On the Zakharov-Mikhailov action: $4$d Chern-Simons origin and covariant Poisson algebra of the Lax connection

Abstract

We derive the $2$d Zakharov-Mikhailov action from $4$d Chern-Simons theory. This $2$d action is known to produce as equations of motion the flatness condition of a large class of Lax connections of Zakharov-Shabat type, which includes an ultralocal variant of the principal chiral model as a special case. At the $2$d level, we determine for the first time the covariant Poisson bracket $r$-matrix structure of the Zakharov-Shabat Lax connection, which is of rational type. The flatness condition is then derived as a covariant Hamilton equation. We obtain a remarkable formula for the covariant Hamiltonian in term of the Lax connection which is the covariant analogue of the well-known formula "$H=Tr{L}^2$".

16 pages. Final authors' version with added references and explanations

References in corpus (7)

• Supersymmetric gauge theory and the Yangian
• Assembling integrable sigma-models as affine Gaudin models
• The Chern-Simons Origin of Superstring Integrability
• Integrable $\mathcal{E}$-Models, 4d Chern-Simons Theory and Affine Gaudin Models. I. Lagrangian Aspects
• A Variational Approach to Lax Representations
• The $\mathsf{C}{\mathsf{P}}^{\mathsf{n}-1}$-model with fermions: a new look
• Multiform description of the AKNS hierarchy and classical r-matrix

Cited by in corpus (14)

• 4-dimensional Chern-Simons theory and integrable field theories
• Sigma models as Gross-Neveu models
• Integrable degenerate $\mathcal{E}$-models from 4d Chern-Simons theory
• Deformed Integrable Models from Holomorphic Chern-Simons Theory
• Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies
• Non-Abelian Toda field theories from a 4D Chern-Simons theory
• Integrable deformed $T^{1,1}$ sigma models from 4D Chern-Simons theory
• Lagrangian multiforms on coadjoint orbits for finite-dimensional integrable systems
• 4D Chern-Simons theory with auxiliary fields
• Higher Gauge Theory and Integrability
• The Courant-Hilbert construction in 4D Chern-Simons theory
• 3-dimensional mixed BF theory and Hitchin's integrable system
• Supersymmetric Grassmannian Sigma Models in Gross-Neveu Formalism
• Integrable models from 4d holomorphic BF theory