[Paper Review] Torsionless T-selfdual Affine NA Toda Models

J. F. Gomes, E. P. Gueuvoghlanian+2 more|arXiv (Cornell University)|Feb 21, 2000

Nonlinear Waves and Solitons4 references3 citations

TL;DR

This paper constructs a class of torsionless, T-selfdual non-abelian affine Toda models using a gauged two-loop WZNW framework, identifying that such models exist only for the affine Kac-Moody algebras $B_n^{(1)}$, $A_{2n}^{(2)}$, and $D_{n+1}^{(2)}$ under specific gradation and constant generator conditions. The zero curvature representation confirms their classical integrability and reproduces models previously proposed by Fateev.

ABSTRACT

A general construction of affine Non Abelian Toda models in terms of gauged two loop WZNW model is discussed. In particular we find the Lie algebraic condition defining a subclass of {\it T-selfdual torsionless NA Toda models} and their zero curvature representation.

Motivation & Objective

• To systematically construct a subclass of singular, torsionless, non-abelian affine Toda models using a gauged two-loop WZNW model framework.
• To identify the Lie algebraic condition under which such models are T-selfdual, i.e., invariant under axial and vector gauging of the $U(1)$ factor in the coset $SL(2)/U(1) \otimes U(1)^{\text{rank}{\cal G}-1}$.
• To provide a unified construction that reproduces the models proposed by Fateev, confirming their classical integrability through a zero curvature representation.
• To explore the generalization of these models to other affine Kac-Moody algebras via different gradations and gauging schemes, including axionic and torsionless models.

Proposed method

• Utilizes the gauged two-loop WZNW model formalism to derive a zero curvature representation for the physical fields in the coset ${\cal G}_0 / {\cal G}_0^0$.
• Applies Hamiltonian reduction by fixing constant grade $\pm 1$ generators $\epsilon_\pm$ and gauging the $H_\pm$ subgroups to reduce the phase space to the coset $H_- \backslash G / H_+$.
• Imposes the condition that the zero-grade subgroup ${\cal G}_0$ is isomorphic to $sl(2) \otimes u(1)^{\text{rank}{\cal G}-1}$, with ${\cal G}_0^0 = U(1)$, to define the torsionless, T-selfdual subclass.
• Derives the action for the physical fields $B \in {\cal G}_0$ by eliminating auxiliary gauge fields $A$ and $\bar{A}$, resulting in a WZNW-like action with additional terms involving $\epsilon_\pm$.
• Constructs the zero curvature representation $\partial A - \bar{\partial} \bar{A} + [A, \bar{A}] = 0$ explicitly for the identified models, confirming integrability.
• Uses the Gauss decomposition $g = NBM$ to separate the physical degrees of freedom and ensures invariance under $g \to \alpha_- g \alpha_+$ with $\alpha_\pm \in H_\pm$.

Experimental results

Research questions

• RQ1 Which affine Kac-Moody algebras support torsionless, T-selfdual non-abelian Toda models under the coset structure ${\cal G}_0 / {\cal G}_0^0 = SL(2)/U(1) \otimes U(1)^{\text{rank}{\cal G}-1}$?
• RQ2 What Lie algebraic condition ensures T-selfduality, i.e., equivalence of axial and vector gauging in the $U(1)$ factor of the coset?
• RQ3 How can the zero curvature representation be explicitly constructed for such models, and what does it imply for classical integrability?
• RQ4 To what extent do these models reproduce or generalize the models proposed by Fateev in the context of strong-coupling limits of 2D field theories?
• RQ5 Can the framework be extended to construct more general integrable models via different gradations and coset structures?

Key findings

• The only affine Kac-Moody algebras supporting torsionless, T-selfdual NA Toda models with ${\cal G}_0 / {\cal G}_0^0 = SL(2)/U(1) \otimes U(1)^{\text{rank}{\cal G}-1}$ are $B_n^{(1)}$, $A_{2n}^{(2)}$, and $D_{n+1}^{(2)}$.
• The T-selfduality condition arises when the choice of $\epsilon_\pm$ and the gradation $Q$ satisfy a specific Lie algebraic constraint, ensuring axial and vector gauging yield identical actions.
• The zero curvature representation is explicitly constructed for all three families, confirming their classical integrability through the flatness of the connection $A$ and $\bar{A}$.
• The action derived from the gauged two-loop WZNW model exactly reproduces the models proposed by Fateev, validating their integrability and providing a systematic derivation.
• The construction generalizes to other coset structures such as $SL(2) \otimes U(1)^{n-1}/U(1)^s$, $SL(2) \otimes SL(2) \otimes U(1)^{n-2}/U(1)$, and $SL(3) \otimes U(1)^{n-2}/U(1)$, suggesting a broader class of integrable models.
• The soliton solutions of these models carry both electric and magnetic (topological) charges, resembling 4D Yang-Mills-Higgs dyons, motivating their study in gauge theory and string theory.