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Classically, the ground states of $\mathcal{N}=4$ supersymmetric Yang-Millstheory on $\mathbb{R}\times S^3/\Gamma$ where $\Gamma$ is a discrete ADEsubgroup of $SU(2)$ are represented by flat Wilson lines winding around the ADEsingularity. By a duality relating such ground states to WZW conformal blocks,the ground state degeneracy cannot be lifted by quantum corrections. Using thesuperconformal index, we compute the supersymmetric Casimir energy of each flatWilson line for $SU(2)$ SYM on different ADE singularities and find that theflat Wilson lines all have the same supersymmetric Casimir energy. We arguethat this exact degeneracy is peculiar to $\mathcal{N}=4$ supersymmetry andshow that the degeneracy is lifted when the number of supersymmetry is reduced.In particular, we uncover a surprising result for the ground state structure ofthe conformal $\mathcal{N}=2$ $SU(2)$ four-flavor theory on $S^3/\Gamma$. For$\mathcal{N}=4$ SYM, S-duality maps the ground state Wilson lines to groundstate t' Hooft lines taking values in the Langlands dual group. We show thatthe supersymmetric Casimir energy of the t' Hooft line ground states is thesame as the Wilson line ground states. This can be viewed as a ground statetest of S-duality.

Exact Degeneracy of Casimir Energy for $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory on ADE Singularities and S-Duality