Why are the extra compact dimensions stable and do not collapse?
I know the anomaly cancellation is the reason why the extra dimensions are necessary.
But I can not visulize how the anomaly cancellation, leaves these extra compact windows open?
Is there any hypothetical mechanism based on anomaly cancellation to extend the size of these small windows, so that I can assign the stability of such extra small dimensions to quantum fluctuations in that context?
Welcome to the world of moduli (fields).
In general, there is really no reason for the size of compact dimensions in a compact dimension to do anything specific. But if the theory we are compactifying has a dynamic metric and if it has the right structure, dynamics for these sizes (or even other aspects of the shape of the extra dimensions) can emerge.
To explain what a moduli field is, the archetypal toy model for compactification is Kaluza-Klein theory, where we compactify gravity in 5 dimensions on a circle, i.e. spacetime has the shape $\mathbb{R}^4\times S^1$. The metric splits as $$ g^{ab} = g^{\mu\nu} + g^{44}$$ (Roman indices for 5d indices, Greek indices for 4d indices) and the circumference of the circle $S^1$ is $$ 2\pi R = \int_{S^1} g^{44}$$ for $R$ the radius of the circle and we say $g^{44}$ (or any function of it) functions as a "moduli field" (this specific kind of modulus is often called a dilaton), meaning the value of this field directly controls the size of the compact dimension (the circle).
Now, in Kaluza-Klein theory the equations of motion don't really fix the dilaton - there's just a kinetic term for it and no potential - so there is no notion of stability here. However, in more complicated theories - in particular many superstring compactifications - there will be potential terms for these moduli fields in the action that lead to equations of motion that fix the (expectation value of/classical solution for) moduli at particular values. Alternatively people will sometimes explicitly add such potential terms - based on more or less well-motivated reasons - to get something that stabilizes the moduli at suitable non-zero values.
Examples for models with such moduli stabilization are the KKLT mechanism ("de Sitter Vacua in String Theory" by Kachru, Kallosh, Linde, Trivedi) or Randall-Sundrum models (a non-string-theoretic example of dimensional reduction). This is very much not an exhaustive list, but the "mechanism" for moduli stabilization will differ in each individual case - the only overall property shared is that there will be some moduli fields that are fixed by some sort of potential.
Indeed Witten has showed that the original version of the KK theory is semi-classically unstable and decays into Minkowski spacetime. Instability of the Kaluza-Klein vacuum
In the same paper he shows that Fermions are necessary for the stability of the KK vacuum.
In a second article, Witten argues for realistic KK theories descending from $11$ dimensional $\mathcal{N}=1$ Supergravity via compactification.(though still supersymmetric)
Search for a realistic Kaluza-Klein theory
Although as he claims, the quantum numbers of fermions are hard to achieve.
Later he shows that singular higher dimensional manifolds can accommodate chiral fermions. And such singularities can be resolved within string theory.
Chiral Fermions from Manifolds of $G_2$ Holonomy
Anomaly Cancellation On Manifolds Of $G_2$ Holonomy
Kowalski-Glikman then searches for SUSic vacuum solutions of such realistic KK theory and he concludes that there are these possibilities:
Either $M^{11}$ is stable(Minkowski) or it decays into compactified $AdS(7)*S^4$ or $AdS(4)*S^7$ via spontaneous compactification, although he also claims that either both $AdS(7)*S^4$ and $AdS(4)*S^7$ are stable, or both are unstable. Vacuum states in supersymmetric Kaluza-Klein theory
I think that the problem of stability is the most important one. In fact if it turns out that only $M^{11}$ is stable, then the Kaluza-Klein ideology will break down. In other words it would be shown, that Kaluza-Klein theory remains only a mathematical trick.
Indeed a conclusive result demands a general proof of the positive energy condition for arbitray dimensions as Witten claims too: A new proof of the positive energy theorem (discussion section)
On the other hand, String theorists argue that the problem of right quantum numbers for fermions and stability can be solved by String Theory. But still if $M^{11}$ turns out stable, KK and SUGRA per se turn out wrong in themselves!
