  The action of the $S$-duality $Sl(2,Z)$ group on the moduli of the Calabi-Yau
manifold $W\IP^{12}_{11226}$ appearing in the rank two dual pair $(K^{3}\times
T^{2}/W\IP^{12}_{11226})$ is defined by interpreting the $N\!=\!4$ to $N\!=\!2$
flow, for $SU(2)$ supersymmetric Yang-Mills, in terms of the Calabi-Yau moduli.
The different singularity loci are mapped in a one to one way, and the
($N\!=\!2$ limit/point particle limit) is obtained in both cases by the same
type of blow up. Moreover, it is shown that the $S$-duality group permutes the
different singularity loci of the moduli of $W\IP^{12}_{11226}$. We study the
transformation under $S$-duality of the Calabi-Yau Yukawa couplings.
