Smooth modules over the N = 1 Bondi–Metzner–Sachs superalgebra

In this paper, we present a determinant formula for a contravariant form on Verma modules over the $N=1$ Bondi–Metzner–Sachs (BMS) superalgebra. This formula establishes a necessary and sufficient condition for the irreducibility of the Verma modules. We then introduce and characterize a class of simple smooth modules that generalize both Verma and Whittaker modules over the $N=1$ BMS superalgebra. We also utilize the Heisenberg–Clifford vertex superalgebra to construct a free field realization for the $N=1$ BMS superalgebra. This free field realization allows us to obtain a family of natural smooth modules over the $N=1$ BMS superalgebra, which includes Fock modules and certain Whittaker modules.