superfield and modify the above action to the following form (for simplicity we ignore chiral fields which are charged under the gauge group), $$\begin{aligned}
{\cal A} & = & -3\int d^{8}z{\bf E}\exp[-\frac{1}{3}K(\phi,\bar{\phi},U)](F(\phi,\bar{\phi},U)+U(S+\bar{S}))\nonumber \\
 &  & +\left(\int d^{8}z\frac{{\bf E}}{2R}[W(\phi)+\frac{1}{4}kS{\cal W}{\cal W}]+h.c.\right).\label{eq:Slinchiral}\end{aligned}$$ Here a trace over the gauge group is implicit in the gauge kinetic term, $F$ is a real function of the chiral fields $\phi$ and the real field $U$, which will be determined by a normalization condition below. $U$ is introduced as the linear superspace dual of the chiral superfield $S$ (for more details see [@Derendinger:1991hq] [@Binetruy:2000zx]) and as we will see this field becomes useful in interpreting threshold corrections in string theory. Note that we have also modified the Kähler potential to include dependence on $U$. Now we may eliminate the chiral superfield $S$ in favor of the real superfield $U$, by using the equation of motion coming from taking the $\delta S$ variation of this action to get; [^1] $$-3(-\frac{1}{4}\bar{\nabla}^{2}+2R)(Ue^{-K/3})+\frac{k}{4}{\cal W}^{2}=0.\label{eq:deltaS}$$ Note that this equation and its conjugate are now effectively constraints on the initially unconstrained superfield $U$. In the absence of the gauge field kinetic term and $K$ this is essentially the linear superfield constraint. Here we 