4d Chern-Simons Theory as a 3d Toda Theory, and a 3d-2d Correspondence

Abstract

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.

28 pages. Presented at "String Math 2020". Further clarifications. To appear in JHEP

References in corpus (1)

• A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories

Cited by in corpus (3)

• 4-dimensional Chern-Simons theory and integrable field theories
• Twistors, the ASD Yang-Mills equations, and 4d Chern-Simons theory
• Integrable degenerate $\mathcal{E}$-models from 4d Chern-Simons theory