1993/05/28 by D. Olive, D. I. Olive, Neil Turok +3
Mathematics · Physics and Astronomy · #Affine Lie algebra #Affine transformation #Algebra over a field #Algebraic number #Algebraic structure #Algebraic structures and combinatorial models #Current algebra #Discrete mathematics #Integrable system #Jordan algebra #Lie algebra #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Operator (biology) #Physics #Pure mathematics #Quantum mechanics #Simple (philosophy) #Soliton #Subalgebra #Toda lattice #Vertex (graph theory) #Vertex operator algebra #hep-th
paper · pdf · doi:10.1016/0550-3213(93)90541-v
published as Nucl.Phys.B409:509-546,1993 · Imperial/TP/92-93/29 SWAT/92-93/5 PU-PH-93/1392, requires newmac
arxiv created 1993/05/28 · openalex publication_date 1993/12/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Affine Toda theories with imaginary couplings associate with any simple Lie algebra \bf g generalisations of Sine Gordon theory which are likewise integrable and possess soliton solutions. The solitons are \lq\lq created" by exponentials of quantities Fi(z) which lie in the untwisted affine Kac-Moody algebra \bf g and ad-diagonalise the principal Heisenberg subalgebra. When \bf g is simply-laced and highest weight irreducible representations at level one are considered, Fi(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 F's, at least when \bf 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.