221 research outputs found
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.Comment: Latex, 9 page
Generating Operator of XXX or Gaudin Transfer Matrices Has Quasi-Exponential Kernel
Let be the tensor product of finite-dimensional polynomial evaluation
Yangian -modules. Consider the universal difference operator whose coefficients are the XXX transfer matrices associated with . We show that the
difference equation for an -valued function has a basis of
solutions consisting of quasi-exponentials. We prove the same for the universal
differential operator whose
coefficients are the Gaudin transfer matrices associated
with the tensor product of finite-dimensional polynomial evaluation
-modules.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Discrete Miura Opers and Solutions of the Bethe Ansatz Equations
Solutions of the Bethe ansatz equations associated to the XXX model of a
simple Lie algebra come in families called the populations. We prove that a
population is isomorphic to the flag variety of the Langlands dual Lie algebra.
The proof is based on the correspondence between the solutions of the Bethe
ansatz equations and special difference operators which we call the discrete
Miura opers. The notion of a discrete Miura oper is one of the main results of
the paper.
For a discrete Miura oper D, associated to a point of a population, we show
that all solutions of the difference equation DY=0 are rational functions, and
the solutions can be written explicitly in terms of points composing the
population.Comment: Latex, 26 page
KZ Characteristic Variety as the Zero Set of Classical Calogero-Moser Hamiltonians
We discuss a relation between the characteristic variety of the KZ equations
and the zero set of the classical Calogero-Moser Hamiltonians
Eigenvalues of Bethe vectors in the Gaudin model
According to the Feigin–Frenkel–Reshetikhin theorem, the eigenvalues of higher Gaudin Hamiltonians on Bethe vectors can be found using the center of an affine vertex algebra at the critical level. We recently calculated explicit Harish-Chandra images of the generators of the center in all classical types. Combining these results leads to explicit formulas for the eigenvalues of higher Gaudin Hamiltonians on Bethe vectors. The Harish-Chandra images can be interpreted as elements of classical W-algebras. By calculating classical limits of the corresponding screening operators, we elucidate a direct connection between the rings of q-characters and classical W-algebras
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