Graphical calculus for quantum vertex operators, II: q-KZB and coordinate Macdonald-Ruijsenaars equations

TL;DR

This paper extends graphical calculus for quantum vertex operators to include dynamical twist functors and fusion operators, providing intuitive derivations of dual q-KZB and Macdonald-Ruijsenaars equations and their extensions.

Contribution

It introduces a graphical framework for dynamical twist functors and fusion operators, enabling new derivations of difference equations for quantum vertex operators.

Findings

Graphical derivation of dual q-KZB equations

Graphical derivation of dual Macdonald-Ruijsenaars equations

Extension to dual coordinate Macdonald-Ruijsenaars equations

Abstract

We extend the graphical calculus developed in the first part of this paper to the parametrising spaces of quantum vertex operators. This involves a graphical implementation of the dynamical twist functor, which is a strict monoidal functor that describes how a morphism acting on the spin space of a quantum vertex operator $\Phi$ is transported to a morphism on the parametrising space of $\Phi$. The monoidal structure of the underlying nonstrict monoidal functor, considered before by Etingof and Varchenko in 1999, is given in terms of dynamical fusion operators, which are operators that describe the fusion of quantum vertex operators on the level of parametrising spaces. In the second part of the paper we use the extended graphical calculus to give intuitive, graphical derivations of various systems of difference equations for universal multipoint weighted trace functions. This includes the dual $q$-Knizhnik-Zamolodchikov-Bernard (KZB) and the dual Macdonald-Ruijsenaars (MR) equations, earlier obtained by Etingof and Varchenko in 2000, as well as an extension of the dual MR equations called dual coordinate MR equations. We use a known symmetry property of the universal weighted trace function, involving the exchange of its geometric and spectral parameter, to derive non-dual versions of these equations.