[Paper Review] Non Abelian Toda models and Constrained KP hierarchies

I. Cabrera-Carnero, J. F. Gomes+3 more|ArXiv.org|Sep 14, 2001

Nonlinear Waves and Solitons2 references3 citations

TL;DR

This paper constructs non-abelian Toda models via gauged two-loop WZNW models, establishing a correspondence between relativistic non-abelian Toda theories and constrained KP hierarchies through positive and negative grading of affine Lie algebras. The key result is that soliton solutions of the relativistic models (e.g., SL(3) case) match those of non-relativistic models like Yajima-Oikawa and generalized non-linear Schrödinger equations, unified under a common algebraic structure of negative flow hierarchies.

ABSTRACT

A general construction of affine Non Abelian Toda models in terms of gauged two loop WZNW model is discussed. Its connection to non relativistic models corresponding to constrained KP hierarchies is established in terms of time evolution associated to positive and negative grading of the Lie algebra.

Motivation & Objective

• To systematically construct non-abelian Toda field theories using a grading operator on affine Lie algebras.
• To establish a correspondence between relativistic non-abelian Toda models and non-relativistic constrained KP hierarchies via positive and negative grading.
• To demonstrate that soliton solutions of the relativistic models match those of non-relativistic integrable systems like Yajima-Oikawa and generalized non-linear Schrödinger models.
• To unify the algebraic structure of these models under a common framework based on zero-grade subalgebras and gauged WZNW actions.

Proposed method

• Utilizes a grading operator Q to decompose the affine Lie algebra 𝔠𝔤 = ⊕𝔠𝔤_i with [Q, 𝔠𝔤_i] = i𝔠𝔤_i.
• Employs Gauss decomposition g = NBM with N, B, M corresponding to negative, zero, and positive grade subspaces.
• Constructs a gauged WZNW action on the coset H_-\G/H_+ with auxiliary gauge fields A and A-bar to enforce H-invariance.
• Derives an effective action for the B-fields (physical degrees of freedom) by integrating out A and A-bar, yielding S = S_WZNW(B) + (k/2π)∫Tr(ε_+ B ε_- B^{-1})dxdt.
• Applies the Riemann-Hilbert problem to generalize the construction to SL(r+1) and relate positive and negative flow hierarchies.
• Uses explicit parametrization of fields ψ, χ, R, φ to match relativistic Toda solutions with known non-relativistic soliton solutions.

Experimental results

Research questions

• RQ1 How can non-abelian Toda models be systematically constructed from affine Lie algebras using grading operators?
• RQ2 What is the algebraic relationship between relativistic non-abelian Toda models and non-relativistic constrained KP hierarchies?
• RQ3 How do soliton solutions of the relativistic Toda models relate to those of non-relativistic integrable systems like Yajima-Oikawa?
• RQ4 Can the conserved charges of the relativistic and non-relativistic models be related through the same algebraic hierarchy?
• RQ5 What role does the negative grading play in generating relativistic equations as first negative flows in a unified integrable hierarchy?

Key findings

• The relativistic SL(3) non-abelian Toda model's soliton solution, parametrized by ψ, χ, R, φ, exactly matches the Yajima-Oikawa model solution after substitution z/γ_i → γ_i²t_2 and x → z̄.
• The effective action for the B-fields is derived as S = S_WZNW(B) + (k/2π)∫Tr(ε_+ B ε_- B^{-1})dxdt, which describes the non-singular non-abelian Toda model.
• The Hamiltonian density 𝒟_rel = (𝒟ψ𝒟χ)/Δ e^{-φ} + (𝒟φ)² with Δ = 1 + (3/4)ψχ e^{-φ} is conserved under the time evolution ∂_t_{-1} = ∂_z, confirming integrability.
• The conserved current for the relativistic model satisfies ∂_t_{-1}𝒟_rel = ∂̄(ψχ e^{-φ}), showing consistency with the non-relativistic hierarchy.
• For SL(4), the model with 𝔠𝔤_0 = SL(2)×SL(2)×U(1) is explicitly constructed, confirming the general framework for higher-rank cases.
• The construction generalizes known relations (e.g., sine-Gordon/mKdV, Lund-Regge/non-linear Schrödinger) to non-abelian Toda and constrained KP hierarchies.