  We show that a hot rotating fluid of relativistic chiral fermions possesses a
new gapless collective mode associated with coherent propagation of energy
density and chiral density waves along the axis of rotation. This mode, which
we call the Chiral Heat Wave, emerges due to a mixed gauge-gravitational
anomaly. At finite density the Chiral Heat Wave couples to the Chiral Vortical
Wave while in the presence of an external magnetic field it mixes with the
Chiral Magnetic Wave. The coupling of the Chiral Magnetic and Chiral Vortical
Waves is also demonstrated. We find that the coupled waves - which are coherent
fluctuations of the vector, axial and energy currents - have generally
different velocities compared to the velocities of the individual waves.
