Chirality of neutrinos modifies the conventional kinetic theory and hydrodynamics, leading to unusual chiral transport related to quantum anomalies in field theory. We argue that these corrections have new phenomenological consequences for hot and dense neutrino gases, especially in core-collapse supernovae. We find that the neutrino density can be converted to the fluid helicity through the chiral vortical effect. This fluid helicity effectively acts as a chiral chemical potential for electrons via the momentum exchange with neutrinos and induces a "helical plasma instability" that generates a strong helical magnetic field. This provides a new mechanism for converting the gravitational energy released by the core collapse to the electromagnetic energy and potentially explains the origin of magnetars. The other possible applications of the neutrino chiral transport theory are also discussed.
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