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

## TL;DR

This paper derives a duality between a 4d N=2 supersymmetric theory with boundary and a 2d complex Toda theory through a series of dimensional reductions of a 6d SCFT.

## Contribution

It provides a novel derivation of a 4d-2d duality connecting boundary N=2 theories to complex Toda models via dimensional reduction and boundary conditions.

## Key findings

- Reduction from 6d to 2d yields a duality between 4d N=2 theory and 2d complex Toda theory.
- Boundary conditions in complex Chern-Simons lead to Toda conformal field theory.
- The work establishes a new link between higher-dimensional SCFTs and integrable 2d models.

## Abstract

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.