Higher Equations of Motion in $N=1$ SUSY Liouville Field Theory

Similarly to the ordinary bosonic Liouville field theory, in its $N=1$ supersymmetric version an infinite set of operator valued relations, the “higher equations of motions”, hold. Equations are in one to one correspondence with the singular representations of the super Virasoro algebra and enumerated by a couple of natural numbers $(m,n)$. We demonstrate explicitly these equations in the classical case, where the equations of type $(1,n)$ survive and can be interpreted directly as relations for classical fields. The general form of higher equations of motion is established in the quantum case, both for the Neveu-Schwarz and Ramond series.