[Paper Review] Classical Integrability of Non Abelian Affine Toda Models

J. F. Gomes, E. P. Gueuvoghlanian+2 more|ArXiv.org|Oct 27, 2000

Nonlinear Waves and Solitons2 references3 citations

TL;DR

This paper constructs non-abelian affine Toda models using the gauged two-loop Wess-Zumino-Witten (WZW) model, demonstrating classical integrability through a zero curvature representation and an r-matrix structure. The models exhibit solitons with nontrivial electric and topological charges due to the non-abelian structure of the zero-grade subalgebra, and the infinite set of conserved charges is proven via the fundamental Poisson bracket relation derived from the r-matrix.

ABSTRACT

A class of non abelian affine Toda models is constructed in terms of the axial and vector gauged WZW model. It is shown that the multivacua structure of the potential together with non abelian nature of the zero grade subalgebra allows soliton solutions with non trivial electric and topological charges. Their zero curvature representation and the classical $r$-matrix are also constructed in order to prove their classical integrability.

Motivation & Objective

• To systematically construct non-abelian affine Toda models based on the coset $ SL(2) \otimes U(1)^{r-1}/U(1) $ using a grading operator and gauged WZW model.
• To demonstrate classical integrability of these models by constructing a zero curvature representation and an r-matrix.
• To show that the non-abelian nature of the zero-grade subalgebra $ \mathfrak{g}_0 = SL(2) \otimes U(1)^{r-1} $ allows for solitons with nontrivial electric and topological charges.
• To derive the fundamental Poisson bracket relation from the classical r-matrix, ensuring involution of conserved charges.

Proposed method

• Utilizes a grading operator $ Q = h'\hat{d} + \sum_{i \neq a} \frac{2\lambda_i \cdot H}{\alpha_i^2} $ to decompose the affine Lie algebra $ \tilde{\mathfrak{G}} = \oplus \mathfrak{G}_i $, with Toda fields parametrizing the zero-grade subspace $ \mathfrak{G}_0 $.
• Constructs the action via the $ G/H $-gauged WZW model, where $ H = H_- \backslash G / H_+ $, with $ H_\pm $ generated by positive/negative grade operators, and introduces auxiliary gauge fields $ A \in \mathfrak{G}_{<} $, $ \bar{A} \in \mathfrak{G}_{>} $.
• Performs Hamiltonian reduction by fixing constant grade $ \pm 1 $ generators $ \epsilon_\pm $, leading to an effective action $ S = S_{WZNW}(B) + \frac{k}{2\pi} \int \mathrm{Tr}(\epsilon_+ B \epsilon_- B^{-1}) d^2x $.
• Derives the zero curvature representation using gauge connections $ A $ and $ \bar{A} $, showing the existence of an infinite number of conserved charges through the flatness condition $ \partial A - \partial \bar{A} + [A, \bar{A}] = 0 $.
• Constructs the classical $ r $-matrix $ r_{\pm} $ from the Casimir operator $ \mathcal{C} $, and verifies the fundamental Poisson bracket relation $ \{ \mathcal{R}_1, \mathcal{R}_2 \} = [r_{12}, \mathcal{R}_1 + \mathcal{R}_2] $.
• Uses dressing transformations on vacuum configurations $ A_{\text{vac}} = \epsilon_- $, $ \bar{A}_{\text{vac}} = -\epsilon_+ - \mu^2 z \hat{c} $ to classify soliton solutions via Heisenberg subalgebra eigenstates and vertex operators.

Experimental results

Research questions

• RQ1 How can non-abelian affine Toda models be systematically constructed from a graded affine Lie algebra with non-abelian zero-grade subalgebra?
• RQ2 What conditions ensure classical integrability in such non-abelian Toda models, and how can this be proven via zero curvature representation and r-matrix structure?
• RQ3 How do the non-abelian structure and multivacua potential lead to solitons with nontrivial electric and topological charges?
• RQ4 What is the role of the $ r $-matrix in ensuring the involution of conserved charges in the classical phase space?
• RQ5 How are soliton solutions classified, and what types of interactions (e.g., 2-soliton, breather) arise from the composition of charged and neutral solutions?

Key findings

• The non-abelian affine Toda models are constructed via the $ G/H $-gauged WZW model with $ H = H_- \backslash G / H_+ $, yielding an effective action that describes integrable perturbations of the $ \mathfrak{g}_0 $-WZNW model.
• The models admit both electric (Noether) and magnetic (topological) charges, with electrically charged topological solitons arising for imaginary coupling $ \beta^2 = -2\pi/k $.
• The zero curvature representation is explicitly constructed, confirming the existence of an infinite number of conserved charges through the flatness of the gauge connection.
• The classical $ r $-matrix is derived from the Casimir operator $ \mathcal{C} $, and the fundamental Poisson bracket relation $ \{ \mathcal{R}_1, \mathcal{R}_2 \} = [r_{12}, \mathcal{R}_1 + \mathcal{R}_2] $ is verified, ensuring the involution of conserved charges.
• Soliton solutions are classified into neutral (abelian $ A_{r-1} $ model), neutral-charged, and charged-charged types, with time delays computed for 2-soliton and breather configurations.
• The dressing method applied to vacuum configurations $ A_{\text{vac}} = \epsilon_- $, $ \bar{A}_{\text{vac}} = -\epsilon_+ - \mu^2 z \hat{c} $ generates solitons via Heisenberg subalgebra eigenstates and vertex operators, yielding explicit solutions for $ A_r^{(1)} $ models.