I know very simple example of anomaly inflow. See section 4.4 in [David Tong: Lectures on Gauge Theory][1]. As I read, such mechanism have some applications in condensed matter and in quantum field theory, but I haven't face with some concrete applications. What is this application?

Essence of example:

 1. One consider 5d fermion with mass term, which depends on fifth coordinate (y):
$$
i\not\partial \psi + i \gamma^5 \partial_y \psi - m(y) \psi = 0
$$

[![][2]][2]

2. Exist **special solution** of this equation, after rewriting Dirac spinor $\psi$ as two Weyl spinors $\psi_+$ and $\psi_-$:
$$
\psi_+ = \exp\left(- \int^y dy^\prime m(y^\prime) \right) \chi_+ (x)
\;\;\;\;\;\;\;\;\;\;
\psi_- = 0
$$
$$
\partial_0\chi_+ - \sigma^i \partial_i \chi_+ = 0
$$
The profile is supported only in the vicinity of the domain wall; it dies off exponentially $∼ e^{−M|y|}$ as $y → ±\infty$. Importantly, there is no corresponding solution for $\psi_−$, since the profile must be of the form $exp(+\int dy^\prime m(y^\prime))$which now diverges exponentially in both directions. So we obtain **chiral fermion zero mode** that lives on domain wall.

3. Chiral anomaly of this zero mode cancel with 5d Chern-Simons term. So theory
$$
S = S^{5d}_{CS} + S^{4d}_{chiral fermion}
$$
is gauge invariant. This known as **anomaly inflow**.

I wanna to understand **more examples of such phenomena**. 

What kind of **useful information** can be extracted from such mechanism?

Could we consider **more exotic gauge symmetries**, what kind of generalisation allow such mechanism?

Also it is related to [this][3] question. As I understand, **appearing of topological theory on domain walls are related to anomaly inflow in discrete symmetry**. Some explanations of this phenomenona will be also appreciated!


  [1]: http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html
  [2]: https://i.sstatic.net/By1pL.jpg
  [3]: https://physics.stackexchange.com/q/540951/