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?

  • 2
    $\begingroup$ Why would they "collapse"? Under which dynamics do they need to be stable? When we do dimensional reduction/compactification, the compact part is in general not dynamic (although there certainly are specific models in which it is), i.e. doesn't change at all, we're just reducing the theory on a static spacetime background. Is there some specific theory you're thinking about here where the compactification manifold is dynamic? $\endgroup$
    – ACuriousMind
    Commented Jan 11, 2023 at 15:38
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    $\begingroup$ @ACuriousMind honestly I just am deeply in trouble with these extra compact dimensions. I'd appreciate if you share some articles that introduce some dynamics maybe within string theory itself or maybe analyze these aspects of compact dimensions. Why the aperture of these small windows remain open at small sizes or does not fluctuate? $\endgroup$ Commented Jan 11, 2023 at 16:24
  • $\begingroup$ good question. $\endgroup$ Commented Jan 13, 2023 at 15:55

2 Answers 2


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.

  • $\begingroup$ Thanks. From what I've studied. Moduli should in general represent directions in the potential that the potential does not change. In all the classical cases I know of, these molduli fieds are blown up at the quantum level. But in Supersymmetric cases, there might be some local minimums but not global due to quantum corrections(both perturbative and none perturbative). What about these moduli fields? Are the minimas after quantum corrections local or global? I expect since susy is broken spontaneously then there's no global minima and any stability should be dur to a local minima. $\endgroup$ Commented Jan 11, 2023 at 17:33
  • $\begingroup$ @BastamTajik I'm not quite sure what you mean by "directions in the potential" - as I say in my answer you don't even necessarily have a potential for the moduli fields in general. $\endgroup$
    – ACuriousMind
    Commented Jan 11, 2023 at 18:03
  • $\begingroup$ You have to be careful, "moduli" is a very popular term that is often applied to various different objects. You seem to be talking about a different notion of "modulus", which is when there are multiple different VEVs for a field - the VEV lies on a vacuum manifold, and this is also called "moduli space" with the modulus indeed being a "direction" in the vacuum manifold, i.e. a direction in which the potential does not change. There is a relation to the usage in my answer, but these two notions are not the same. $\endgroup$
    – ACuriousMind
    Commented Jan 11, 2023 at 18:03
  • $\begingroup$ Yes I chose the wrong word, I mean flat directions in the vacuum manifold. Given this are these directions lifted in this case and local minimas appear? @ACuriousMind $\endgroup$ Commented Jan 12, 2023 at 13:12

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!


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