
Dynamical elliptic Bethe algebra, KZB eigenfunctions, and thetapolynomialsPages: 78 – 125 AuthorsAbstractLet \((\otimes_{j=1}^nV_j)[0]\) be the zero weight subspace of a tensor product of finitedimensional irreducible \(\mathfrak{sl}_2\)modules. The dynamical elliptic Bethe algebra is a commutative algebra of differential operators acting on \((\otimes_{j=1}^nV_j)[0]\)valued functions on the Cartan subalgebra of \(\mathfrak{sl}_2\). The algebra is generated by values of the coefficient \(S_2(x)\) of a certain differential operator D = \(\partial_x^2 + S_2(x)\), defined by V. Rubtsov, A. Silantyev, D. Talalaev in 2009. We express \(S_2(x)\) in terms of the KZB operators introduced by G. Felder and C. Wieszerkowski in 1994. We study the eigenfunctions of the dynamical elliptic Bethe algebra by the Bethe ansatz method. Under certain assumptions we show that such Bethe eigenfunctions are in onetoone correspondence with ordered pairs of thetapolynomials of certain degree. The correspondence between Bethe eigenfunctions and twodimensional spaces, generated by the two thetapolynomials, is an analog of the nondynamical nonelliptic correspondence between the eigenvectors of the \(\mathfrak{gl}_2\) Gaudin model and the twodimensional subspaces of the vector space \(\mathbb{C}[x]\), due to E. Mukhin, V. Tarasov, A. Varchenko. We obtain a counting result for, equivalently, certain solutions of the Bethe ansatz equation, certain fibers of the elliptic Wronski map, or ratios of theta polynomials, whose derivative is of a certain form. We give an asymptotic expansion for Bethe eigenfunctions in a certain limit, and deduce from that that the Weyl involution acting on Bethe eigenfunctions coincides with the action of an analytic involution given by the transposition of thetapolynomials in the associated ordered pair.
Received: 19 February 2019; Accepted: 29 August 2019. 
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