Significance of total divergence anomaly term

Question

What is the significance of the fact that the anomany term (calculated from the triangle diagram) is a total divergence? Or, in other words, what is the significance of $$\partial_\mu j^\mu_A\sim Tr(W\tilde{W}) =\text{a total divergence}$$ for global anomalies. I think this fact is related to why baryon number violation in standard model cannot be a perturbative process. Perhaps someone can illuminate.

Answer 1

Leaving out numerical factors, we have that

$$ \mathrm{d}j_A = \mathrm{Tr}(F \wedge F)$$

This already shows that we are dealing with a topological quantity, since the RHS is the second Chern character of the gauge field (or rather, the principal bundle associated to it). Now, there is also the (3D) Chern-Simons form

$$ \omega = \mathrm{Tr}(F \wedge A - \frac{1}{3}A \wedge A \wedge A)$$

and it is easily calculated that $\mathrm{d}\omega = \mathrm{Tr}(F \wedge F)$, and so, $\mathrm{d}j_A = \mathrm{d}\omega$. Now, we can obtain the Noether charge by taking a spacelike three-dimensional slice $\Sigma$ in our four-dimensional spacetime and integrating the current over it. Such a spacelike slice will, for ordinary Minkowski space, always be the boundary of some four-dimensional region $M$, and so we find:

$$ \int_\Sigma j_A = \int_{\partial M}j_A = \int_M \mathrm{d}j_A = \int_M \mathrm{d}\omega = \int_\Sigma \omega$$

The RHS of this is now a topological (and gauge) invariant quantity, since the Chern-Simons form does not depend on the choice of a metric on spacetime, and is well-known to produce a topological field theory.

Thus, the Noether charge does only depend on the topological structure of the gauge bundle over this slice, and the topological structure of the bundle is precisely what instantons describe (for more on instantons, topology and vacua, see my answer here). Instantons are non-perturbative because each of them is its own local minimum of the action, i.e. they are all vacua, while perturbative things are only ever stemming from fluctuations around a single vacuum.

Additionally, the integral $\int\mathrm{Tr}(F\wedge F)$ is a discrete term, taking on values $8\pi^2 k$ with integer $k \in \mathbb{Z}$, so it is not a smooth function of anything, but jumps discontinuously when the topology changes. In contrast, perturbative results should vary smoothly when sending the perturbatively small parameter to zero, which cannot be the case here.