Ward identities and relations between conductivities and viscosities in holography

We derive relations between viscosities and momentum conductivity in $2+1$ dimensions by finding a generalization of holographic Ward identities for the energy-momentum tensor. The generalization is novel in the sense that it goes beyond the usual identities obtained from holographic renormalization. Our results are consistent with previous field theory analysis. The main tools we use are a constant `probability current' in the gravity dual, that we are able to define for any system of linear ODEs, and parity symmetry.