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A. Varchenko

Publications and source records attributed to A. Varchenko.

At least 37 records · Page 2Linked to original sources
arXiv2012

Bethe Algebra of Homogeneous XXX Heisenberg Model Has Simple Spectrum

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-dimensional $gl_2$-modules. As a byproduct we show that there exist exactly $\binom {n}{l}-\binom{n}{l-1}$ two-dimensional vector subspaces $V \subset \C[u]$ with a basis $f,g\in V$ such that $°f = l, °g = n-l+1$ and $f(u)g(u-1) - f(u-1)g(u) = (u+1)^n$.

math.QA↗
arXiv2011

Norms of eigenfunctions to trigonometric KZB operators

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 differential operators acting on $V[0]$-valued functions on the Cartan subalgebra of $g$. Meromorphic eigenfunctions to the operators are constructed by the Bethe ansatz. We introduce a scalar product on a suitable space of functions such that the operators become symmetric, and the square of the norm of a Bethe eigenfunction equals the Hessian of the master function at the corresponding critical point.

math.QA↗
arXiv2011

Reality property of discrete Wronski map with imaginary step

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 coefficients of the discrete Wronskian are real and for every its roots the imaginary part is at most |h|, then the complex span of this set of quasi-exponentials has a basis consisting of quasi-exponentials with real coefficients. This result is a generalization of the statement of the B. and M. Shapiro conjecture on spaces of polynomials. The proof is based on the Bethe ansatz for the XXX model.

math.QA↗
arXiv2011

Path count asymptotics and Stirling numbers

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 numbers of the first and second kinds.

math.CO↗
arXiv2011

Conformal blocks and equivariant cohomology

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 bundles of conformal blocks have rank one, this construction may be compared with the old integral formulas of the second and the third author. The proportionality coefficients are some Selberg type integrals which are computed. Finally, a geometric construction of the tensor products of vector representations of the Lie algebra $\frak{gl}(m)$ is proposed.

math-ph↗
arXiv2009

Critical points and resonance of hyperplane arrangements

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 defined by the vanishing of the one-form w = d log F, and the resonance of y. For arrangements satisfying certain conditions, we show that if y is resonant in dimension p, then the critical set of F has codimension at most p. These include all free arrangements and all rank 3 arrangements.

math.AG↗
arXiv2009

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

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 space of that decomposition a commutative (Bethe) algebra of linear operators acting on the multiplicity space. The Bethe ansatz method is a method to find eigenvectors and eigenvalues of the Bethe algebra. One starts with a critical point of a suitable (master) function and constructs an eigenvector of the Bethe algebra. In this paper we consider the algebra of functions on the critical set of the associated master function and show that the action of this algebra on itself is isomorphic to the action of the Bethe algebra on a suitable subspace of the multiplicity space. As a byproduct we prove that the Bethe vectors corresponding to different critical points of the master function are linearly independent and, in particular, nonzero.

math.QA↗
arXiv2009

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

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 the natural symmetric measure for the Bratteli-Vershik dynamical system based on the Euler graph is the unique fully supported invariant ergodic Borel probability measure. Key ingredients of the proof are a two-dimensional induction argument and a one-to-one correspondence between most paths from two vertices at the same level to another vertex.

math.DS↗
arXiv2009

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

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 the commuting Gaudin Hamiltonians and the algebra of functions on a suitable space of N-th order differential operators. In this paper we introduce a third side of the correspondence: the algebra of functions on the critical set of a master function. We construct isomorphisms of the third algebra and the first two. A new object is the Bethe vector averaging maps.

math.QA↗
arXiv2009

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

For distinct complex numbers $z_1,...,z_{2N}$, we give a polynomial $P(y_1,...,y_{2N})$ in the variables $y_1,...,y_{2N}$, which is homogeneous of degree $N$, linear with respect to each variable, $sl_2$-invariant with respect to a natural $sl_2$-action, and is of order $N-1$ at $(y_1,...,y_{2N})=(z_1,...,z_{2N})$. We give also a Selberg integral type formula for the associated one-dimensional space of conformal blocks.

math.QA↗
arXiv2009

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

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 generators of the Bethe algebra in terms of the Gaudin Hamiltonians does not depend on N. Moreover, this formula coincides with Wilson's formula for the stationary Baker-Akhiezer function on the adelic Grassmannian.

math.QA↗
arXiv2008

On reality property of Wronski maps

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 this set of quasi-exponentials has a basis consisting of quasi-exponentials with real coefficients. This result generalizes the B. and M.Shapiro conjecture about spaces of polynomials. The proof is based on the Bethe ansatz method for the XXX model.

math.QA↗