Two-dimensional N=(2,2) supersymmetry (2D $\mathcal{N}=(2,2)$ SUSY)

Two-dimensional $\mathcal{N}=(2,2)$ supersymmetry has independent left- and right-moving complex supercharges. Its superspace uses ${\theta }^{\pm },{\bar{\theta }}^{\pm }$ and supports both chiral and twisted chiral superfields.

A twisted chiral superfield obeys ${\bar{D}}_{+}\Sigma =D_{-}\Sigma =0$. In an Abelian gauge theory the field-strength multiplet $\Sigma ={\bar{D}}_{+}{D}_{-}V$ is gauge invariant and twisted chiral.

A twisted superpotential is a holomorphic function of twisted chiral superfields integrated over $d{\theta }^{+}d{\bar{\theta }}^{-}$ superspace. Its highest component varies by a spacetime total derivative under two-dimensional N=(2,2) supersymmetry.