Holographic KMS relations for finite density fermions

We extend the holographic Schwinger-Keldysh prescription introduced in \cite{Glorioso:2018mmw} to charged black branes, with a view towards studying Hawking radiation in these backgrounds. Equivalently we study real time fluctuations of the dual CFT held at finite temperature and finite chemical potential. We check our prescription using charged Dirac probe fields. We solve the Dirac equation in a boundary derivative expansion extending the results in \cite{Loganayagam:2020eue}. The Schwinger-Keldysh correlators derived using this prescription automatically satisfy the appropriate KMS relations with Fermi--Dirac factors.