qKZB Equations Research Articles

TRACES OF INTERTWINERS FOR QUANTUM AFFINE ALGEBRAS AND DIFFERENCE EQUATIONS (AFTER ETINGOF–SCHIFFMANN–VARCHENKO)

We modify and give complete proofs for the results of Etingof-Schiffmann-Varchenko on traces of intertwiners of untwisted quantum affine algebras in the opposite coproduct and the standard grading. More precisely, we show that certain normalized generalized traces for $U_q(\hat{\mathfrak{g}})$ solve four commuting systems of q-difference equations: the Macdonald-Ruijsenaars, dual Macdonald-Ruijsenaars, q-KZB, and dual q-KZB equations. In addition, we show a symmetry property for these renormalized trace functions. Our modifications are motivated by their appearance in recent work of the author.

Dynamical Gelfand–Zetlin algebra and equivariant cohomology of grassmannians

We consider the rational dynamical quantum group [Formula: see text] and introduce an [Formula: see text]-module structure on [Formula: see text], where [Formula: see text] is the equivariant cohomology algebra [Formula: see text] of the cotangent bundle of the Grassmannian [Formula: see text] with coefficients extended by a suitable ring of rational functions in an additional variable [Formula: see text]. We consider the dynamical Gelfand–Zetlin algebra which is a commutative algebra of difference operators in [Formula: see text]. We show that the action of the Gelfand–Zetlin algebra on [Formula: see text] is the natural action of the algebra [Formula: see text] on [Formula: see text], where [Formula: see text] is the shift operator. The [Formula: see text]-module structure on [Formula: see text] is introduced with the help of dynamical stable envelope maps which are dynamical analogs of the stable envelope maps introduced by Maulik and Okounkov [Quantum Groups and quantum cohomology, preprint (2012) 1–276, arXiv:1211.1287]. The dynamical stable envelope maps are defined in terms of the rational dynamical weight functions introduced in [G. Felder, V. Tarasov and A. Varchenko, Solutions of the elliptic QKZB equations and Bethe ansatz I, in Topics in Singularity Theory V. I. Arnold’s 60th Anniversary Collection, Advances in the Mathematical Sciences, AMS Translations, Series 2, Vol. 180 (AMS, 1997), pp. 45–76.] to construct q-hypergeometric solutions of rational qKZB equations. The cohomology classes in [Formula: see text] induced by the weight functions are dynamical variants of Chern–Schwartz–MacPherson classes of Schubert cells.

Resonance Relations, Holomorphic Trace Functions and Hypergeometric Solutions to qKZB and Macdonald-Ruijsenaars Equations

The resonance relations are identities between coordinates of functions ψ(λ) with values in tensor products of representations of the quantum group 𝒰q(sl2)⁠. We show that the space of hypergeometric solutions of the associated qKZB equations is characterized as the space of functions of Baker–Akhiezer type, satisfying the resonance relations. We give an alternative representation–theoretic construction of this space, using the traces of regularized intertwining operators for the quantum group 𝒰q(sl2)⁠, and thus establish the equivalence between hypergeometric and trace function solutions of the qKZB equations. We define the quantum conformal blocks as distinguished Weylanti–invariant hypergeometric solutions of the qKZB equations with values in a tensor product of finite–dimensional 𝒰q(sl2) -modules. We prove that for generic q the dimension of the space of quantum conformal blocks equals the dimension of 𝒰q(sl2)-invariants, and when q is a root of unity is computed by the Verlinde algebra.

Resonance Relations, Holomorphic Trace Functions and Hypergeometric Solutions to qKZB and Macdonald-Ruijsenaars Equations

The resonance relations are identities between coordinates of functions with values in tensor products of representations of the quantum group Uq(sl2). We show that the space of hypergeometric solutions of the associated qKZB equations is characterized as the space of functions of Baker-Akhiezer type, satisfying the resonance relations. We give an alternative representation-theoretic construction of this space, using the traces of regularized intertwining operators for the quantum group, and thus establish the equivalence between hypergeometric and trace function solutions of the qKZB equations. We define the quantum conformal blocks as distinguished Weyl anti-invariant hypergeometric qKZB solutions with values in a tensor product of finite-dimensional modules. We prove that for generic q the dimension of the space of quantum conformal blocks equals the dimension of the quantum group invariants, and is computed by the Verlinde algebra when q is a root of unity.

Twisted Traces of Quantum Intertwiners¶and Quantum Dynamical R-Matrices¶Corresponding to Generalized Belavin–Drinfeld Triples

This paper is a continuation of math.QA/9907181 and math.QA/9908115. We consider traces of intertwiners between certain representations of the quantized enveloping algebra associated to a semisimple complex Lie algebra g, which are twisted by a ``generalized Belavin-Drinfeld triple'', i.e a triple consisting of two subdiagrams of the Dynkin diagram of g together with an isomorphism between them. The generating functions F(lambda,mu) for such traces depend on two weights lambda and mu. We show that F(lambda,mu) satisfy two sets of difference equations in the variable lambda: the Macdonald-Ruijsenaars (MR) equations and the quantum Knizhnik-Zamolodchikov (qKZB) equations. These equations involve as a main ingredient the quantum dynamical R-matrices constructed in math.QA/9912009. When the generalized Belavin-Drinfeld triple is an automorphism, we show that F(lambda,mu) satisfy another two sets of difference equations with respect to the weight mu. These dual MR and dual qKZB equations involve the usual Felder's dynamical R-matrix. These results were first obtained by the first author and A. Varchenko in the special case of the trivial Belavin-Drinfeld triple. However, the symmetry between lambda and mu which exists in that case is destroyed in the twisted setting. At the end, we brielfly treat the (simialr) case of Kac-Moody algebras g and derive the classical limits of all the previous results.

The q -Deformed Knizhnik-Zamolodchikov-Bernard Heat Equation

We introduce a q-deformation of the genus one sl_2 Knizhnik-Zamolodchikov-Bernard heat equation. We show that this equation for the dependence on the moduli of elliptic curves is compatible with the qKZB equations, which give the dependence on the marked points.

The Elliptic Gamma Function and SL(3, [formula omitted])⋉[formula omitted

The elliptic gamma function is a generalization of the Euler gamma function and is associated to an elliptic curve. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function, respectively. The elliptic gamma function appears in Baxter's formula for the free energy of the eight-vertex model and in the hypergeometric solutions of the elliptic qKZB equations. In this paper, the properties of this function are studied. In particular we show that elliptic gamma functions are generalizations of automorphic forms of G=SL(3, Z)⋉Z3 associated to a non-trivial class in H3(G, Z).

MONODROMY OF SOLUTIONS OF THE ELLIPTIC QUANTUM KNIZHNIK–ZAMOLODCHIKOV–BERNARD DIFFERENCE EQUATIONS

The elliptic quantum Knizhnik–Zamolodchikov–Bernard (qKZB) difference equations associated to the elliptic quantum group Eτ,η(sl2) is a system of difference equations with values in a tensor product of representations of the quantum group and defined in terms of the elliptic R-matrices associated with pairs of representations of the quantum group. In this paper we solve the qKZB equations in terms of elliptic hypergeometric functions and describe the monodromy properties of solutions. It turns out that the monodromy transformations of solutions are described in terms of elliptic R-matrices associated with pairs of representations of the "dual" elliptic quantum group Ep,η(sl2), where p is the step of the difference equations. Our description of the monodromy is analogous to the Kohno–Drinfeld description the monodromy group of solutions of the KZ differential equations associated to a simple Lie algebra in terms of the corresponding quantum group.

RESONANCE RELATIONS FOR SOLUTIONS OF THE ELLIPTIC QKZB EQUATIONS, FUSION RULES, AND EIGENVECTORS OF TRANSFER MATRICES OF RESTRICTED INTERACTION-ROUND-A-FACE MODELS

Conformal blocks for the WZW model on tori can be represented by vector valued Weyl anti-symmetric theta functions on the Cartan subalgebra satisfying vanishing conditions on root hyperplanes. We introduce a quantum version of these vanishing conditions in the sl2 case. They are compatible with the qKZB equations and are obeyed by the hypergeometric solutions as well as by their critical level counterpart, which are Bethe eigenfunctions of IRF row-to-row transfer matrices. In the language of IRF models the vanishing conditions turn out to be equivalent to the sl2 fusion rules defining restricted models.

RESONANCE RELATIONS FOR SOLUTIONS OF THE ELLIPTIC QKZB EQUATIONS, FUSION RULES, AND EIGENVECTORS OF TRANSFER MATRICES OF RESTRICTED INTERACTION-ROUND-A-FACE MODELS

Conformal blocks for the WZW model on tori can be represented by vector valued Weyl anti-symmetric theta functions on the Cartan subalgebra satisfying vanishing conditions on root hyperplanes. We introduce a quantum version of these vanishing conditions in the sl(2) case. They are compatible with the qKZB equations and are obeyed by the hypergeometric solutions as well as by their critical level counterpart, which are Bethe eigenfunctions of IRF row-to-row transfer matrices. In the language of IRF models the vanishing conditions turn out to be equivalent to the sl(2) fusion rules defining restricted models.