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

## TL;DR

This paper identifies the 2D conformal field theory structures underlying 4D $ ext{N}=1$ class $ ext{S}_k$ gauge theories, linking conformal blocks to Seiberg-Witten curves and instanton partition functions.

## Contribution

It establishes the specific 2D CFT symmetry algebra and its representations that correspond to 4D $ ext{N}=1$ class $ ext{S}_k$ theories, including non-unitary representations.

## Key findings

- Conformal blocks reproduce Seiberg-Witten curves.
- Blocks involve non-unitary $W_{kN}$ algebra representations.
- Predicts instanton partition functions for 4D $ ext{N}=1$ SCFTs.

## Abstract

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 $\mathcal{S}_k$.