Affine Toda field theory and exact S-matrices (original) (raw)
The masses and three-point couplings for all affine Toda theories are calculated. The exact factorisable S-matrices are conjectured on the basis of the classical masses and couplings and found, in the case of theories based on simply-laced algebras, to give consistent solutions to the bootstrap. An investigation of the properties of the exact S-matrices in perturbation theory is begun but non-perturbative methods will be required to understand the conjectured duality between weak and strong coupling which appears to be a striking feature of these theories. * This conclusion has also been reached, from a different starting point, by Eguchi and Yang [6]. H. W. Braden et al. / Toda field theory effectively the scale of the roots ß in an unusual way ; in fact /3 is pure imaginary [4]. For example, in the case of the Ising model (or rather the conformal field theory for which c = z), studied by Zamolodchikov, one is required to take ß z /4-r7 =-î2 to obtain the correct value of c for the theory in which the set of simple roots is chosen to be the one belonging to the Lie algebra of Es , and normalised so that a? = 2. From the point of view of the S-matrices we consider in sect. 3, such a choice of scale seems distinctly odd, as it does from the point of view of perturbative field theory. It is a challenge to make sense of it from the perturbative point of view. If in eq. (2.2) we shifted the fields by setting f,->~P~+-ln 2 , ß f ai the field equation would remain the same except that C would be replaced by the Cartan matrix of the algebra : 2a ; , aj C; j = * There are slight changes only for the untwisted non-simply-laced cases, described later. ** Lower case letters denote the affine diagram or algebra; upper case is reserved for the corresponding finite algebra.
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