Non-Commutative Gauge Theory at the Beach

The KP equation is perhaps the most famous example of a three-dimensional integrable system. Here we show that a non-commutative five-dimensional Chern-Simons theory living on the projective spinor bundle of three-dimensional space-time compactifies to a Lagrangian formulation of the KP equation. Essential to the definition of the theory is a 2-form pulled back from minitwistor space. The dispersionless limit of the KP equation is similarly described by Poisson-Chern-Simons theory. We further show that, consistent with integrability, all tree level amplitudes vanish. The universal vertex algebra living on a two-dimensional surface defect in $5d$ is $W_{1+\infty}$, and its operator products coincide with collinear splitting functions on space-time. Taking the dispersionless limit contracts the vertex algebra to $w_{1+\infty}$.