and is very different from the linear multiplet formulation where it was precisely the constraint on $U$ that enabled us to write a non-holomorphic gauge coupling.

What this illustrates is the general expectation that the dualization makes sense only when there when the dualized field pair represents a massless state. The linear superpotential case studied at the beginning of this subsection is thus an exception.

Generalization
--------------

The discussion in the previous two subsections can be easily generalized to the case when there is a duality relation between several chiral multiplets $S^{i},\, i=0,\ldots,N$ and linear multiplets $U_{i}$. For future reference in the type IIB string theory case we rewrite these fields as $S\equiv S^{0}$ and $T^{i}\equiv S^{i},\, i=1,\ldots,N$. We also choose to dualize only the $T'$s and write the dual fields as $U_{i},\, i=1,\ldots,N$. The gauge field kinetic term (in the chiral formulation) is now taken to be $f_{a}{\cal W}_{a}^{2}$ where a sum over the different gauge groups is implied, and $$f_{a}=k_{a}S+\alpha_{ai}T^{i}$$ One follows the same steps from eqn through replacing the appropriate products by dot products etc. to get after dualization of the $T'$s the gauge coupling function, $$k_{a}S+\Gamma_{a},\,\Gamma_{a}=\Gamma^{i}\alpha_{ai},\,\Gamma^{i}=-F_{U_{i}}+\frac{1}{3}K_{U_{i}}.\label{eq:Gamma3}$$ Note that we’ve set the normalization function $$N\equiv\frac{F-U_{i}F_{U_{i}}}{1-\frac{1}{3}U_{i}K_{U_{i}}}=1,\label{eq:N}$$ in the 