Liouville mode in Gauge/Gravity Duality

TL;DR

This paper constructs inhomogeneous black hole solutions in AdS space using Liouville equations, analyzes their transport properties, and explores implications for strongly coupled dual media and condensed matter physics.

Contribution

It introduces new inhomogeneous black hole solutions with Liouville-type horizon geometries and studies their transport coefficients within gauge/gravity duality.

Findings

Exponential suppression of diffusion coefficient and shear viscosity ratio due to inhomogeneity.

Identification of trial distributions matching experimental shear viscosity ratios.

Extension of solutions to higher-dimensional AdS spaces and analysis of conductivity ratios.

Abstract

We establish solutions corresponding to AdS4 static charged black holes with inhomogeneous two-dimensional horizon surfaces of constant curvature. Depending on the choice of the 2D constant curvature space, the metric potential of the internal geometry of the horizon satisfies the elliptic wave/elliptic Liouville equations. We calculate the charge diffusion and transport coefficients in the hydrodynamic limit of gauge/gravity duality and observe the exponential suppression in the diffusion coefficient and in the shear viscosity-per-entropy density ratio in the presence of an inhomogeneity on black hole horizons with planar, spherical, and hyperbolic geometry. We discuss the subtleties of the approach developed for a planar black hole with inhomogeneity distribution on the horizon surface in more detail and find, among others, a trial distribution function, which generates values of the shear viscosity-per-entropy density ratio falling within the experimentally relevant range. The solutions obtained are also extended to higher-dimensional AdS space. We observe two different DC conductivities in 4D and higher-dimensional effective strongly coupled dual media and formulate conditions under which the appropriate ratio of different conductivities is qualitatively the same as that observed in an anisotropic strongly coupled fluid. We briefly discuss ways of how the Liouville field could appear in condensed matter physics and outline prospects of further employing the gauge/gravity duality in CMP problems.