Why do we solve the Wess-Zumino consistency condition using the method of descent? Consider a quantum field theory in $d$ dimensions with a symmetry $G$. For the purpose of this discussion let's say that $d$ is even and $G$ is a compact, connected Lie group. We say that the symmetry has a 't Hooft anomaly if it is not possible to gauge $G$. To detect this we couple the theory to a background gauge field $A$ for the symmetry $G$ and look for gauge dependence of the partition function. Variation of $\mathrm{log} \ Z(A)$ with gauge parameter $\epsilon$ defines a functional $I(\epsilon, A)$. It is a simple exercise to check that this functional has to satisfy the so called Wess-Zumino consistency conditions. 
Solutions of the consistency condition can be constructed using the method of descent: we start from an invariant polynomial in curvature in $(d+2)$-dimensions, to which one can associate a Chern-Simons theory in $(d+1)$-dimensions. Then the gauge variation of the Chern-Simons action on a $(d+1)$-dimensional manifold with a boundary is a local functional on the boundary (which is of dimension $d$) and satisfies the consistency condition for the same reason as the sought after $I(\epsilon, A)$ does. 
It is my impression that many people believe that these solutions of the consistency condition are the only ones (or perhaps at least the only ones which are relevant in the context of anomalies). Is this correct? If so, why?
 A: The case of perturbative anomalies is straightforward: the only such anomalies are of the ABJ type, i.e., they appear in even dimensions and are associated to chiral fermions. These are completely known, cf.
$$
I\sim \operatorname{tr} F^{d/2}+\operatorname{tr} R^{d/2}+\text{subleading}
$$
where $F$ is the strength tensor of your background gauge field, and $R$ the curvature tensor of the background metric. The subleading terms represent flavour-gravity mixed anomalies. One can derive the expression above by e.g. the well-known Fujikawa computation (cf. this PSE post).
As the anomaly is known, it can be checked by brute force that it indeed can be expressed through a local function in one (or two) higher dimension, namely
$$
I\sim [\hat A e^F]_{d+2}
$$
where $\hat A$ is the "A-roof genus" and $e^F$ the Chern character. A very detailed discussion can be found in ref.1., where e.g. in section 9 it is proved that $I$ satisfies the Wess-Zumino conditions, and it is in fact the only function that does (modulo a global coefficient, and modulo BRST exact counterterms).
The case of global anomalies is much more subtle. To the best of my knowledge, this is still an open question, i.e., it is not fully known whether "anomaly inflow" captures all the possible anomalies a quantum system may have. If this is the case, then anomalies are classified by bordism groups. For more on this, see e.g. refs.2–4.

References.


*

*Lectures on Anomalies, Adel Bilal, https://arxiv.org/abs/0802.0634.

*Fermionic Symmetry Protected Topological Phases and Cobordisms, Kapustin et al, https://arxiv.org/abs/1406.7329.

*Dai-Freed anomalies in particle physics, Iñaki García-Etxebarria, Miguel Montero, https://arxiv.org/abs/1808.00009.

*Anomaly Inflow and the $\eta$-Invariant, Edward Witten, Kazuya Yonekura, https://arxiv.org/abs/1909.08775.
