Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 044, 56 pages      arXiv:2509.20643      https://doi.org/10.3842/SIGMA.2026.044

Non-Commutative Gauge Theory at the Beach

Roland Bittleston a, Simon Heuveline bc, Surya Raghavendran d and David Skinner b
a) Perimeter Institute for Theoretical Physics, 31 Caroline Street, Waterloo, Ontario, Canada
b) Department of Applied Maths & Theoretical Physics, University of Cambridge, Wilberforce Road, UK
c) Center for the Fundamental Laws of Nature & Black Hole Initiative, Harvard University, Cambridge, USA
d) Department of Mathematics, Yale University, 219 Prospect St, New Haven, USA

Received October 16, 2025, in final form April 23, 2026; Published online May 05, 2026

Abstract
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 $w_{1+\infty}$.

Key words: integrable; KP equation; non-commutative; topological-holomorphic field theory; W-algebra.

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