UC Berkeley

Classically, the ground states of $\mathcal{N}=4$ Super Yang-Mills Theory (SYM) on $\mathbb{R}\times S^3/\Gamma$ where $\Gamma$ is one of the ADE subgroup of $SU(2)$ are flat Wilson lines winding around the ADE singularity. S-duality acts on this finite-dimensional ground state Hilbert space and its action is the same as the $S$ operator in a certain dual Chern-Simons theory on $T^2$. The dual Chern-Simons theory arises out of the only long-range interaction in a string/M theory construction by considering a stack of D3(M5) branes on ADE singularity. This SYM/Chern-Simons duality is verified by matching the ground state Hilbert spaces of both theories and by comparing the S-duality operators of both theories.

To one-loop order, the SYM ground state degeneracy is exact. A detailed computation using the superconformal index shows that each classical SYM ground state acquires the same supersymmetric Casimir energy. S-duality maps the SYM ground state Wilson lines to ground state t' Hooft lines taking values in the Langlands dual group. The number of t' Hooft lines are shown to agree with that of the Wilson lines. In addition, the t' Hooft lines have the same supersymmetric Casimir energy as the corresponding Wilson lines. These two facts provide a ground state test of S-duality.

The SYM/Chern-Simons duality has an important extension to the class S theory obtained from compactifying M5 branes on a Riemann surface $\mathcal{R}$. The ground states of class S theory on $\mathbb{R}\times S^3/\Gamma$ are dual to the states of the dual Chern-Simons theory on $\mathcal{R}$. In particular, we uncover a surprising result that there is only one unique ground state for the conformal $\mathcal{N}=2$ $SU(2)$ four-flavor theory on $\mathbb{R}\times S^3/\Gamma$. Finally, we apply the SYM/Chern-Simons duality to a non-Lagrangian class S theory and find that its ground states obey the fusion rule of the current algebra of the dual Chern-Simons theory.