On AGT conjecture for pure super Yang-Mills and W-algebra

Recently Alday, Gaiotto and Tachikawa have proposed relation between 2- and 4-dimensional conformal field theories. The relation implies that the Nekrasov partition functions of \( \mathcal{N} = 2 \) superconformal gauge theories are equal to conformal blocks associated with the conformal algebra. Likewise, a counterpart in pure super Yang-Mills theory exists in conformal field theory. We propose a simple relation between the Shapovalov matrix of the \( {\mathcal{W}_3} \)-algebra and the Nekrasov partition function of \( \mathcal{N} = 2 \) SU(3) Yang-Mills theory.