About 30 Years of Integrable Chiral Potts Model, Quantum Groups at Roots of Unity and Cyclic Hypergeometric Functions

TL;DR

This paper reviews 30 years of the integrable chiral Potts model, highlighting its connections to quantum groups at roots of unity, hypergeometric functions, and knot theory, and discusses new identities for hypergeometric series arising from integrability.

Contribution

It introduces new summation identities for hypergeometric series derived from the integrable chiral Potts model and explores its relation to quantum groups at roots of unity.

Findings

Star-triangle equation as a summation formula for ${}_4\Phi_3$ series

Periodic summands enable summation of hypergeometric series at roots of unity

Connections established between integrable models, hypergeometric functions, and quantum groups

Abstract

In this paper we discuss the integrable chiral Potts model, as it clearly relates to how we got befriended with Vaughan Jones, whose birthday we celebrated at the Qinhuangdao meeting. Remarkably we can also celebrate the birthday of the model, as it has been introduced about 30 years ago as the first solution of the star-triangle equations parametrized in terms of higher genus functions. After introducing the most general checkerboard Yang--Baxter equation, we specialize to the star-triangle equation, also discussing its relation with knot theory. Then we show how the integrable chiral Potts model leads to special identities for basic hypergeometric series in the $q$ a root-of-unity limit. Many of the well-known summation formulae for basic hypergeometric series do not work in this case. However, if we require the summand to be periodic, then there are many summable series. For example, the integrability condition, namely, the star-triangle equation, is a summation formula for a well-balanced ${}_4\Phi_3$ series. We finish with a few remarks about the relation with quantum groups at roots of unity.