Affine Toda Solitons and Vertex Operators

Abstract

Affine Toda theories with imaginary couplings associate with any simple Lie algebra $\mathbf{g}$ generalisations of Sine Gordon theory which are likewise integrable and possess soliton solutions. The solitons are \lq\lq created" by exponentials of quantities ${\hat{F}}^i(z)$ which lie in the untwisted affine Kac-Moody algebra $\hat{\mathbf{g}}$ and ad-diagonalise the principal Heisenberg subalgebra. When $\mathbf{g}$ is simply-laced and highest weight irreducible representations at level one are considered, ${\hat{F}}^i(z)$ can be expressed as a vertex operator whose square vanishes. This nilpotency property is extended to all highest weight representations of all affine untwisted Kac-Moody algebras in the sense that the highest non vanishing power becomes proportional to the level. As a consequence, the exponential series mentioned terminates and the soliton solutions have a relatively simple algebraic expression whose properties can be studied in a general way. This means that various physical properties of the soliton solutions can be directly related to the algebraic structure. For example, a classical version of Dorey's fusing rule follows from the operator product expansion of two $\hat{F}$'s, at least when $\mathbf{g}$ is simply laced. This adds to the list of resemblances of the solitons with respect to the particles which are the quantum excitations of the fields.

Imperial/TP/92-93/29 SWAT/92-93/5 PU-PH-93/1392, requires newmac

References in corpus (7)

• Quantum Solitons in Affine Toda Field Theories
• Hirota's Solitons in the Affine and the Conformal Affine Toda Models
• Affine Toda Solitons and Automorphisms of Dynkin Diagrams
• Duality in Quantum Toda theory and W-algebras
• Connection between the Affine and Conformal Affine Toda Models and their Hirota's Solution
• Crossing and Antisolitons in Affine Toda Theories
• The Quantum Spectrum of the Conserved Charges in Affine Toda Theories

Cited by in corpus (63)

• Yangians, Integrable Quantum Systems and Dorey's rule
• Affine Toda Systems Coupled to Matter Fields
• Tau-functions and Dressing Transformations for Zero-Curvature Affine Integrable Equations
• On purely transmitting defects in affine Toda field theory
• Semi-classical spectrum of the Homogeneous sine-Gordon theories
• Solitons, Tau-functions and Hamiltonian Reduction for Non-Abelian Conformal Affine Toda Theories
• Vertex Operators and Soliton Time Delays in Affine Toda Field Theory
• On a Solitonic Specialisation for the General Solutions of Some Two-Dimensional Completely Integrable Systems
• Affine Toda Solitons and Systems of Calogero-Moser Type
• Multi--Soliton Solutions of Affine Toda Models
• Exact S-Matrices for Bound States of $a_2^{(1)}$ Affine Toda Solitons
• Exact S-matrices for d_{n+1}^{(2)} affine Toda solitons and their bound states
• Non-unitarity in quantum affine Toda theory and perturbed conformal field theory
• Tau-Functions generating the Conservation Laws for Generalized Integrable Hierarchies of KdV and Affine-Toda type
• Confinement, solitons and the equivalence between the sine-Gordon and massive Thirring models
• Quantum mass corrections for affine Toda solitons
• Soliton-preserving boundary condition in affine Toda field theories
• Conserved charges and soliton solutions in affine Toda theory
• Folding defect affine Toda field theories
• p-Branes as Composite Antisymmetric Tensor Field Theories
• Negative Even Grade mKdV Hierarchy and its Soliton Solutions
• Confinement and soliton solutions in the SL(3) Toda model coupled to matter fields
• Inverse scattering and solitons in $A_{n-1}$ affine Toda field theories
• Soliton Solutions for the Super mKdV and sinh-Gordon Hierarchy
• Toda Soliton Mass Corrections and the Particle--Soliton Duality Conjecture
• Minimal surfaces in AdS space and Integrable systems
• Scattering of compact oscillons
• Solitons and Vertex Operators in Twisted Affine Toda Field Theories
• Electrically Charged Topological Solitons
• A Moyal Quantization of the Continuous Toda Field
• Exact quantum S-matrices for solitons in simply-laced affine Toda field theories
• BPS Z(N) String Tensions, Sine Law and Casimir Scaling and Integrable Field Theories
• Negative flows of generalized KdV and mKdV hierarchies and their gauge-Miura transformations
• Poisson involutions, spin Calogero-Moser systems associated with symmetric Lie subalgebras and the symmetric space spin Ruijsenaars-Schneider models
• Quantum Mass corrections for C_2^(1) Affine Toda theory solitons
• The algebraic structure behind the derivative nonlinear Schroedinger equation
• Nonvanishing boundary condition for the mKdV hierarchy and the Gardner equation
• Multicharged Dyonic Integrable Models
• Inverse scattering and solitons in $A_{n-1}$ affine Toda field theories II
• Defect fusing rules in affine Toda field theory
• Abelian Toda solitons revisited
• Finite Dimensional Representations of Quantum Affine Algebras
• Abelian Toda field theories on the noncommutative plane
• On decay of shock-like waves into compact oscillons
• Remarks on excited states of affine Toda solitons
• Integrable field theory with boundary conditions
• Vertex Operators and Soliton Solutions of Affine Toda Model with U(2) Symmetry
• $A_n^{(1)}$ Toda solitons and the dressing symmetry
• Restricted Quantum Theory of Affine Toda Solitons
• Reality of Complex Affine Toda Solitons
• Dressing approach to the nonvanishing boundary value problem for the AKNS hierarchy
• More non-Abelian loop Toda solitons
• Z(2) monopoles in SU(n) Yang-Mills-Higgs theories
• Vertex operators, semiclassical limit for soliton S-matrices and the number of bound states in Affine Toda Field Theories
• Solitons, Boundaries, and Quantum Affine Algebras
• Solving non-abelian loop Toda equations
• Orthogonal Decomposition of Some Affine Lie Algebras in Terms of their Heisenberg Subalgebras
• Static multi-soliton solutions in the affine su(N+1) Toda models
• Real Hamiltonian Forms of Affine Toda Models Related to Exceptional Lie Algebras
• New negative grade solitonic sector for supersymmetric KdV and mKdV hierarchies
• Vertex Operator Representation of the Soliton Tau Functions in the $A_n^{(1)}$ Toda Models by Dressing Transformations
• Soliton solutions of the classical lattice sine-Gordon system
• Real Hamiltonian forms of affine Toda field theories: spectral aspects