function of the chiral superfields $\Phi$. However threshold effects in string theory appeared to give non-holomorphic moduli dependent corrections to the gauge coupling function. The resolution lay in the introduction of the linear multiplet formulation of the gauge coupling function. The key observation here is that string theory moduli and the dilaton naturally arise in the string theory context as (components of) linear multiplets. This is because axionic partners of the scalar moduli are in fact second rank tensor fields. Thus for instance the axio-dilaton $S$, which is often identified in 4D as a chiral scalar, has its origins in a multiplet containing an antisymmetric second rank tensor $b_{\mu\nu}$, and thus naturally belongs to a linear multiplet.

Let us first focus on this case where in the usual formulation the gauge coupling function is given by $f=kS$ where $S$ is the dilaton chiral superfield i.e. $\bar{\nabla}^{\dot{\alpha}}S=0.$ Let $\{\phi\}$ denote all the other chiral superfileds in the theory. We take (for the moment) the superpotential $W$ to be independent of $S$ as is the case in perturbative string theory (except in IIB where it can be linear in $S$ in the presence of internal fluxes). Let $U$ be an unconstrained real 