Four-dimensional mathcal{N} = 2 supersymmetric theory with boundary as a two-dimensional complex Toda theory

Abstract

We perform a series of dimensional reductions of the 6d, mathcal{N} = (2, 0) SCFT on S2 × Σ × I × S1 down to 2d on Σ. The reductions are performed in three steps: (i) a reduction on S1 (accompanied by a topological twist along Σ) leading to a supersymmetric Yang-Mills theory on S2 × Σ × I, (ii) a further reduction on S2 resulting in a complex Chern-Simons theory defined on Σ × 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 S2 and S1, 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 Σ. As the reduction of the 6d theory on Σ would give rise to an mathcal{N} = 2 supersymmetric theory on S2 × I × S1, our results imply a 4d-2d duality between four-dimensional mathcal{N} = 2 supersymmetric theory with boundary and two-dimensional complex Toda theory.