M5 branes on ADE singularities: BPS spectrum and partition functions

The dynamics of a stack of M5 branes probing a transverse multi-centered Taub-NUT space are described by a class of 6d $\mathcal{N}=(1,0)$ superconformal field theories known as the M-string orbifold SCFTs. We determine the equivariant partition functions for this class of theories on a geometric background of type $T^2\times\mathbb{C}^2/\Gamma$, where $\Gamma \in\{\mathcal{C}_N,\mathcal{Q}_N, \mathcal{T},\mathcal{O},\mathcal{I}\}$ is an arbitrary finite subgroup of $SU(2)$. The partition functions are built out of contributions from BPS strings as well as BPS particles that arise upon putting the 6d theory on a circle. We find that BPS particle contributions can be expressed in terms of $\Gamma$-covariant Hilbert series which count holomorphic sections of vector bundles on the orbifold singularity with monodromy specified by an irreducible representation of $\Gamma$. The BPS string contributions, on the other hand, are given by the elliptic genera of 2d $\mathcal{N}=(0,4)$ $\Gamma$-dressed quiver gauge theories, obtained by stacking Kronheimer-Nakajima quivers of type $\Gamma$ between interfaces that support current algebras for the McKay dual affine Lie algebra $\widehat{\mathfrak{g}}$. We obtain explicit expressions for the elliptic genera of arbitrary BPS string configurations corresponding to fractional instanton strings on $\mathbb{C}^2/\Gamma$, and for the case of star-shaped quivers of type $\Gamma\in\{\mathcal{Q}_4,\mathcal{T},\mathcal{O},\mathcal{I}\}$ we give a prescription to compute the elliptic genera by gluing 2d analogues of Gaiotto and Witten's $T[SU(N)]$ theories.