Twistors, the ASD Yang-Mills equations, and 4d Chern-Simons theory

Abstract

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.

40+14 pages; v2: substantial clarifications, typos corrected, appendix added; v3: published in JHEP

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• Integrable Deformations of Sigma Models
• Gauge independent kinematic algebra of self-dual Yang-Mills theory
• Flat Self-dual Gravity
• Integrable $\mathcal{E}$-Models, 4d Chern-Simons Theory and Affine Gaudin Models. I. Lagrangian Aspects
• Deformed Integrable Models from Holomorphic Chern-Simons Theory
• Integrable Deformations from Twistor Space
• Geometry of the spectral parameter and renormalisation of integrable $\sigma$-models
• Higher Gauge Theory and Integrability
• Gauging The Diamond: Integrable Coset Models from Twistor Space
• The Courant-Hilbert construction in 4D Chern-Simons theory
• Integrability in Gravity from Chern-Simons Theory
• A Twistor Space Action for Yang-Mills Theory
• Four-Dimensional Chern-Simons and Gauged Sigma Models
• A Lax Operator for $d=2$ $N=2$ Supergravity
• Integrable models from 4d holomorphic BF theory
• Warped Product Einstein Manifolds in Four Dimensions