N=1 theories of class Sk

We construct classes of N=1$$ \mathcal{N}=1 $$ superconformal theories elements of which are labeled by punctured Riemann surfaces. Degenerations of the surfaces correspond, in some cases, to weak coupling limits. Different classes are labeled by two integers (N, k). The k = 1 case coincides with AN − 1N=2$$ \mathcal{N}=2 $$ theories of class S$$ \mathcal{S} $$ and simple examples of theories with k > 1 are ℤk$$ {\mathrm{\mathbb{Z}}}_k $$ …