  The exponential amplification of initial seed magnetic fields in relativistic
plasmas is a very important topic in astrophysics, from the conditions in the
early Universe to the interior of neutron stars. While dynamo action in a
turbulent plasma is often invoked, in the last years a novel mechanism of
quantum origin has gained increasingly more attention, namely the Chiral
Magnetic Effect (CME). This has been recognized in semi-metals and it is most
likely at work in the quark-gluon plasma formed in heavy-ion collision
experiments, where the highest magnetic fields in nature, up to B~10^18 G, are
produced. This effect is expected to survive even at large hydrodynamical/MHD
scales and it is based on the chiral anomaly due to an imbalance between left-
and right-handed relativistic fermions in the constituent plasma. Such
imbalance leads to an electric current parallel to an external magnetic field,
which is precisely the same mechanism of an alpha-dynamo action in classical
MHD. Here we extend the close parallelism between the chiral and the dynamo
effects to relativistic plasmas and we propose a unified, fully covariant
formulation of the generalized Ohm's law. Moreover, we derive for the first
time the 3+1 general relativistic MHD equations for a chiral plasma both in
flat and curved spacetimes, in view of numerical investigation of the CME in
compact objects, especially magnetars, or of the interplay among the non-ideal
magnetic effects of dynamo, the CME and reconnection.
