Variational cohomology and topological solitons in Yang-Mills-Chern-Simons theories
Ekkehart Winterroth
TL;DR
This work shows that cohomological obstructions arising in the variational formulation of gauge theories, captured by $[\Xi \rfloor \eta_{CS}]$, can prevent global solutions for Chern–Simons and Yang–Mills–Chern–Simons theories on odd-dimensional manifolds. Using Krupka's variational sequence on jet bundles, it demonstrates that the obstruction vanishes precisely when the $p$th Chern character $ch_p(\boldsymbol{P})$ vanishes, linking solvability to topological data of the bundle. Consequently, the Euler–Lagrange equations for YM–CS theories on compact $(2p{+}1)$-manifolds typically admit no solutions whenever $0 \neq ch_p(\boldsymbol{P})$, a result with strong implications for five-dimensional holographic QCD models. The paper further discusses how these obstructions interact with holographic constructions (e.g., Sakai–Sugimoto) and the need for careful treatment of noncompact extensions to ensure finite action and consistent soliton physics. These findings constrain the use of YM–CS frameworks in holographic QCD and illuminate the boundary–bulk interplay from a cohomological perspective.
Abstract
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 $\geq 5$ 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 $I\!\!R^{5}$) are discussed.