Super string theory establishes that at each point of our 4 dimensional spacetime, there is attached a 6-dimensional Calabi-Yau (CY) manifold which encodes the geometry of the additional compactified spatial dimensions needed to formulate the theory. It is postulated that the size of these compactified dimensions is of the order of the Planck scale.

On the other hand, I've always understood that in quantum gravity, we are confronted with the big problem of wild quantum fluctuations -due to the Heisenberg principle- at the Planck scale.These fluctuations should lead to wild shifts and fluctuations in the metric coefficients and hence, they would make hard to properly talk in the terms of differential geometry (manifold, distance, metric, curvature) that are the terms in which we talk about CY manifolds.

I'd like to know how string theory solves this contradiction that has arisen due to my lack of knowledge of the full technical details of the theory. Is it possible that, the conformal invariance required for the string worldsheet during its propagation, prevents the pathological effects of these quantum fluctuations allowing us to talk in terms of geometry even at the Planck scale?


2 Answers 2


There are a two levels for the answer:

First, perturbative string theory actually has two different scales: the string scale in which the extended nature of the string comes into play (even classically strings are different than particles), and the Planck scale in which the quantum mechanics of the string (including quantum gravity) becomes important. The string energy scale is lower than the Planck energy scale, their ratio is the string coupling constant (which is proportional to $\hbar$), so it is meaningful to talk about stringy physics in the sub-Planckian regime. Your mental picture then depends on the size of the compact geometry.

When the CY manifolds are larger the Planck length you can use perturbative string theory, in which case the notion of a manifold still kind of makes sense. When reaching the Planck scale the string theory is no longer perturbative and large fluctuations in the geometry are part of the story. Since we know quite a bit about non-perturbative string theory (through dualities and non-renormalization theorems) these expected phenomena can be observed and analyzed precisely. For example, topology change of the CY manifold is a smooth process that can be discussed very precisely in this framework.

(Incidentally, in this context we see that in the transition region between two well-defined geometries there is no geometrical description at all - contrary to some people's intuition about quantum geometry and spacetime foam and all the rest of that. The geometrical description fails and there is some well-defined procedure to do any well-defined calculation without any reference to any type of geometry whatsoever).

The second level of answer is that even when discussing perturbative strings on sub-Planckian CY manifolds, you have to keep in mind that this is a shorthand for something more precise and technical - (2,2) SCFT which has a limit in which it becomes a sigma model with a CY manifold as a target space. In English this means that even classical strings in some target space probe the geometry in different ways than point particles, and to describe their physics you need more complicated machinery than just differential geometry. As you suspected the fact that the string is extended means that it is more forgiving to non-smooth features of the manifold, even classically. This is encoded in the fact that the SCFT is better behaved than differential geometry on that space, and stays well defined even in the presence of some types of singularities (orbifolds of space are a famous example).

So, in summary "string theory on CY" really means string theory whose classical point particle limit (valid when the manifold is very large and smooth) reduces to CY manifold. Good to remember in general that there is some poetic license taken sometimes when physicists describe their work.

  • $\begingroup$ Thanks for the answer @Moshe. It solves my doubts as well as points me to some technicalities I'd like to learn a bit more. $\endgroup$
    – xavimol
    Commented Mar 16, 2011 at 17:54
  • 1
    $\begingroup$ This is a pretty non-technical answer. I'd recommend Brian Greene's TASI lectures "string theory on CY manifolds" as one source where you can learn the basic technicalities. $\endgroup$
    – user566
    Commented Mar 16, 2011 at 17:56
  • $\begingroup$ @Moshe. Yes but that's precisely what I needed. First a non technical answer to allocate myself in the way of thinking about this. Thanks for the reference. $\endgroup$
    – xavimol
    Commented Mar 16, 2011 at 18:01
  • $\begingroup$ @Moshe, +1 and this is why - this is probably the first time i've read an answer (or at least the suggestion that some answer is known) to the question of how topology change is treated mathematically in these theories. Now, from your answer i infer there is a dual description to the geometrical one that we can use to temporarily switch between two topologically inequivalent geometries? is this something that can be generically applied to jump the singularities of topological transitions or is specific to CY manifolds? $\endgroup$
    – lurscher
    Commented Mar 16, 2011 at 20:36
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    $\begingroup$ @lurscher: The general picture is the following: there is a mathematical description that is valid everywhere, which reduces in some limit to CY with specific geometry, and in other limit to another CY with a different topology. In the middle, there is no meaningful sense in which string theory is "on CY manifold", or any other geometry. Note that there is no "jump", it is just a smooth transition from equivalent descriptions of the same physics - one more useful in one limit, the other in the opposite limit. Either one of this geometries is always an approximation to something more detailed. $\endgroup$
    – user566
    Commented Mar 16, 2011 at 20:46

A compactified direction in $10$ dimension has some circle so that $x^9~=~x^9~+~2\pi R$ results in a huge mass. For the $n^{th}$ quantized momentum state this mass is $m~=~n/R$. So the large quantum fluctuations compete with the huge mass. For $R~\simeq~\ell_s$ $=~4\pi\sqrt{\alpha'}$ The tension of the string is $T~\simeq~1/\alpha'$ and the energy of the string is given by the winding number $w$ multiplied bythe energy $E~=~\oint Tdr$ or $$ E~=~2\pi wTR~=~\frac{wR}{\alpha'}, $$ which contributes to the mass in a T-duality which is invariant under $R~\rightarrow~\alpha'/R$. This mass is the energy associated with the Heisenberg uncertainty principle and the tiny localization of the $x^9$ coordinate direction. The interchange means this is equivalent to a theory much larger than the string scale. This exchanges type IIA and type IIB string theories.

  • $\begingroup$ Dear Lawrence, just a simple comment explaining to you why your comment can't possibly have anything to do with the question: Calabi-Yau spaces are simply connected so there can't be any nonzero winding $w$. So you have just written some stuff that is vaguely related to string theory but has nothing to do with the question. $\endgroup$ Commented Mar 16, 2011 at 20:58
  • $\begingroup$ CY manifolds can be tori or projective spaces which wrap strings or branes. So I don’t see what is wrong with this little argument. $\endgroup$ Commented Mar 16, 2011 at 22:04
  • $\begingroup$ @Luboš: why do you need CY to be simply connected? According to some definitions they need not be and can have finite fundamental group (e.g. Enriques surface with cover being a K3 surface). Is it the case that this class of manifolds is just not used in string theory for some reason? $\endgroup$
    – Marek
    Commented Mar 17, 2011 at 8:23

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