Topology, rigid cosymmetries and linearization instability in higher gauge theories

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.