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

We study a set of cohomology classes which emerge in the cohomological formulations of the calculus of variations as obstructions to the existence of (global) solutions of the Euler-Lagrange equations of Chern-Simons gauge theories in higher dimensions $2p+1 > 3$. It seems to be 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 in odd dimensions $> 5$ we find a non trivial obstruction which leads to a quite strong non existence theorem for topological solitons/instantons. Applied to holographic QCD this reveals then a possible mathematical inconsistency. For solitons in the important Sakai-Sugimoto model this inconsistency takes the form that their $\mathfrak{u}_1$-component cannot decay sufficiently fast to "extend to infinity" like the $\mathfrak{su}_n$-component.