6
$\begingroup$

In topology, there is a concept of compactification which is defined as follows.

A space $Z$ is a compactification of $X$ if $Z$ is compact Hausdorff and there exists an embedding $j:X \rightarrow Z $ such that $\overline{j(X)} = Z$.

I would like to know if the compactification in String theory is the same compactification defined in topology.

Improve this question
$\endgroup$
2
  • $\begingroup$ An embedding is a homeomorphism onto its image, therefore $j(X)\subsetneq Z$. $\endgroup$
    – Samuel Adrian Antz
    Commented Aug 14, 2022 at 23:00
  • $\begingroup$ @SamuelAdrianAntz Thanks for the response. I'm familiar with the concept of embedding but would like to know if it is related to compactification in String Theory. $\endgroup$
    – htr
    Commented Aug 14, 2022 at 23:04

2 Answers 2

Reset to default
4
$\begingroup$

No, different concepts.

The compactification in string theory is the process of taking a theory with extra infinitely extended dimensions (w.r.t. the canonical 3+1 dimensions) and modifying it so that the extra dimensions are finite.

Intuitively, compactness in topology does not refer to the size of a space, but to properties of not having "gaps" or "missing endpoints". In particular a compact space may very well extend infinitely, for instance the real line including the endpoints $+\infty$, $-\infty$.

Improve this answer
$\endgroup$
2
  • 1
    $\begingroup$ But how different are they? The real number line has no point at $\infty$. $\infty$ is unreachable. Compactification adds a point at $\infty$, making it reachable. Furthermore, $-\infty$ the same point. The real number line becomes a giant circle. The number plane becomes a giant sphere. This changes the properties of the space. For example, all continuous functions on the space are bounded. A compact space shares important properties with a finite space. $\endgroup$
    – mmesser314
    Commented Aug 15, 2022 at 2:49
  • 1
    $\begingroup$ It is true that string theory curls these dimensions up to microscopic sizes. But I think the string theory term compactification comes from the same term in topology, not from the size. $\endgroup$
    – mmesser314
    Commented Aug 15, 2022 at 2:49
4
$\begingroup$

I don't know much about compactification in string theory, but the idea is based on the compactification in (4+1) D Kaluza-Klein theory, proposed in the years after the theory of general relativity was published to unite gravity and electromagnetism. (The weak and strong interaction were not discovered yet.)

The idea proposed by Kaluza was to generalize the metric tensor to five dimensions to add the electromagnetic vector potential as the components of the fifth dimension.

The idea proposed by Klein was that this dimension cannot be observed because it is compactified and here we have the connection to the topological one-point or Alexandroff compactification: $\mathbb{R}$ can be mapped to $\mathbb{S}^1\setminus\{*\}$ by the stereographic projection (which is your map $j$) and then be compactified to $\mathbb{S}^1$ by adding the missing point. Spacetime is therefore not $\mathbb{R}^{1,4}$, but $\mathbb{R}^{1,3}\times\mathbb{S}^1$, the same topological space as the Anti-deSitter space $\operatorname{AdS}_5$.

The extension of this extra dimension curled up is described by another scalar field $\phi$ added to the theory and so small, that only microscopic objects can observe it. (Just like a rope looks like a one-dimensional line to us, but has a two-dimensional surface to an ant.) Electric charge for example can be described as the velocity of the particle along the curled up fifth dimension, but is far above that of light so the theory failed.

Decades later, string theory reused the idea again to compactify even more dimensions using six-dimensional Calabi-Yau manifolds to reduce the dimensions from ten to four.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.