4d Chern-Simons theory as a 3d Toda theory, and a 3d-2d correspondence

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 is three-dimensional analogue of Toda novel 3d W-algebra symmetry. By embedding in partial twist five-dimensional maximally supersymmetric Yang-Mills on manifold corners, we argue this two-dimensional topological sigma model A-branes moduli space solutions Bogomolny equations. This furnishes 3d-2d correspondence, which, among other mathematical implications, also reveals modules are for quantized algebra certain holomorphic functions space.