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

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