Coinvariants for Yangian Doubles and Quantum Knizhnik-zamolodchikov Equations

We present a quantum version of the construction of the KZ system of equations as a flat connection on the spaces of coinvariants of representations of tensor products of Kac-Moody algebras. We consider here representations of a tensor product of Yangian doubles and compute the coinvariants of a deformation of the subalgebra generated by the regular functions of a rational curve with marked points. We observe that Drinfeld’s quantum Casimir element can be viewed as a deformation of the zero-mode of the Sugawara tensor in the Yangian double. These ingredients serve to define a compatible system of difference equations, which we identify with the quantum KZ equations introduced by I. Frenkel and N. Reshetikhin. Introduction. The Knizhnik-Zamolodchikov (KZ) system is a set of differential equations satisfied by correlation functions in Wess-Zumino-Witten theories ([12]). These equations can be interpreted as the equations satisfied by matrix elements of intertwining operators associated with representations of affine KacMoody algebras ([14]). They define a local system on the configuration space of n distinct marked points on the rational curve CP . This local system also has another interpretation: to a complex curve with marked points, a system of weights of a semisimple Lie algebra and a positive integer is associated the vector space of conformal blocks. It is the space of coinvariants of a representation of a product of Kac-Moody algebras attached to the points of the curve, with respect to the subalgebra formed by the rational functions on the curve, regular outside the points. The conformal blocks form a vector bundle on the moduli space of curves with marked points. There exists a natural connection on this vector bundle, the KZB (for KZ-Bernard) connection. It is provided by the action of the Sugawara field. This field is a generating functional for elements of the enveloping algebra of the Kac-Moody algebra. One shows that certain combinations of these elements conjugate the regular algebras associated to nearby elements of the moduli space (see [2, 15]). In [11], I. Frenkel and N. Reshetikhin introduced q-deformed analogues of the KZ equations. This qKZ system is a difference system obeyed by matrix elements of intertwining operators of quantum affine algebras. Later, elliptic analogues of this system were defined and studied ([9, 10]). If one wished to understand qdeformed versions of the KZB connection in higher genus, it would be important