Supersymmetric BMS$_4$ Algebras Revisited: Electric/Magnetic Superalgebras and Free Field Realization

In this work, we present a systematic classification of supersymmetric extensions of the BMS$_4$ algebra and their realizations in free field theories. By requiring that supercharges admit finite-dimensional subsectors, we identify ten distinct electric super BMS$_4$ algebras and six magnetic ones. The electric case is characterized by supercharge anticommutators closing on supertranslations, while the magnetic case necessarily involves superrotations. To realize these algebras in free field theories, we follow a constructive procedure: first identify the modes bilinears of which generate the symmetry algebra, then determine the fields with appropriate transformation properties under the BMS$_4$ algebra, and finally construct consistent theories whose equations of motion admit the desired supersymmetry. Notably, $R$-symmetry with nonvanishing spin is essential for the Type II-II, Type I-II, and Type II-I magnetic super BMS$_4$ algebras, shedding new light on spacetime structure of string theory for flat holography. Moreover, in the Type I-I theory, $R$-symmetry relates electric and magnetic scalars, indicating their equal significance. In addition, $R$-symmetry maps the electric scalar to a spin-$1$ field and vice versa, offering a novel perspective on supersymmetric extensions of soft theorems.