions, t…

math-ph2012

Extended Joseph polynomials, quantized conformal blocks, and a q-Selberg type integral

R. RimÃ¡nyi, V. Tarasov, A. Varchenko +1

We consider the tensor power $V = (C^{N})^{\otimes n}$ of the vector representation of $g l_{N}$ and its weight decomposition $V = \oplus_{I^» = (I^»_{1}, ..., I^»_{N})} V [I^»]$ . For $Î»= (Î»_1 \geq ...…

math.QA2000

Differential Equations Compatible with KZ Equations

G. Felder, Y. Markov, V. Tarasov +1

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…

math.QA2012

Bethe Algebra of Homogeneous XXX Heisenberg Model Has Simple Spectrum

E. Mukhin, V. Tarasov, A. Varchenko

We show that the algebra of commuting Hamiltonians of the homogeneous XXX Heisenberg model has simple spectrum on the subspace of singular vectors of the tensor product of two-dime…

A generalization of the Capelli identity

E. Mukhin, V. Tarasov, A. Varchenko

We prove a generalization of the Capelli identity. As an application we obtain an isomorphism of the Bethe subalgebras actions under the (gl(N),gl(M)) duality.

math.AG2009

Critical points and resonance of hyperplane arrangements

D. Cohen, G. Denham, M. Falk +1

If F is a master function corresponding to a hyperplane arrangement A and a collection of weights y, we investigate the relationship between the critical set of F, the variety defi…

Selberg Type Integrals Associated with $s l_{3}$

V. Tarasov, A. Varchenko

We present several formulae for the Selberg type integrals associated with the Lie algebra $s l_{3}$ .

math.AG2014

Trigonometric weight functions as K-theoretic stable envelope maps for the cotangent bundle of a flag variety

R. Rimanyi, V. Tarasov, A. Varchenko

We consider the cotangent bundle $T^{*} F_{I^} »$ of a $G L_{n}$ partial flag variety, $I^» = (I^»_{1}, ..., I^»_{N})$ , $∣ I^»∣ = \sum_{i} I^»_{i} = n$ , and the torus $T=(\C^\times)^{n+1}$ equivariant K-theory alg…

math.RT2005

Poincare-Birkhoff-Witt expansions of the canonical elliptic differential form

G. Felder, R. Rimanyi, A. Varchenko

We study the canonical U(\n)-valued elliptic differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of el…

math.CO2005

Combinatorics of rational functions and Poincare-Birkhoff-Witt expansions of the canonical U(n-)-valued differential form

R. Rimanyi, L. Stevens, A. Varchenko

We study the canonical U(n-)-valued differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of KZ-type dif…

math.AG2017

Critical points of master functions and the mKdV hierarchy of type A^2_2

A. Varchenko, T. Woodruff, D. Wright

We consider the population of critical points generated from the critical point of the master function with no variables, which is associated with the trivial representation of the…

math.QA2013

Bethe subalgebras of the group algebra of the symmetric group

E. Mukhin, V. Tarasov, A. Varchenko

We introduce families of maximal commutative subalgebras, called Bethe subalgebras, of the group algebra of the symmetric group. Bethe subalgebras are deformations of the Gelfand-Z…

hep-th1994

Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors

V. Schechtman, H. Terao, A. Varchenko

In this note we strenghten a theorem by Esnault-Schechtman-Viehweg which states that one can compute the cohomology of a complement of hyperplanes in a complex affine space with co…

math.QA1998

Functorial properties of the hypergeometric map

E. Mukhin, A. Varchenko

The quantized Knizhnik-Zamolodchikov equation is a difference equation defined in terms of rational $R$ matrices. We describe all singularities of hypergeometric solutions to the q…

Hypergeometric solutions of the quantum differential equation of the cotangent bundle of a partial flag variety

V. Tarasov, A. Varchenko

We describe hypergeometric solutions of the quantum differential equation of the cotangent bundle of a gl_n partial flag variety. These hypergeometric solutions manifest the Landau…

Bethe eigenvectors of higher transfer matrices

E. Mukhin, V. Tarasov, A. Varchenko

We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra $g l_{N}$ . The models are defined on a tensor product irreducible $g l_{N}$ -modules. For eac…

Spaces of quasi-exponentials and representations of the Yangian Y(gl_N)

E. Mukhin, V. Tarasov, A. Varchenko

We consider a tensor product $V(b)= \otimes_{i=1}^n\C^N(b_i)$ of the Yangian $Y (g l_{N})$ evaluation vector representations. We consider the action of the commutative Bethe subalgebra…