  Euler hydrodynamics of perfect fluids can be viewed as an effective bosonic
field theory. In cases when the underlying microscopic system involves Dirac
fermions, the quantum anomalies should be properly described. In 1+1 dimensions
the action formulation of hydrodynamics at zero temperature is reconsidered and
shown to be equal to standard field-theory bosonization. Furthermore, it can be
derived from a topological gauge theory in one extra dimension, which
identifies the fluid variables through the anomaly inflow relations. Extending
this framework to 3+1 dimensions yields an effective field theory/hydrodynamics
model, capable of elucidating the mixed axial-vector and axial-gravitational
anomalies of Dirac fermions. This formulation provides a platform for
bosonization in higher dimensions. Moreover, the connection with 4+1
dimensional topological theories suggests some generalizations of fluid
dynamics involving additional degrees of freedom.
