An integral arising from the chiral sl(n) Potts model

TL;DR

This paper expresses a complex integral related to the $sl(n)$ Potts model in terms of hypergeometric functions, linking statistical mechanics, Mahler measures, and special functions.

Contribution

It provides a new hypergeometric representation of an integral from the $sl(n)$ Potts model, connecting it to Mahler measures and special functions.

Findings

Integral expressed in terms of ${_5F_4}$ hypergeometric functions

Links between Potts model free-energy and Mahler measures established

Explicit formula for the integral $J(t)$ at specific points

Abstract

We show that the integral $J(t) = (1/\pi^3) \int_0^\pi \int_0^\pi \int_0^\pi dx dy dz \log(t - \cos{x} - \cos{y} - \cos{z} + \cos{x}\cos{y}\cos{z})$, can be expressed in terms of ${_5F_4}$ hypergeometric functions. The integral arises in the solution by Baxter and Bazhanov of the free-energy of the $sl(n)$ Potts model, which includes the term $J(2)$. Our result immediately gives the logarithmic Mahler measure of the Laurent polynomial $k - (x+1/x) - (y+1/y) - (z+1/z) + 1/4(x+1/x) (y+1/y) (z+1/z)$ in terms of the same hypergeometric functions.