Browsing by Subject "Bethe ansatz equations"

Now showing 1 - 4 of 4
  • Thumbnail Image
    Item
    Bethe Ansatz Equations for Orthosymplectic Lie Superalgebras and Self-dual Superspaces
    (Springer, 2021-12) Lu, Kang; Mukhin, Evgeny; Mathematical Sciences, School of Science
    We study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras $$\mathfrak {osp}_{2m+1|2n}$$and $$\mathfrak {osp}_{2m|2n}$$. Given a solution, we define a reproduction procedure and use it to construct a family of new solutions which we call a population. To each population we associate a symmetric rational pseudo-differential operator $$\mathcal R$$. Under some technical assumptions, we show that the superkernel W of $$\mathcal R$$is a self-dual superspace of rational functions, and the population is in a canonical bijection with the variety of isotropic full superflags in W and with the set of symmetric complete factorizations of $$\mathcal R$$. In particular, our results apply to the case of even Lie algebras of type D$${}_m$$corresponding to $$\mathfrak {osp}_{2m|0}=\mathfrak {so}_{2m}$$.
  • Thumbnail Image
    Item
    Completeness of the Bethe Ansatz for the Periodic Isotropic Heisenberg Model
    (World Scientific, 2018-09) Tarasov, V.; Mathematical Sciences, School of Science
    For the periodic isotropic Heisenberg model with arbitrary spins and inhomogeneities, we describe the system of algebraic equations whose solutions are in bijection with eigenvalues of the transfer-matrix. The system describes pairs of polynomials with the given discrete Wronskian (Casorati determinant) and additional divisibility conditions on discrete Wronskians with multiple steps. If the polynomial of the smaller degree in the pair is coprime with the Wronskian, this system turns into the standard Bethe ansatz equations. Moreover, if the transfer-matrix is diagonalizable, then its spectrum is necessarily simple modulo natural degeneration.
  • Thumbnail Image
    Item
    Finite Type Modules and Bethe Ansatz Equations
    (Springer, 2017-08) Feigin, Boris; Jimbo, Michio; Miwa, Tetsuji; Mukhin, Eugene; Department of Mathematical Sciences, School of Science
    We introduce and study a category OfinbObfin of modules of the Borel subalgebra UqbUqb of a quantum affine algebra UqgUqg, where the commutative algebra of Drinfeld generators hi,rhi,r, corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional UqgUqg modules. In particular, we classify the irreducible objects, discuss their properties, and describe the combinatorics of the q-characters. We study transfer matrices corresponding to modules in OfinbObfin. Among them, we find the Baxter QiQi operators and TiTi operators satisfying relations of the form TiQi=∏jQj+∏kQkTiQi=∏jQj+∏kQk. We show that these operators are polynomials of the spectral parameter after a suitable normalization. This allows us to prove the Bethe ansatz equations for the zeroes of the eigenvalues of the QiQi operators acting in an arbitrary finite-dimensional representation of UqgUqg.
  • Thumbnail Image
    Item
    On the Gaudin model associated to Lie algebras of classical types
    (AIP, 2016-10) Lu, Kang; Mukhin, Eugene; Varchenko, A.; Department of Mathematical Sciences, School of Science
    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 and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic.