Ground states of Class \( \mathcal{S} \) theory on ADE singularities and dual Chern-Simons theory

In radial quantization, the ground states of a gauge theory on ADE singularities ℝ⁴/Γ are characterized by flat connections that are maps from Γ to the gauge group. We study Class $\mathcal{S}$ \( \mathcal{S} \) theory of type ${\mathfrak{a}}_1$ \( {\mathfrak{a}}_1 \) = $\mathfrak{su}\left(2\right)$ \( \mathfrak{su}(2) \) on a Riemann surface of genus g > 1, without punctures. The fundamental building block of Class $\mathcal{S}$ \( \mathcal{S} \) 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 $\mathcal{S}$ \( \mathcal{S} \) 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 Γ = 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 Γ = Dic₂, corresponding to Chern-Simons gauge group SO(8).