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

Abstract

We show that the integral $J(t)=(1/{\pi }^3){\int }_0^{\pi }{\int }_0^{\pi }{\int }_0^{\pi }dxdydz\log (t-\cos x-\cos y-\cos z+\cos x\cos y\cos z)$, can be expressed in terms of ${}_5{F}_4$ 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.

References in corpus (1)

• Spanning tree generating functions and Mahler measures