Elliptic pfaffians and solvable lattice models

TL;DR

This paper introduces elliptic pfaffians and demonstrates their application in expressing partition functions of solvable lattice models, revealing new connections between elliptic functions, pfaffians, and integrable models.

Contribution

It presents twelve new multivariable theta functions defined by pfaffians and links them to partition functions of the eight-vertex-solid-on-solid and three-colour models.

Findings

Partition function for the eight-vertex-solid-on-solid model expressed as a sum of two pfaffians.

Partition function for the three-colour model expressed as a sum of two Hankel determinants.

Solutions to the TQ-equation related to the supersymmetric eight-vertex model expressed via elliptic pfaffians.

Abstract

We introduce and study twelve multivariable theta functions defined by pfaffians with elliptic function entries. We show that, when the crossing parameter is a cubic root of unity, the domain wall partition function for the eight-vertex-solid-on-solid model can be written as a sum of two of these pfaffians. As a limit case, we express the domain wall partition function for the three-colour model as a sum of two Hankel determinants. We also show that certain solutions of the TQ-equation for the supersymmetric eight-vertex model can be expressed in terms of elliptic pfaffians.