Dynkin Diagrams and Integrable Models Based on Lie Superalgebras

Abstract

An analysis is given of the structure of a general two-dimensional Toda field theory involving bosons and fermions which is defined in terms of a set of simple roots for a Lie superalgebra. It is shown that a simple root system for a superalgebra has two natural bosonic root systems associated with it which can be found very simply using Dynkin diagrams; the construction is closely related to the question of how to recover the signs of the entries of a Cartan matrix for a superalgebra from its Dynkin diagram. The significance for Toda theories is that the bosonic root systems correspond to the purely bosonic sector of the integrable model, knowledge of which can determine the bosonic part of the extended conformal symmetry in the theory, or its classical mass spectrum, as appropriate. These results are applied to some special kinds of models and their implications are investigated for features such as supersymmetry, positive kinetic energy and generalized reality conditions for the Toda fields. As a result, some new families of integrable theories with positive kinetic energy are constructed, some containing a mixture of massless and massive degrees of freedom, others being purely massive and supersymmetric, involving a number of coupled sine/sinh-Gordon theories.

31 pages; plain TeX, macros included; 5 main Figs., more in tables; v2: minor but confusing inaccuracy corrected in statement of one proposition (already corrected in published version)

References in corpus (12)

• W-symmetry in Conformal Field Theory
• Quantum Solitons in Affine Toda Field Theories
• Exact S-Matrices for Nonsimply-Laced Affine Toda Theories
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• S-matrix of N=2 Supersymmetric Sine-Gordon Theory
• Integrable N=2 Supersymmetric Field Theories
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• Supersymmetric, Integrable Toda Field Theories: The B(1,1) Model
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Cited by in corpus (6)

• Quantum Integrable Systems from Supergroup Gauge Theories
• Classical r-matrices of two and three dimensional Lie super-bialgebras and their Poisson-Lie supergroups
• Quantum Supersymmetric Toda-mKdV Hierarchies
• Conserved Charges in the Principal Chiral Model on a Supergroup
• Quantum Integrability of Coupled N=1 Super Sine/Sinh-Gordon Theories and the Lie Superalgebra D(2,1;\A)
• Generalized Calogero-Moser system and supergroup gauge origami