We consider a class of non-linear PDE systems, whose equations possess
Noether identities (the equations are redundant), including non-variational
systems (not coming from Lagrangian field theories), where Noether identities
and infinitesimal gauge transformations need not be in bijection. We also
include theories with higher stage Noether identities, known as higher gauge
theories (if they are variational). Some of these systems are known to exhibit
linearization instabilities: there exist exact background solutions about which
a linearized solution is extendable to a family of exact solutions only if some
non-linear obstruction functionals vanish. We give a general, geometric
classification of a class of these linearization obstructions, which includes
as special cases all known ones for relativistic field theories (vacuum
Einstein, Yang-Mills, classical N=1 supergravity, etc.). Our classification
shows that obstructions arise due to the simultaneous presence of rigid
cosymmetries (generalized Killing condition) and non-trivial de Rham cohomology
classes (spacetime topology). The classification relies on a careful analysis
of the cohomologies of the on-shell Noether complex (consistent deformations),
adjoint Noether complex (rigid cosymmetries) and variational bicomplex
(conserved currents). An intermediate result also gives a criterion for
identifying non-linearities that do not lead to linearization instabilities.Comment: v2: 33 pages, added an important reference to earlier work of
Arms-Anderson, close to published versio