Shear mode transport coefficients from multiple polylogarithms

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 coefficients of the dispersion relation for N = 4 super Yang-Mills theory, by performing a bulk computation in the five-dimensional black hole background up to order q 10 , which extends the results previously available in the literature. We then generalize the procedure to d + 1 dimensions, characterizing the mathematical structure of the resulting transport coefficient expressions.