SOS model partition function and the elliptic weight functions

TL;DR

This paper links the partition function of the SOS model with domain-wall boundary conditions to elliptic current algebra, providing explicit integral representations and generalizing previous results from the 6-vertex model.

Contribution

It extends the connection between partition functions and quantum affine algebra currents to the elliptic case, with explicit integral formulas.

Findings

Derived integral transform representations of the partition function.

Proved the kernel of the transform is proportional to the SOS model partition function.

Generalized previous results from the 6-vertex model to elliptic current algebra.

Abstract

We generalize a recent observation [arXiv:math/0610433] that the partition function of the 6-vertex model with domain-wall boundary conditions can be obtained by computing the projections of the product of the total currents in the quantum affine algebra $U_{q}(\hat{\mathfrak{sl}}_{2})$ in its current realization. A generalization is proved for the the elliptic current algebra [arXiv:q-alg/9703018,arXiv:q-alg/9601022]. The projections of the product of total currents are calculated explicitly and are represented as integral transforms of the product of the total currents. We prove that the kernel of this transform is proportional to the partition function of the SOS model with domain-wall boundary conditions.