Exact results for class $S_k$

We will introduce a large class of $\mathcal{N}=1$ superconformal theories, called
$S_k$, which is obtained from Gaiotto's $\mathcal{N}=2$ class $S$ via orbifolding. We
will study the Coulomb branch of the theories in the class by
constructing and analyzing their spectral curves. Using our experience
from the $\mathcal{N}=2$ \textsc{agt} correspondence we will search for a 2d/4d relations
(\textsc{agt}${}_{k}$) for the $\mathcal{N}=1$ theories of class $S_k$. From the curves we will
identify the 2d \textsc{cft} symmetry algebra and its representations, namely
the conformal blocks of the Virasoro/W-algebra, that underlie the 2d
theory and reproduce the Seiberg-Witten curves of the $\mathcal{N} = 1$ gauge
theories. We find that the blocks corresponding to the ${\rm SU}(N)$ $S_k$ gauge
theories involve fields in certain non-unitary representations of the
$W_{k,N}$ algebra. These conformal blocks give a prediction for the
instanton partition functions of the 4d $\mathcal{N} = 1$ \textsc{scft}s of class $S_k$.