The holomorphic effective action in $N$=2 $D$=4 supergauge theories with various gauge groups

We consider the theory of hypermultiplets in arbitrary representations of arbitrary semisimple gauge groups coupled to gauge superfields. Using the $N=2$ harmonic superspace formulation of these models, we find the general structure of the holomorphic effective action depending on the gauge superfield with values in the Cartan subalgebra of the gauge algebra. We find explicit expressions for the effective actions in the cases where the hypermultiplets are in the fundamental and adjoint representations of $SU(n)$, $SO(n)$, and $Sp(2n)$.