Noether symmetries for fields and branes in backgrounds with Killing vectors

Abstract

Within the formulation of classical fields and branes in curved spacetime, we first discuss the Belinfante construction of an improved energy-momentum tensor density in curved backgrounds, in analogy to the case of flat spacetime. A novelty of our approach is the use of Noether identities for generally covariant theories as a guiding instrument. In this part of the paper, our basic results, which hold irrespective of the background being dynamical or fixed, agree with those previously obtained in Gamboa Saravi (2004 J. Phys. A: Math. Gen. 37 9573). We illustrate with the example of Maxwell theory the role played by the Belinfante tensor in the construction of the Noether conserved currents and symmetry generators associated with the Killing symmetries of the background, which become Noether symmetres for the matter Lagrangian. We also give some details on the existing obstructions to the realization of part of these symmetries in phase space. Next we study extended objects (p-branes) either in a dynamic or in a fixed background, and obtain the Noether identities associated both with target spacetime and world volume diffeomorphisms. We show that in field theory as well as with extended objects, the Killing symmetries of the background become ordinary rigid Noether symmetries of the theory in this fixed background.