  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.
