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The Yang-Baxter Sigma Model from Twistor Space

Meer Ashwinkumar, Jitendra Pal

TL;DR

This work constructs a novel $4d$ integrable field theory (IFT$_4$) from $6d$ holomorphic Chern--Simons theory on twistor space, with dynamics governed by a skew-symmetric operator on a Lie algebra and boundary data that, upon specialization to the modified classical Yang--Baxter equation, yields a $4d$ Yang--Baxter sigma model. The IFT$_4$ equations of motion embed the anti-self-dual Yang--Mills equations, revealing an ASDYM embedding of the $2d$ YB sigma model. By performing symmetry reductions from $6d$ to $4d$ CS with disorder surface defects, the work realises the YB sigma model within this twistor framework, forming a diamond of reductions relating $6d$, $4d$, and $2d$ integrable theories. A complete reduction to $2d$ IFT is obtained from $4d$ CS, reproducing the standard YB sigma model action through a precise relation between the YB operator and a boundary P-operator, thereby embedding the $2d$ YB EOM into ASDYM and clarifying the role of semi-local symmetries and the modified YB equation in the construction.

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.

The Yang-Baxter Sigma Model from Twistor Space

TL;DR

This work constructs a novel integrable field theory (IFT) from holomorphic Chern--Simons theory on twistor space, with dynamics governed by a skew-symmetric operator on a Lie algebra and boundary data that, upon specialization to the modified classical Yang--Baxter equation, yields a Yang--Baxter sigma model. The IFT equations of motion embed the anti-self-dual Yang--Mills equations, revealing an ASDYM embedding of the YB sigma model. By performing symmetry reductions from to CS with disorder surface defects, the work realises the YB sigma model within this twistor framework, forming a diamond of reductions relating , , and integrable theories. A complete reduction to IFT is obtained from CS, reproducing the standard YB sigma model action through a precise relation between the YB operator and a boundary P-operator, thereby embedding the YB EOM into ASDYM and clarifying the role of semi-local symmetries and the modified YB equation in the construction.

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.
Paper Structure (13 sections, 175 equations, 1 figure)

This paper contains 13 sections, 175 equations, 1 figure.

Table of Contents

  1. Introduction
  2. 4d IFT from 6d CS on Twistor Space
  3. Derivation of 4d IFT
  4. Equations of Motion and Anti-Self-Dual Yang -Mills Equations
  5. 4d ASDYM and Lax formulation
  6. Symmetry reduction of 6d CS to 4d CS
  7. Modified Classical Yang--Baxter Equation, Semi-local Symmetry and a 4d Yang--Baxter Sigma Model
  8. Symmetry Reduction of $\textrm{IFT}_4$ to $\textrm{IFT}_2$
  9. Reduction to Yang--Baxter Sigma Model
  10. 2d IFT from 4d CS
  11. Twistor Conventions
  12. Some Useful Identities
  13. Proof of Equation \ref{['YB Operator1']}

Figures (1)

  • Figure 1: Relationship between 4d Chern-Simons theory and 6d Chern-Simons theory on twistor space, as well as 4d and 2d integrable field theories, forming a "diamond".