The Yang-Baxter Sigma Model from Twistor Space

TL;DR

This paper derives a new integrable 4d field theory from 6d twistor space Chern-Simons theory, connecting it to the Yang-Baxter sigma model and embedding its equations into anti-self-dual Yang-Mills equations.

Contribution

It introduces a novel 4d integrable field theory from twistor space and relates it to the Yang-Baxter sigma model via symmetry reduction and geometric embedding.

Findings

Derived a 4d integrable field theory from 6d twistor space Chern-Simons theory.

Connected the 4d theory to the 2d Yang-Baxter sigma model through symmetry reduction.

Embedded the 2d Yang-Baxter sigma model equations into anti-self-dual Yang-Mills equations.

Abstract

We derive a novel two-field four-dimensional integrable field theory (IFT) from 6d holomorphic Chern-Simons theory on twistor space. The four-dimensional IFT depends on a skew-symmetric linear operator acting on a Lie algebra, and when this operator is specialised to a solution of the modified classical Yang-Baxter equation, the IFT develops a semi-local symmetry associated with this solution. The resulting 4d analogue of the Yang-Baxter sigma model is related by symmetry reduction to the well-known 2d Yang-Baxter sigma model. An important implication that we find is the embedding of the equations of motion of the 2d Yang-Baxter sigma model in the anti-self-dual Yang-Mills equations. The 6d Chern-Simons theory on twistor space can alternatively be symmetry reduced to a 4d Chern-Simons theory configuration with disorder surface defects. The latter realises the Yang-Baxter sigma model, implying a "diamond" for the Yang-Baxter sigma model obtained from twistor space.