On the existence of superconformal 2-tori and doubly periodic affine Toda fields
Superconformal surfaces are harmonic surfaces in \(\mathbb{C} P^n\) associated to the 2D Toda field equations. In particular, each superconformal 2-torus corresponds to a doubly periodic solution of the elliptic Toda equations. The paper under review is a continuation of two previous papers of the author [\textit{I. McIntosh}, Nonlinearity 7, No. 1, 85-108 (1994; Zbl 0840.58023) and Aspects Math. E23, 205-220 (1994; Zbl 0810.35110)], establishing the conditions required for the existence of superconformal 2-tori. The doubly periodic solutions of the Toda equations are known to possess a spectral curve (or, in other words, they are of finite type). The author uses the monodromy to identify among all finite type Toda solutions those which are doubly periodic. It is proved, first, that the monodromy can be expressed in terms of a scalar function (the Baker function) and then it is shown that the periodicity conditions can be expressed using differentials on the spectral curve. It turns out that, while there always exist doubly periodic Toda solutions, there are not enough free parameters to ensure the existence of superconformal 2-tori, except for the cases of \(\mathbb{C}P^1\) and \(\mathbb{C}P^2\). The situation is getting better if the harmonic maps have extra symmetries (for instance, maps with \(S^1\) symmetry may provide superconformal tori in \(\mathbb{C}P^3\)). In the final part of the paper it is shown that the finite type Toda solutions are those with finite dimensional orbit for the action of the infinite dimensional abelian group of higher flows and, moreover, the orbit is nothing else but the Jacobian of the spectral curve. The paper has also an appendix, where it is shown how the spectral curve can be constructed by using polynomial Killing fields, emphasizing the fact that the algebra of these Killing fields is, in fact, the coordinate ring of the spectral curve.
scientific article; zbMATH DE number 1115762
scientific article