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Giovanni Felder

Publications and source records attributed to Giovanni Felder.

84 records · Page 5Linked to original sources
arXiv1996

Solutions of the elliptic qKZB equations and Bethe ansatz I

We give an integral representation for solutions of the elliptic quantum Knizhnik-Zamolodchikov-Bernard difference equations, in the case of sl(2). The result is based on a geometric construction of highest weight representations of the elliptic quantum group associated to sl(2). We also obtain Bethe ansatz eigenfunctions for the corresponding integrable systems of difference operators, and prove their completeness in some cases.

q-alg↗
arXiv1996

Algebraic Bethe ansatz for the elliptic quantum group $E_{τ,η}(sl_2)$

To each representation of the elliptic quantum group $E_{τ,η}(sl_2)$ is associated a family of commuting transfer matrices. We give common eigenvectors by a version of the algebraic Bethe ansatz method. Special cases of this construction give eigenvectors for IRF models, for the eight-vertex model and for the two-body Ruijsenaars operator. The latter is a $q$-deformation of Hermite's solution of the Lamé equation.

q-alg↗
arXiv1995

Three formulas for eigenfunctions of integrable Schroedinger operators

We give three formulas for meromorphic eigenfunctions (scattering states) of Sutherland's integrable N-body Schroedinger operators and their generalizations. The first is an explicit computation of the Etingof-Kirillov traces of intertwining operators, the second an integral representation of hypergeometric type, and the third is a formula of Bethe ansatz type. The last two formulas are degenerations of elliptic formulas obtained previously in connection with the Knizhnik-Zamolodchikov-Bernard equation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctions are parametrized by a ``Hermite-Bethe'' variety, a generalization of the spectral variety of the Lame' operator. We also give the q-deformed version of our first formula. In the scalar sl_N case, this gives common eigenfunctions of the commuting Macdonald-Rujsenaars difference operators.

hep-th↗
arXiv1995

Integral representation of solutions of the elliptic Knizhnik--Zamolodchikov--Bernard equations

We give an integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations for arbitrary simple Lie algebras. If the level is a positive integer, we obtain formulas for conformal blocks of the WZW model on a torus. The asymptotics of our solutions at critical level gives eigenfunctions of Euler-Calogero-Moser integrable $N$-body systems. As a by-product, we obtain some remarkable integral identities involving classical theta functions.

hep-th↗
arXiv1994

Elliptic quantum groups

This note for the Proceedings of the International Congress of Mathematical Physics gives an account of a construction of an ``elliptic quantum group'' associated with each simple classical Lie algebra. It is closely related to elliptic face models of statistical mechanics, and, in its semiclassical limit, to the Wess-Zumino-Witten model of conformal field theory on tori.

hep-th↗
arXiv1994

Conformal blocks on elliptic curves and the Knizhnik--Zamolodchikov--Bernard equations

We give an explicit description of the vector bundle of WZW conformal blocks on elliptic curves with marked points as subbundle of a vector bundle of Weyl group invariant vector valued theta functions on a Cartan subalgebra. We give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperplanes. In some cases, explicit calculation are possible and confirm the conjecture. The Friedan--Shenker flat connection is calculated, and it is shown that horizontal sections are solutions of Bernard's generalization of the Knizhnik--Zamolodchikov equation.

hep-th↗
arXiv1994

Elliptic Dunkl operators, root systems, and functional equations

We consider generalizations of Dunkl's differential-difference operators associated with groups generated by reflections. The commutativity condition is equivalent to certain functional equations. These equations are solved in many cases. In particular, solutions associated with elliptic curves are constructed. In the $A_{n-1}$ case, we discuss the relation with elliptic Calogero-Moser integrable $n$-body problems, and discuss the quantization ($q$-analogue) of our construction.

hep-th↗
arXiv1994

A simple construction of elliptic $R$-matrices

We show that Belavin's solutions of the quantum Yang--Baxter equation can be obtained by restricting an infinite $R$-matrix to suitable finite dimensional subspaces. This infinite $R$-matrix is a modified version of the Shibukawa--Ueno $R$-matrix acting on functions of two variables.

hep-th↗