  We show that the approaches to integrable systems via 4d Chern-Simons theory
and via symmetry reductions of the anti-self-dual Yang-Mills equations are
closely related, at least classically. Following a suggestion of Kevin
Costello, we start from holomorphic Chern-Simons theory on twistor space,
defined with the help of a meromorphic (3,0)-form $\Omega$. If $\Omega$ is
nowhere vanishing, it descends to a theory on 4d space-time with classical
equations of motion equivalent to the anti-self-dual Yang-Mills equations.
Examples include a 4d analogue of the Wess-Zumino-Witten model and a theory of
a Lie algebra valued scalar with a cubic two derivative interaction. Under
symmetry reduction, these yield actions for 2d integrable systems. On the other
hand, performing the symmetry reduction directly on twistor space reduces
holomorphic Chern-Simons theory to the 4d Chern-Simons theory with disorder
defects studied by Costello & Yamazaki. Finally we show that a similar
reduction by a single translation leads to a 5d partially holomorphic
Chern-Simons theory describing the Bogomolny equations.
