 Two-dimensional N=(1,1) superspace

In light-cone coordinates $x^{\pm }$ on two-dimensional Minkowski spacetime, add two real odd Grassmann variables ${\theta }^{\pm }$. With left Grassmann derivatives, the supercharges and supersymmetric covariant derivatives can be represented asThe identity $\{{\partial }_{\theta },\theta \}=1$ gives $Q_{\pm }^2=i{\partial }_{\pm }$, $D_{\pm }^2=-i{\partial }_{\pm }$, $\{Q_{+},Q_{-}\}=\{D_{+},D_{-}\}=0$, and $\{Q_s,D_t\}=0$ for every $s,t$. A Lorentz boost acts as $x^{\pm }\mapsto e^{\pm \omega }{x}^{\pm }$, ${\theta }^{\pm }\mapsto e^{\pm \omega /2}{\theta }^{\pm }$. Thus $D_{\pm }$ have boost weights $\mp 1/2$, and $D_{-}{D}_{+}$ is a Lorentz scalar. Since it commutes with both supercharges, an equation $i{D}_{-}{D}_{+}\Phi =W(\Phi )$ for an even scalar superfield is Lorentz invariant and supersymmetry covariant.