Lax equations in 10-dimensional supersymmetric classical Yang--Mills theories

J. Gervais

Teoretičeskaâ i matematičeskaâ fizika, Tome 123 (2000) no. 2, pp. 189-197

Résumé

Saveliev and the author recently showed that there exists an on-shell light-cone gauge where the nonlinear part of the field equations reduces to a (super) version of the Yang equations that can be solved using methods inspired by those previously developed for the self-dual Yang–Mills equations in four dimensions. Here, the analogy between these latter theories and the present ones is extended by writing a set of super linear partial differential equations that have consistency conditions derivable from the supersymmetric Yang–Mills equations in 10 dimensions and are analogues of the Belavin–Zakharov–Lax pair. In the simplest example of the two-pole ansatz, the same solution-generating techniques work as in the derivation of the multi-instanton solutions in the late 1970s. The present Lax representation, however, is only a consequence of (instead of being equivalent to) the field equations, in contrast to the Belavin–Zakharov–Lax pair.