The supercurrent and anomaly supermultiplets
P. Freund · Cambridge University Press eBooks · 1986
When the fields that appear in the lagrangian of a classical theory undergo the transformations dictated by an ordinary Lie algebra, the change of the action itself, according to the celebrated theorem of Emmy Noether, can be cast into the form of a space–time integral of the divergence of some currents. In particular, when the action is invariant under the effect of the Lie algebra these Noether currents (their number equal to the Lie algebra's dimension) are conserved. Not surprisingly, this theorem admits a straightforward generalization to the case of Lie superalgebras, i.e., to the supersymmetric case. Along with the vectorial and tensorial Bose currents, there will exist spinorial Fermi currents in this case (see chapter 1). In both ordinary and supersymmetric quantum theories, anomalies can break the classical conservation laws. We shall discuss here the classical conservation laws and their quantum demise via anomalies in a supersymmetric theory. As we shall see, the existence of a supermultiplet containing both the energy–momentum tensor and the chiral current at the classical level can lead to some paradoxical results in the quantum theory, unless sufficient care is exercised. Consider a field theory which classically is both N = 1 Poincaré supersymmetric and conformally invariant, i.e., invariant under the full N = 1 superconformal algebra su (2,2|1) (see chapter 4).