# Topological theta-term as a background electric/magnetic field?

The topological $$\theta$$-term in the Schwinger model (1+1-dimensional QED) can be interpreted as a background electric field, as explained in Chapter 7.1.2 of Tong's lecture notes. The same holds true for 1+1-dimensional QCD, as pointed out in this paper by Witten.

Now my question is: (how) does this change when we go to 3+1 dimensions?

I've read in several papers (e.g. by Lüscher or Dvali) that the $$\theta$$-term in 4-dimensional QCD gives rise to an (electric) non-Abelian background field. If this statement is true, why don't we get a non-Abelian magnetic background field as well? What happens if we go to compact 4-dimensional QED, which also exhibits a physical $$\theta$$-term, as pointed out in this paper by Polyakov?

The $$\theta$$-term exists in all even $$2n$$ dimensions and the integral is over $$F^n$$, where the exponent denotes the multiplicity of the exterior product of the 2-form $$F$$ with itself. Locally, this $$F^n$$ is the derivative of a Chern-Simons form and if you read e.g. Dvali's paper that you link closely, it is the "constant electric field" of this Chern-Simons form interpreted as a $$2n-1$$-dimensional analogue of the usual gauge potential that the $$\theta$$-angle corresponds to.
In 2 dimensions, the $$2n-1 = 1$$-dimensional Chern-Simons form is merely $$A$$, i.e. the vector potential itself, so we indeed obtain a true electric field interpretation of the $$\theta$$-angle.
In higher dimensions, the Chern-Simons form is not the same as the vector potential, but instead a $$2n-1$$-dimensional higher gauge field and correspondingly, the "constant electric field" associated to it has nothing to do with any constant electromagnetic or gluonic field strength. It is a purely formal analogy.