Supersymmetric Liouville equation ($i{D}_{-}{D}_{+}\Phi =e^{\Phi }$)

In two-dimensional N=(1,1) superspace, one convention for the supersymmetric Liouville equation is $i{D}_{-}{D}_{+}\Phi =e^{\Phi }$. With the ordered expansion $\Phi =\varphi +i{\theta }^{-}{\psi }_{+}+i{\theta }^{+}{\psi }_{-}+i{\theta }^{-}{\theta }^{+}F$ and left Grassmann derivatives, its left side isWriting $N=\Phi -\varphi$, Grassmann parity gives $N^2=2{\theta }^{-}{\theta }^{+}{\psi }_{+}{\psi }_{-}$ and $N^3=0$. HenceComparing coefficients eliminates the auxiliary field as $F=e^{\varphi }$ and gives ${\partial }_{-}{\psi }_{-}=e^{\varphi }{\psi }_{+}$, ${\partial }_{+}{\psi }_{+}=-e^{\varphi }{\psi }_{-}$, ${\partial }_{-}{\partial }_{+}\varphi =-e^{2\varphi }+i{e}^{\varphi }{\psi }_{+}{\psi }_{-}$. The fermion bilinear sign depends on the stated ordering and derivative convention.