Geometry of q-Hypergeometric Functions as a Bridge between Yangians and Quantum Affine Algebras

Abstract

The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the Lie algebra $sl_2$ is a system of linear difference equations with values in a tensor product of $sl_2$ Verma modules. We solve the equation in terms of multidimensional $q$-hypergeometric functions and define a natural isomorphism between the space of solutions and the tensor product of the corresponding quantum group $U_q(sl_2)$ Verma modules, where the parameter $q$ is related to the step $p$ of the qKZ equation via $q=e^{pi i/p}$. We construct asymptotic solutions associated with suitable asymptotic zones and compute the transition functions between the asymptotic solutions in terms of the trigonometric $R$-matrices. This description of the transition functions gives a new connection between representation theories of Yangians and quantum loop algebras and is analogous to the Kohno-Drinfeld theorem on the monodromy group of the differential Knizhnik-Zamolodchikov equation. In order to establish these results we construct a discrete Gauss-Manin connection, in particular, a suitable discrete local system, discrete homology and cohomology groups with coefficients in this local system, and identify an associated difference equation with the qKZ equation.

66 pages, amstex.tex (ver. 2.1) and amssym.tex are required; misprints are corrected