Unconstrained $\mathrm{SU}\left(2\right)\mathrm{}$ Yang-Mills theory with a topological term in the long-wavelength approximation

The Hamiltonian reduction of $\mathrm{SU}\left(2\right)\mathrm{}$ Yang-Mills theory for an arbitrary $\theta$ angle to an unconstrained nonlocal theory of a self-interacting positive definite symmetric $3\times 3$ matrix field $S\left(x\right)\mathrm{}$ is performed. It is shown that, after exact projection to a reduced phase space, the density of the Pontryagin index remains a pure divergence, proving the $\theta$ independence of the unconstrained theory obtained. An expansion of the nonlocal kinetic part of the Hamiltonian in powers of the inverse coupling constant and truncation to lowest order, however, lead to violation of the $\theta$ independence of the theory. In order to maintain this property on the level of the local approximate theory, a modified expansion in the inverse coupling constant is suggested, which for a vanishing $\theta$ angle coincides with the original expansion. The corresponding approximate Lagrangian up to second order in derivatives is obtained, and the explicit form of the unconstrained analogue of the Chern-Simons current linear in derivatives is given. Finally, for the case of degenerate field configurations $S\left(x\right)\mathrm{}$ with $\mathrm{rank}\Vert S\Vert =1,$ a nonlinear σ-type model is obtained, with the Pontryagin topological term reducing to the Hopf invariant of the mapping from the three-sphere $S^3$ to the unit two-sphere $S^2$ in the Whitehead form.