$U(1)$ current from the AdS/CFT: diffusion, conductivity and causality

Understand

For a holographically defined finite temperature theory, we derive an off-shell constitutive relation for a global $U(1)$ current driven by a weak external non-dynamical electromagnetic field.

• The constitutive relation involves an all order gradient expansion resummed into three momenta-dependent transport coefficient functions: diffusion, electric conductivity, and "magnetic" conductivity.
• These transport functions are first computed analytically in the hydrodynamic limit, up to third order in the derivative expansion, and then numerically for generic values of momenta.
• We also compute a diffusion memory function, which, as a result of all order gradient resummation, is found to be causal.