Hidden supersymmetry as a key to constructing low-energy superfield effective actions

In this review paper, we outline and exemplify the general method of constructing the supefield low-energy quantum effective action of supersymmetric Yang-Mills (SYM) theories with extended supersymmetry in the Coulomb phase, grounded upon the requirement of invariance under the non-manifest (hidden) part of the underlying supersymmetry. In this way we restore the $\mathcal{N}=4$ supersymmetric effective actions in $4D,\mathcal{N}=4$ SYM, $\mathcal{N}=2$ supersymmetric effective actions in $5D,\mathcal{N}=2$ SYM and $\mathcal{N}=(1,1)$ supersymmetric effective actions in $6D,\mathcal{N}=(1,1)$ SYM theories. The manifest off-shell fractions of the full supersymmetry are, respectively, $4D,\mathcal{N}=2$, $5D,\mathcal{N}=1$ and $6D,\mathcal{N}=(1,0)$ supersymmetries. In all cases the effective actions depend on the corresponding covariant superfield SYM strengths and the hypermultiplet superfields. The whole construction essentially exploits a power of the harmonic superspace formalism.