Improvement of a conserved current density versus adding a total derivative to a Lagrangian density

Understand

For classical relativistic field theory in Minkowski space-time, the addition of a superpotential term to a conserved current density is trivial in the sense that it does not modify the local conservation law nor change the conserved charge, though it may allow us to obtain a current density with some improved properties.

• The addition of a total derivative term to a Lagrangian density is also trivial in the sense that it does not modify the equations of motion of the theory.
• These facts suggest that both operations are related and possibly equivalent to each other for any global symmetry of an action functional.
• We address this question following the study of two quite different (and well known) instances: the Callan-Coleman-Jackiw improvement of the canonical energy-momentum tensor for scalar and vector fields (providing an on-shell traceless energy-momentum tensor) and the construction of a current density satisfying a zero curvature condition for two-dimensional sigma models on deformed spaces (notably the squashed three-sphere and warped AdS spaces).