Variational cohomology and topological solitons in Yang-Mills-Chern-Simons theories

In cohomological formulations of the calculus of variations obstructions to the existence of (global) solutions of the Euler-Lagrange equations can arise in principle. It seems, however, quite common to assume that such obstructions always vanish, at least in the cases of interest in theoretical physics. This is not so: for Yang-Mills-Chern-Simons theories on compact manifolds in odd dimensions $\ge 5$ we find a non trivial obstruction which leads to a quite strong non existence theorem for topological solitons/instantons. The consequences of this result for the Yang-Mills-Chern-Simons theories of holographic QCD (on $I\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}{R}^5$) are discussed.