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arXiv · 1208.3345

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

Anthony J. Guttmann·Mathew D. Rogers
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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.

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Anthony J. Guttmann, Mathew D. Rogers. 2012-08-16. An integral arising from the chiral sl(n) Potts model. https://doi.org/10.1088/1751-8113/46/4/045202

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