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

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 $\Sigma$. The reductions are performed in three steps: (i) a reduction on $S^1$ (accompanied by a topological twist along $\Sigma$) leading to a supersymmetric Yang-Mills theory on $S^2\times\Sigma\times I$, (ii) a further reduction on $S^2$ resulting in a complex Chern--Simons theory defined on $\Sigma\times I$, with the real part of the complex Chern-Simons level being zero, and the imaginary part 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 with the Nahm pole boundary condition at both ends of the interval $I$, which gives rise to a complex Toda CFT on the Riemann surface $\Sigma$. As the reduction of the 6d theory on $\Sigma$ would give rise to an $\mathcal{N}=2$ supersymmetric theory on $S^2\times I\times S^1$, our results imply a 4d-2d duality between four-dimensional $\mathcal{N}=2$ supersymmetric theory with boundary and two-dimensional complex Toda theory.