Local charges in involution and hierarchies in integrable sigma-models

Abstract

Integrable $\sigma$-models, such as the principal chiral model, ${\mathbb{Z}}_T$-coset models for $T\in {\mathbb{Z}}_{\ge 2}$ and their various integrable deformations, are examples of non-ultralocal integrable field theories described by (cyclotomic) $r/s$-systems with twist function. In this general setting, and when the Lie algebra $\mathfrak{g}$ underlying the $r/s$-system is of classical type, we construct an infinite algebra of local conserved charges in involution, extending the approach of Evans, Hassan, MacKay and Mountain developed for the principal chiral model and symmetric space $\sigma$-model. In the present context, the local charges are attached to certain `regular' zeros of the twist function and have increasing degrees related to the exponents of the untwisted affine Kac-Moody algebra $\hat{\mathfrak{g}}$ associated with $\mathfrak{g}$. The Hamiltonian flows of these charges are shown to generate an infinite hierarchy of compatible integrable equations.

67 pages, published version

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Cited by in corpus (15)

• Integrable Deformations of Sigma Models
• Assembling integrable sigma-models as affine Gaudin models
• Integrable coupled sigma-models
• Yang-Baxter deformations of the Principal Chiral Model plus Wess-Zumino term
• Sigma models with local couplings: a new integrability -- RG flow connection
• New integrable coset sigma models
• Integrability vs. RG flow in $G\times G$ and $G\times G/H$ sigma models
• Universal 1-loop divergences for integrable sigma models
• Integrable Deformations from Twistor Space
• Constrained affine Gaudin models and diagonal Yang-Baxter deformations
• An elliptic integrable deformation of the Principal Chiral Model
• Quartic Hamiltonians, and higher Hamiltonians at next-to-leading order, for the affine ${\mathfrak{sl}}_2$ Gaudin model
• On a class of conformal $\mathcal{E}$-models and their chiral Poisson algebras
• On the Integrable Structure of the SU(2) Wess-Zumino-Novikov-Witten Model
• Integrable models from 4d holomorphic BF theory