Ground States of Class S Theory on ADE Singularities and dual Chern-Simons theory

In radial quantization, the ground states of a gauge theory on ADE singularities $\mathbb{R}^4/\Gamma$ are characterized by flat connections that are maps from $\Gamma$ to the gauge group. We study Class $\mathcal{S}$ theory of type $\mathfrak{a}_1=\mathfrak{su}(2)$ on a Riemann surface of genus $g>1$, without punctures. The fundamental building block of Class $\mathcal{S}$ theory is the trifundamental Trinion theory - a low energy limit of two M5 branes compactified on the three-punctured Riemann sphere. We show, through the superconformal index, that the supersymmetric Casimir energy of the trifundamental theory imposes a constraint on the set of allowed flat connections, which agrees with the prediction of a duality relating the ground state Hilbert space of Class $\mathcal{S}$ on ADE singularities to the Hilbert space of a certain dual Chern-Simons theory whose gauge group is given by the McKay correspondence. The conjecture is shown to hold for $\Gamma=\mathbb{Z}_k$, agreeing with the previous results of Benini et al. and Alday et al. A non-abelian generalization of this duality is analyzed by considering the example of the dicyclic group $\Gamma=\text{Dic}_2$, corresponding to Chern-Simons gauge group SO$(8)$.