Shear mode transport coefficients from multiple polylogarithms

TL;DR

The paper analyzes the analytic structure of shear sector transport coefficients in holographic theories by solving bulk gravitational perturbations around planar Schwarzschild anti-de Sitter backgrounds. It introduces a recursive construction in the small momentum parameter $\mathfrak{q}$ that expresses bulk solutions in terms of logs and multiple polylogarithms in several variables, enabling explicit coefficients up to $\mathfrak{q}^{10}$ in $d=4$ ($\mathcal{N}=4$ SYM) and generalizing to $d+1$ dimensions. The coefficients $\mathfrak{w}_k$ are shown to belong to colored multiple zeta values of weight $\le k-1$, with the level determined by the spacetime dimension, and a concrete five-term result for the five-dimensional case includes a new $\mathfrak{w}_5$. The work provides a unifying algebraic framework to organize higher-order hydrodynamic data in holography and exposes the central role of polylogarithmic structures in transport theory.

Abstract

We present an analytical study of the transport coefficients associated with the shear sector of gravitational perturbations around asymptotically anti-de Sitter black branes. In the long-wavelength, low-frequency limit, the wave solutions admit a structure that is fully described in terms of multiple polylogarithms in several variables. We focus primarily on computing the transport coefficients for $\mathcal{N}=4$ SYM, by performing a bulk computation in the five-dimensional black hole background up to order $\mathfrak{q}^{10}$, which extends the results previously available in the literature. We then generalise the procedure to $d+1$ dimensions, characterising the mathematical structure of the resulting transport coefficient expressions.