  We show that the four-dimensional Chern-Simons theory studied by Costello,
Witten and Yamazaki, is, with Nahm pole-type boundary conditions, dual to a
boundary theory that is a three-dimensional analogue of Toda theory with a
novel 3d W-algebra symmetry. By embedding four-dimensional Chern-Simons theory
in a partial twist of the five-dimensional maximally supersymmetric Yang-Mills
theory on a manifold with corners, we argue that this three-dimensional Toda
theory is dual to a two-dimensional topological sigma model with A-branes on
the moduli space of solutions to the Bogomolny equations. This furnishes a
novel 3d-2d correspondence, which, among other mathematical implications, also
reveals that modules of the 3d W-algebra are modules for the quantized algebra
of certain holomorphic functions on the Bogomolny moduli space.
