Instanton counting in Class $\mathcal{S}_k$

We compute the instanton partition functions of $\mathcal{N}=1$ SCFTs in class $\mathcal{S}_k$. We obtain this result via orbifolding Dp/D(p-4) brane systems and calculating the partition function of the supersymmetric gauge theory on the worldvolume of $K$ D(p-4) branes. Starting with D5/D1 setups probing a $\mathbb{Z}_\ell\times \mathbb{Z}_k$ orbifold singularity we obtain the $K$ instanton partition functions of 6d $(1,0)$ theories on $\mathbb{R}^4 \times T^2$ in the presence of orbifold defects on $T^2$ via computing the 2d superconformal index of the worldvolume theory on $K$ D1 branes wrapping the $T^2$. We then reduce our results to the 5d and to the 4d instanton partition functions. For $k=1$ we check that we reproduce the known elliptic, trigonometric and rational Nekrasov partition functions. Finally, we show that the instanton partition functions of $SU(N)$ quivers in class $\mathcal{S}_k$ can be obtained from the class $\mathcal{S}$ mother theory partition functions with $SU(kN)$ gauge factors via imposing the `orbifold condition' $a_{\mathcal{A}} \rightarrow a_A e^{2\pi i j/k}$ with $\mathcal{A}=jA$ and $A=1,\dots, N$, $j=1,\dots, k$ on the Coulomb moduli and the mass parameters.