On Symmetries and Cohomological Invariants of Equations Possessing Flat Representations

We study the equation Efc of flat connections in a given fiber bundle and discover a specific geometric structure on it, which we call a flat representation. We generalize this notion to arbitrary PDE and prove that flat representations of an equation E are in 1-1 correspondence with morphisms φ : E → Efc, where E and Efc are treated as submanifolds of infinite jet spaces. We show that flat representations include several known types of zero-curvature formulations of PDEs. In particular, the Lax pairs of the self-dual Yang–Mills equations and their reductions are of this type. With each flat representation φ we associate a complex Cφ of vectorvalued differential forms such that H(Cφ) describes infinitesimal deformations of the flat structure, which are responsible, in particular, for parameters in Bäcklund transformations. In addition, each higher infinitesimal symmetry S of E defines a 1-cocycle cS of Cφ. Symmetries with exact cS form a subalgebra reflecting some geometric properties of E and φ. We show that the complex corresponding to Efc itself is 0-acyclic and 1-acyclic (independently of the bundle topology), which means that higher symmetries of Efc are exhausted by generalized gauge ones, and compute the bracket on 0-cochains induced by commutation of symmetries.