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For the stated two-dimensional N=(2,2) supersymmetry conventions, define the supersymmetric covariant derivativesThe terms in which a Grassmann derivative hits the explicit Grassmann coordinate cancel the spacetime-derivative terms, while derivatives involving different signs act on independent coordinates. Therefore

The twisted chiral superfield constraints $D_{-}U={\bar{D}}_{+}U=0$ are solved by the twisted chiral coordinatesThe superfield depends only on $({\tilde{y}}^{\pm };{\theta }^{+},{\bar{\theta }}^{-})$ and has the finite Grassmann variable expansionwhere every component on the right is evaluated at $\tilde{y}$; numerical $\sqrt{2}$ factors may be absorbed into the component definitions.

For one chiral superfield $\Phi$ and one twisted chiral superfield $U$, the most general local two-derivative supersymmetric action isHere $K$ is real, $W$ is a holomorphic superpotential, and $\tilde{W}$ is a holomorphic twisted superpotential. A full superspace integral, a chiral F-term, and a twisted F-term each vary by a spacetime or Berezin total derivative, so all three are supersymmetric.

Choose the Vector R-symmetry and Axial R-symmetry conventionswith conjugate coordinates transforming oppositely, and assign compatible charges to $\Phi$ and $U$. The D-term is invariant when $K$ is neutral, up to a generalized Kähler transformation. The chiral measure has vector R-charge $-2$ and axial charge zero, whereas the twisted chiral measure has axial R-charge $-2$ and vector charge zero. Hence classical invariance requirestogether with neutrality of $K$. Equivalently, $W$ and $\tilde{W}$ must be quasi-homogeneous with these charges; absent suitable charge assignments, the corresponding superpotential breaks that R-symmetry.

Let $\Lambda$ be a chiral gauge parameter. One consistent Abelian supergauge transformation convention isfor which ${\bar{\Phi }}_i{e}^{q_{i}V}{\Phi }_i$ is invariant. Wess-Zumino gauge uses the nonordinary components of $\Lambda$ to remove the superfluous scalar and spinor components of $V$, leaving the photon, gauginos, complex scalar, and real auxiliary field of the two-dimensional vector multiplet. Ordinary $U(1)$ gauge transformations remain.

Define the field-strength multiplet, up to conventional normalization, byGauge invariance follows because chirality, antichirality, ${\bar{D}}_{+}^2=D_{-}^2=0$, and $\{{\bar{D}}_{+},D_{-}\}=0$ annihilate the variation of $V$. The same identities giveso $\Sigma$ is a twisted chiral superfield. The Fayet–Iliopoulos term is a twisted F-term. In the standard axial convention its measure $d{\theta }^{+}d{\bar{\theta }}^{-}$ has axial charge $-2$, so invariance requiresReversing all axial-charge conventions reverses both signs but leaves this statement unchanged: the field and measure have opposite charges.

The charged matter fermions are chiral with respect to the axial symmetry. In a background with gauge flux, their functional measure has the two-dimensional axial anomalyup to orientation and current normalization. Equivalently, a Fujikawa transformation multiplies the torus path integral by a phase proportional to $\alpha ({\sum }_i{q}_i)\int F$. Since the background flux may be nonzero, the continuous axial $U(1)$ survives quantum mechanically precisely when