# Holographic DC conductivity and Onsager relations

## TL;DR

This paper derives a variational principle for holographic DC conductivity, enabling straightforward derivation of Onsager relations and analysis of anomalous Hall effects in four-dimensional gravity theories with broken translation symmetry.

## Contribution

It introduces a novel variational principle for holographic DC conductivity, facilitating the derivation of Onsager relations and extending the analysis to theories with topological terms and broken translation symmetry.

## Key findings

- Derived a variational principle for holographic DC conductivity.
- Obtained Stokes equations for gravity theories with a $ heta F ilde{F}$ term.
- Analyzed anomalous Hall conductivity and solved for holographic lattices.

## Abstract

Within holography the DC conductivity can be obtained by solving a system of Stokes equations for an auxiliary fluid living on the black hole horizon. We show that these equations can be derived from a novel variational principle involving a functional that depends on the fluid variables of interest as well as the time reversed quantities. This leads to simple derivation of the Onsager relations for the conductivity. We also obtain the relevant Stokes equations for bulk theories of gravity in four dimensions including a $\vartheta F\wedge F$ term in the Lagrangian, where $\vartheta$ is a function of dynamical scalar fields. We discuss various realisations of the anomalous Hall conductivity that this term induces and also solve the Stokes equations for holographic lattices which break translations in one spatial dimension.