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.

Quantum supersymmetric Toda–mKdV hierarchies

“…It was already shown explicitly on the concrete examples that the hierarchies, based on the partly bosonic simple root systems are not invariant under the supersymmetry transformation (see e.g. [10], [11], [24]). The affine superalgebras which allow such root systems are of the following type [12]: A(m, m) (1) = sl(m+1, m+1) (1) , A(2m, 2m) (4) = sl(2m+1, 2m+1) (4) , D(2, 1, α) (1) .…”

QUANTUM INTEGRABILITY OF COUPLED N=1 SUPER SINE/SINH–GORDON THEORIES AND THE LIE SUPERALGEBRA D(2,1;α)

“…The reality properties of the fields can be changed by a trick analogous to (1.3) whenever there is a reflection symmetry of the affine Dynkin diagram, with different choices of reality conditions essentially corresponding to different real forms of a given complex algebra. In the simplest cases above, the algebra is of complex type A (1) 1 , with the sine-Gordon and sinh-Gordon theories corresponding to the real forms su (2) (1) and sl(2) (1) respectively. Supersymmetric versions of (1.1) and (1.2) are readily constructed.…”

“…But for other Toda models, with more than one field, the situation is rather subtle. The most systematic approach is to generalize the Lax-pair construction, replacing the underlying affine Lie algebra with an affine Lie superalgebra (see [1] for a review). Even then the theory is supersymmetric only if the system of simple roots used is totally fermionic.…”

“…The first systematic analysis of affine superalgebra theories along these lines was recently carried out in [1], to which we refer the reader for additional background, including more detailed references to earlier work. A number of cases emerged as being of special interest, and it is our aim in this paper to study the most novel and important of these, corresponding to the exceptional Lie superalgebra D(2, 1; α) (1) (the others can actually be recovered in various limits).…”

“…To pass from D(2, 1; α) to its affine extension D(2, 1; α) (1) , we must add an additional generator of non-zero grade, rendering the algebra infinite-dimensional. For our purposes it is sufficient to consider the projection of the new root system onto the horizontal or zero-grade subspace spanned by the original simple roots of D(2, 1; α).…”