Quantum Knizhnik-Zamolodchikov equations and holomorphic vector bundles

TL;DR

This paper provides a geometric interpretation of quantum KZ equations via holomorphic vector bundles on elliptic curves, linking solutions to sections of these bundles and exploring their topological and stability properties.

Contribution

It introduces a novel geometric framework connecting quantum KZ equations with holomorphic vector bundles on elliptic curves, extending to generalized equations for affine root systems.

Findings

Solutions correspond to sections of holomorphic vector bundles

Bundles are topologically nontrivial and described by crystal bases

Bundles are semistable for generic parameters in the quantum sl(2) case

Abstract

The paper introduces a new geometric interpretation of the quantum Knizhnik-Zamolodchikov equations introduced in 1991 by I.Frenkel and N.Reshetikhin. It turns out that these equations can be linked to certain holomorphic vector bundles on the N-th Cartesian power of an elliptic curve. These bundles are naturally constructed by a gluing procedure from a system of trigonometric quantum affine $R$-matrices. Meromorphic solutions of the quantum KZ equations are interpreted as sections of such a bundle. This interpretation is an analogue of the interpretation of solutions of the classical KZ equations as sections of a flat vector bundle. Matrix elements of intertwiners between representations of the quantum affine algebra correspond to regular (holomorphic) sections. The vector bundle obtained from the quantum KZ system is topologically nontrivial. Its topology can be completely described in terms of crystal bases, using the crystal limit ``q goes to 0''. In the case N=2, this bundle is is essentially a bundle on an elliptic curve which is shown to be semistable (for the case of quantum sl(2)) if the parameters take generic values. The proof makes use of the crystal limit ``q goes to 0''. Finally, we give a vector bundle interpretation of the generalized quantum KZ equations for arbitrary affine root systems defined recently by Cherednik.