Lax equations in ten dimensional supersymmetric classical Yang-Mills theories

TL;DR

This paper extends the analogy between self-dual Yang-Mills equations and ten-dimensional supersymmetric Yang-Mills theories by formulating a Lax pair and solution-generating techniques, revealing new integrability structures.

Contribution

It introduces a set of super partial linear differential equations as a Lax pair for ten-dimensional SUSY Yang-Mills, generalizing known methods from four-dimensional theories.

Findings

Super partial linear differential equations serve as a Lax pair.

Solution-generating techniques from instanton theory apply.

Lax representation is a consequence, not equivalent, of the field equations.

Abstract

In a recent paper (hep-th/9811108), Saveliev and the author showed that there exits an on-shell light cone gauge where the non-linear part of the field equations reduces to a (super) version of Yang's equations which may be solved by methods inspired by the ones previously developed for self-dual Yang-Mills equations in four dimensions. Here, the analogy between these latter theories and the present ones is pushed further by writing down a set of super partial linear differential equations whose consistency conditions may be derived from the SUSY Y-M equations in ten dimensions, and which are the analogues of the Lax pair of Belavin and Zakharov. On the simplest example of the two pole ansatz, it is shown that the same solution-generating techniques are at work, as for the derivation of the celebrated multi-instanton solutions carried out in the late seventies. The present Lax representation, however, is only a consequence of (instead of being equivalent to) the field equations, in contrast with the Belavin Zakharov Lax pair.