$\widehat{\mathfrak{s}\mathfrak{l}}(n{)}_N$ WZW conformal blocks from $SU(N)$ instanton partition functions on ${\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_n$

Generalizations of the AGT correspondence between 4D $\mathcal{N}=2$ $SU(2)$ supersymmetric gauge theory on ${\mathbb{C}}^2$ with $\mathrm{\Omega }$-deformation and 2D Liouville conformal field theory include a correspondence between 4D $\mathcal{N}=2$ $SU(N)$ supersymmetric gauge theories, $N=2,3,\dots$, on ${\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_n$, $n=2,3,\dots$, with $\mathrm{\Omega }$-deformation and 2D conformal field theories with $$\mathcal{W}$^{, para}_{N, n}$($n$-thparafermion$$\mathcal{W}$_N$)symmetryand$\widehat{\mathfrak{sl}}(n)_N$ symmetry. In this work, we trivialize the factor with ${\mathcal{W}}_{N,n}^{para}$ symmetry in the 4D $SU(N)$ instanton partition functions on ${\mathbb{C}}^2\mathrm{/}{\mathbb{Z}}_n$ (by using specific choices of parameters and imposing specific conditions on the $N$-tuples of Young diagrams that label the states), and extract the 2D $\widehat{\mathfrak{s}\mathfrak{l}}(n{)}_N$ WZW conformal blocks, $n=2,3,\dots$, $N = 1, 2, \ldots, .$