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 superfield 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.