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.

As for why there appears no "magnetic" field: magnetic fields are sourced by currents in Maxwell's equations, but there are no local sources or sinks for the Chern-Simons form - its theory is exactly analogous to that of the eletromagnetic field in 1+1 dimensions, and it has no propagating degrees of freedom.


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