167 citations · 1.9k across the 134 of their papers we have counts for
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Solutions of the qKZ equations modulo an integer
Evgeny Mukhin, Alexander Varchenko
We study the qKZ difference equations with values in the -th tensor power of the vector representation , variables and integer step . For any intege…
Frobenius-like structure in Gaudin model
Evgeny Mukhin, Alexander Varchenko
We introduce a Frobenius-like structure for the Gaudin model. Namely, we introduce potential functions of the first and second kind. We describe the Shapovalov form i…
Self-dual Grassmannian, Wronski map, and representations of , ,
Kang Lu, E. Mukhin, A. Varchenko
We define a -stratification of the Grassmannian of planes . The -stratification consists of strata labeled by…
Cyclotomic discriminantal arrangements and diagram automorphisms of Lie algebras
Alexander Varchenko, Charles A. S. Young
Recently a new class of quantum integrable models, the cyclotomic Gaudin models, were described in arXiv:1409.6937, arXiv:1410.7664. Motivated by these, we identify a class of affi…
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…
Populations of Solutions to Cyclotomic Bethe Equations
Alexander Varchenko, Charles A. S. Young
We study solutions of the Bethe Ansatz equations for the cyclotomic Gaudin model of [Vicedo B., Young C.A.S., arXiv:1409.6937]. We give two interpretations of such solutions: as cr…
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…
Norms of eigenfunctions to trigonometric KZB operators
E. Jensen, A. Varchenko
Let be a simple Lie algebra and the zero weight subspace of a tensor product of -modules. The trigonometric KZB operators are commuting di…
Quantum Integrable Model of an Arrangement of Hyperplanes
Alexander Varchenko
The goal of this paper is to give a geometric construction of the Bethe algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra. More precisely, in this pape…
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…
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…
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…
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…
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.
A Selberg Integral Type Formula for an sl_2 One-Dimensional Space of Conformal Blocks
A. Varchenko
For distinct complex numbers , we give a polynomial in the variables , which is homogeneous of degree , linear with respect t…
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_{λ^{(s)}})_λ, the weight subspace of weight of the tensor product of k polynomial irreducible gl_N-modules…
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-…
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…
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…
Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel
Evgeny Mukhin, Vitaly Tarasov, Alexander Varchenko
Let be the tensor product of finite-dimensional polynomial evaluation Yangian -modules. Consider the universal difference operator $D = \sum_{k=0}^N (-1)^k T_k(u) e^{-…
Combinatorial Formulae for Nested Bethe Vectors
Vitaly Tarasov, Alexander Varchenko
We give combinatorial formulae for vector-valued weight functions (off-shell nested Bethe vectors) for tensor products of irreducible evaluation modules over the Yangian $Y({\mathf…
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…
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.
Bispectral and (gl_N, gl_M) Dualities, Discrete Versus Differential
E. Mukhin, V. Tarasov, A. Varchenko
Let be a space of quasi-polynomials in of dimension . The regularized fundamental differential operator of…