V}}$
   2$\frac{1}{g_{a}^{2}}$   $k_{a}(S+\bar{S})+\alpha_{ai}(T^{i}+\bar{T}^{i})$   $k_{a}(S+\bar{S})+\alpha_{ai}(\frac{2}{3}\frac{c^{ijk}U_{j}U_{k}}{\hat{{\cal V}}}-V^{i})$

$$$$

It is important to note that the functions $V^{i}$ do not depend on the chiral moduli $T$ which have been dualized. Also even though these functions are arbitrary at the level of these duality transformation, the usefulness of these results lies in the fact that such functions may arise in string perturbation theory. In this case of course (at one loop level) they are expected to be independent of $S$.

Relation to KL formula
----------------------

What is the relation if any to the KL formula of the above?

The KL formula [@Kaplunovsky:1994fg] for the gauge coupling constant for a (simple or $U(1)$ gauge group $G_{a}$) is $$\begin{aligned}
\frac{1}{g_{a}^{2}(\Phi,\bar{\Phi};\mu^{2})} & = & \Re f_{a}(\Phi)+\frac{b_{a}}{16\pi^{2}}\ln\frac{\Lambda^{2}}{\mu^{2}}+\frac{c_{a}}{16\pi^{2}}K(\Phi,\bar{\Phi})\nonumber \\
 &  & +\frac{T(G_{a})}{8\pi^{2}}\ln\frac{1}{g_{a}^{2}(\Phi,\bar{\Phi};\mu^{2})}-\sum_{r}\frac{T_{a}(r)}{8\pi^{2}}\ln\det Z_{(r)}(\Phi,\bar{\Phi};g^{2}(\mu^{2})).\label{eq:KL}\end{aligned}$$ Here $T_{a}(r)={\rm Tr}_{(r)}T_{a}^{2},\, T(G_{a})=T_{a}(r={\rm adjoint})$, $$\begin{aligned}
b_{a} & = & \sum_{r}n_{r}T_{a}(r)-3T(G_{a}),\label{eq:ba}\\
c_{a} & = & \sum_{r}n_{r}T_{a}(r)-T(G_{a}).\label{eq:ca}\end{aligned}$$ $\Phi$ stands for a set of neutral fields that will be identified with string theory moduli $(M)$ and axio-dilaton $(S)$. The total Kaehler potential has been expanded in powers of the charged matter chiral fields $Q$ which also defines the metric $Z_{(r)}$ of their kinetic terms in the representation $r$ of the gauge group, i.e. $$K(\Phi,\bar{\Phi};Q,\bar{Q})=\kappa^{-2}K(\Phi,\bar{\Phi})+\sum_{r}Z_{(r)\bar{I}J}(\Phi,\bar{\Phi})\bar{Q}_{(r)}^{\bar{I}}e^{2V}Q_{(r)}^{J}+\ldots.\label{eq:Kexpn}$$ The above formula was 