Why are gauge anomalies so important for any model?
Secondly, any model has to respect the gauge anomalies cancellation requirement? If this isn't true, then why does one check their model to look for gauge anomalies cancellations?
Ok I am by no means an expert on the topic but can hopefully give a bit on insight on your question:
- It depends on what you mean by so important but I can sketch a few things out for you:
I am not aware where you stumbled across anomalies but there are in essence different types of anomalies:
Purely gauge, Mixed gauge global & Purely global, also sometimes known as t'Hooft anomalies.
Now note that in essence, they all work exactly the same calculation wise (up to switching out a few group factors here and there) but their interpretation is widely different.
Let's first sum up what the problem of these anomalies is in general (at least for the non pure global ones): These anomalies normally tell you that under a symmetry transformation of a certain group the path integral measure will pick up a phase factor depending solely on the group parameters, particle content, and a topological term (in 4-d this term depends on $F\wedge F$ and the manifold you are integrating over, where $F$ is the field strength tensor).
Now what does this tell us? First of all, these anomalies are purely Quantum effects as they come about from the path integral, something which classical physics can not depend upon. Second of all this makes it seem like these anomalies lead to the symmetry not being a symmetry of the The second point now depends on the type of anomaly we are talking about:
Let us first look at gauge anomalies your mentioned:
The above argument seems to imply that a gauge transformation is no longer a symmetry of our quantum theory if the anomaly does not vanish! But that can never be the case as a gauge symmetry is by definition a redundancy of our theory and as such can never be broken! Therefore we better make sure that this is never broken! i.e that the anomaly of our theory equates to $0$! I think this also answers your second question we do always have to check if the gauge anomaly canceles for our theory to valid as a quantum theory.
Now onto the other types of anomalies:
The mixed gauge, global anomaly has the same problem that it looks like one of the symmetries involved here is no longer valid in the quantum case, but this is no longer a problem, it simply tells us that the global symmetry group we are looking at is no longer valid in the quantum theory! As such these types of anomalies can give us insight on what symmetries survive quantization.
One concrete example of this is the U(1) axial symmetry in QED, which is a symmetry of the classical but not quantum theory.
Now the last type of anomaly is the most subtle but probably also the most powerful one:
t'Hooft anomalies at first just seem to tell us that if we were to gauge one of the symmetry groups that were involved in the calculation, we would either break a global symmetry or just have a sick theory altogether but what happens if you do not plan on doing so? Are they still useful?
The answer is yes and in my opinion they turn out the be the most useful because of the following fact:
Anomalies do not change under RG-flow (up to spontaneous symmetry breaking). Thus the anomalies of the IR- theory have to match those of the UV theory. Why is this useful? Look for example at QCD a theory that flows to strong coupling in the IR. This is a regime where perturbation theory fails completely and it is a major open problem in physics to understand what happens there. We have very few tools to talk about this theory but one of them are anomalies. We know that whatever pyhsics we conjecture to take place, has to have the same t'Hooft anomalies. For example people expect that this theory confines in a certain way and one way to check if that is that that the anomalies have to match! Now this is by no means a proof for this statement but puts a very strong restriction upon the types of allowed models!
If you want to read more about this or see some actual calculations of this stuff I can heavily recommend reading the sections on anomalies in David Tongs notes on Supersymmetry or Gauge theories, both of which can be found here.