Variational cohomology and Chern-Simons gauge theories in higher dimensions.

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 commonly assumed that such obstructions always vanish, at least in the cases of interest in theoretical physics. This is not so, neither for Chern-Simons theories of gravity nor for Yang-Mills-Chern-Simons theories. In the case of unitary Chern-Simons gravity we show that there are non trivial obstructions and that they are sharp, i.e. there are (global) solutions if and only these obstruction vanish; the procedure employed can be applied in principle also in the case of special orthogonal Chern-Simons gravity, at least for constructing specific solutions. For Yang-Mills-Chern-Simons theories in odd dimensions $> 5$ we find a quite strong non existence theorem which raises doubts about the mathematical consistency of the use of solitons/instantons in holographic QCD.