be unity. Substituting into gives the chiral field formulation with the standard chiral gauge coupling function i.e. . Note also that eliminating $(S+\bar{S})$ between the last two displayed equations gives the relation $$1-\frac{1}{3}UK_{U}=F-UF_{U}\label{eq:norm2}$$ so that the function $N=1$ (see ) and the expression for $\Gamma$ becomes $$\Gamma=-F_{U}+\frac{1}{3}K_{U}.\label{eq:Gamma2}$$

Equation can be rewritten as a differential equation for F, $$-U^{2}\frac{d}{dU}(U^{-1}F)=1-\frac{1}{3}UK_{U}.$$ This has the solution $$F=1+\frac{U}{3}\int^{U}\frac{dU'}{U'}K_{U'}+V(\phi,\bar{\phi})U.\label{eq:Ssoln}$$ A simple example that is relevant to string theory is obtained by putting $$K=\hat{K}(\phi,\bar{\phi})+\alpha\ln U.\label{eq:Kexample}$$ In this case $F=1-\frac{\alpha}{3}+VU$ and $\frac{\alpha}{3}U^{-1}=S+\bar{S}+V(\phi,\bar{\phi})$. Then in the chiral multiplet formulation we have the action with $f=kS$ and $$K=\hat{K}(\phi,\bar{\phi})-\alpha\ln(S+\bar{S}+V(\phi,\bar{\phi})).\label{eq:KexampleCMF}$$ In the string theory case where $S$ is the four dimensional dilaton chiral superfield $\alpha=1$ and $V$ is a one-loop effect.

On the other hand we have in the linear multiplet formulation the action with Kaehler potential , the gauge coupling (see ) given by (putting $\alpha=1)$ $$\Gamma=-\frac{1}{3U}+V(\phi,\bar{\phi}).\label{eq:Gammamodel}$$ The moral of this story is that if one wants to accommodate a non-harmonic gauge coupling function (as in the above relation) one must choose the LM formulation. On the other hand this is equivalent to taking the chiral multiplet dual field $S$ as the gauge coupling function in the CM formulation but with 