Topological Terms and Diffeomorphism Anomalies in Fluid Dynamics and Sigma Models

The requirement of diffeomorphism symmetry for the target space can lead to anomalous commutators for the energy-momentum tensor for sigma models and for fluid dynamics, if certain topological terms are added to the action. We analyze several examples . A particular topological term is shown to lead to the known effective hydrodynamics of a dense collection of vortices, i.e. the vortex fluid theory in 2+1 dimensions. The possibility of a similar vortex fluid in 3+1 dimensions, as well as a fluid of knots and links, with possible extended diffeomorphism algebras is also discussed.