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, N = (2, 0) SCFT on S-2 x Sigma x I x 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 x Sigma x I, (ii) a further reduction on S-2 resulting in a complex Chern-Simons theory defined on Sigma x 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 N = 2 supersymmetric theory on S-2 x I x S-1, our results imply a 4d-2d duality between four-dimensional N = 2 supersymmetric theory with boundary and two-dimensional complex Toda theory.