 Solution

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