An effective theory of anomalous charge diffusion from holography

We consider a probe $U\left(1\right)$ Maxwell-Chern-Simons theory in 5-dimensions, and analyze the vector perturbations around a classical charged black-brane background. We solve the equations of motion for these perturbations in a derivative expansion. By computing the boundary current, we find that time and spatial derivatives can be interpreted as the induced electric and magnetic field respectively, and the Chern-Simons term contributes to a nonzero divergence of the boundary current which indicates a quantum anomaly. Using holography, we construct a two-derivative effective action for the vector perturbations. By complexifying the radial coordinate, and using appropriate transformation, we construct the full solution on the complexified bulk contour. By computing the on-shell action for the full Schwinger-Keldysh geometry, we obtain the Keldysh functional. We find that the single boundary on-shell action mixes parity, whereas the Keldysh functional does not depend on the Chern-Simons term up to the quadratic orders in derivative expansion.