# Diffusion constant of slowly rotating black three-brane

## TL;DR

This paper investigates the diffusion properties of a slowly rotating black three-brane by analyzing gauge field perturbations, revealing a complex diffusion constant linked to angular momentum and hydrodynamic damping.

## Contribution

It introduces the calculation of a complex diffusion constant in a rotating black brane background, connecting supergravity solutions to hydrodynamic behavior in the dual field theory.

## Key findings

- The diffusion constant is complex, with an imaginary part due to angular momentum.
- Fick's laws are derived in the long wavelength regime for the rotating black brane.
- The static limit recovers the known diffusion constant for non-rotating branes.

## Abstract

In this paper, we take the slowly rotating black three-brane background and perturb it by introducing a vector gauge field. We find the components of the gauge field through Maxwell equations and Bianchi identities. Using currents and some ansatz we find Fick's first law at long wavelength regime. An interesting result for this non-trivial supergravity background is that the diffusion constant on the stretched horizon which emerges from Fick's first law is a complex constant. The pure imaginary part of the diffusion constant appears because the black three-brane has angular momentum. By taking the static limit of the corresponding black brane the well known diffusion constant will be recovered. On the other hand, from the point of view of the Fick's second law, we have the dispersion relation $\omega=-iDq^{2}$ and we found a damping of hydrodynamical flow in the holographically dual theory. Existence of imaginary term in the diffusion constant introduces an oscillating propagation of the gauge field in the dual field theory.