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We perform a series of dimensional reductions of the 6d, $\mathcal{N}=(2,0)$SCFT on $S^2\times\Sigma\times I\times S^1$ down to 2d on $\Sigma$. Thereductions are performed in three steps: (i) a reduction on $S^1$ (accompaniedby a topological twist along $\Sigma$) leading to a supersymmetric Yang-Millstheory on $S^2\times\Sigma\times I$, (ii) a further reduction on $S^2$resulting in a complex Chern--Simons theory defined on $\Sigma\times I$, withthe real part of the complex Chern-Simons level being zero, and the imaginarypart being proportional to the ratio of the radii of $S^2$ and $S^1$, and (iii)a final reduction to the boundary modes of complex Chern--Simons theory withthe Nahm pole boundary condition at both ends of the interval $I$, which givesrise to a complex Toda CFT on the Riemann surface $\Sigma$. As the reduction ofthe 6d theory on $\Sigma$ would give rise to an $\mathcal{N}=2$ supersymmetrictheory on $S^2\times I\times S^1$, our results imply a 4d-2d duality betweenfour-dimensional $\mathcal{N}=2$ supersymmetric theory with boundary andtwo-dimensional complex Toda theory.

Four-Dimensional N=2 Supersymmetric Theory with Boundary as a Two-Dimensional Complex Toda Theory