2D CFT blocks for the 4D class ${\mathcal{S}}_k$ theories

This is the first in a series of papers on the search for the 2D CFT description of a large class of 4D $\mathcal{N}$ = 1 gauge theories. Here, we identify the 2D 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 SU(N) ${\mathcal{S}}_k$ gauge theories involve fields in certain non-unitary representations of the W$_{kN}$ algebra. These conformal blocks give a prediction for the instanton partition functions of the 4D $\mathcal{N}$ = 1 SCFTs of class ${\mathcal{S}}_k$