On connection matrices of quantum Knizhnik-Zamolodchikov equations based on Lie super algebras

TL;DR

This paper introduces a novel method for computing connection matrices of quantum Knizhnik-Zamolodchikov equations linked to super algebra-based integrable models, exemplified by the $ ext{U}_q( ext{gl}(2|1))$ Perk-Schultz model, revealing elliptic solutions to supersymmetric dynamical Yang-Baxter equations.

Contribution

It develops a new approach to compute connection matrices for quantum KZ equations using decomposition into principal series modules, applied to super algebra models.

Findings

Connection matrices described by elliptic solutions of supersymmetric dynamical Yang-Baxter equations.

Method successfully applied to the $ ext{U}_q( ext{gl}(2|1))$ Perk-Schultz model.

Provides a systematic way to handle super algebra symmetries in integrable models.

Abstract

We propose a new method to compute connection matrices of quantum Knizhnik-Zamolodchikov equations associated to integrable vertex models with super algebra and Hecke algebra symmetries. The scheme relies on decomposing the underlying spin representation of the affine Hecke algebra in principal series modules and invoking the known solution of the connection problem for quantum affine Knizhnik-Zamolodchikov equations associated to principal series modules. We apply the method to the spin representation underlying the $\mathcal{U}_q\bigl(\hat{\mathfrak{gl}}(2|1)\bigr)$ Perk-Schultz model. We show that the corresponding connection matrices are described by an elliptic solution of a supersymmetric version of the dynamical quantum Yang-Baxter equation with spectral parameter.