Quantum flux operators in the fermionic theory and their supersymmetric extension

We construct quantum flux operators with respect to the Poincar\'e symmetry in the massless Dirac theory at future null infinity. An anomalous helicity flux operator emerges from the commutator of the superrotation generators. The helicity flux operator corresponds to the local chiral symmetry which is the analog of superduality in the gauge theories. We also find its relation to the non-closure of the Lie transport of the spinor field around a loop. We discuss various algebras formed by these operators and constrain the test functions by the requirement of eliminating the non-local terms and satisfying the Jacobi identities. Furthermore, we explore their $\mathcal{N}=1$ supersymmetric extension in the Wess-Zumino model. There are four kinds of quantum flux operators, which correspond to the supertranslation, superrotation, superduality and supersymmetry, respectively. Interestingly, besides the expected supertranslation generator, a helicity flux operator will also emerge in the commutator between the superflux operators. We check that our flux algebra can give rise to the super-BMS and super-Poincar\'e algebras with appropriate choice of parameters. In the latter reduction, we find the helicity flux reduces to behaving like a $R$ symmetry generator in the commutator with the superflux. For completion, we derive the $R$ flux which also includes a charge flux for complex scalar besides the helicity flux for spinor field.