qKZ equations and ground state of the O(1) loop model with open boundary conditions

TL;DR

This paper solves the qKZ equations related to the two boundaries Temperley-Lieb algebra, connecting their solutions to the ground state of the dense O(1) loop model with open boundaries, and provides a new construction and sum rule.

Contribution

It introduces a recursive solution to the qKZ equations at a specific parameter and links it to the ground state of the O(1) loop model with open boundaries, including a novel sum rule.

Findings

Solution reduces to the ground state at the combinatorial point

Sum rule expressed as a product of four symplectic characters

Alternative construction based on mixed boundary conditions

Abstract

We consider the qKZ equations based on the two boundaries Temperley Lieb algebra. We construct their solution in the case $s=q^{-3/2}$ using a recursion relation. At the combinatorial point $q^{1/2}= e^{-2\pi i/3}$ the solution reduces to the ground state of the dense O(1) loop model on a strip with open boundary conditions. We present an alternative construction of such ground state based on the knowledge of the ground state of the same model with mixed boundary conditions and prove that the sum rule as of its components is given by the product of four symplectic characters.