Triangle anomalies, thermodynamics, and hydrodynamics

We consider $3+1$-dimensional fluids with U(1)³ anomalies. We use Ward identities to constrain low-momentum Euclidean correlation functions and obtain differential equations that relate two and three-point functions. The solution to those equations yields, among other things, the chiral magnetic conductivity. We then compute zero-frequency functions in hydrodynamics and show that the consistency of the hydrodynamic theory also fixes the anomaly-induced conductivities.