M-theory says that there's a Calabi-Yau manifold, representing $n = 7$ extra spatial dimensions (here simplified to $n = 3$; check out animated video) curled up and compactified inside every 3D Planck volume of our Universe.

enter image description here

The way I understood compactification from various popular sources is that extra dimensions in these manifolds are indeed finite - i.e., extra dimensions are not continuing outside each manifold (inside each 3D Planck volume), as if the product of their compactification literally looked like a "blob" from the picture above.

Therefore I believe that different manifolds are not connected to each other, and not interacting with each other - separated by their boundary conditions, which remain them stable inside their Planck volume.

So, here're my questions:

  1. How Uncertainty principle is okay with such precise positioning of CY manifolds (inside 1 Planck volume) - ?.. Shouldn't each of these manifolds obey superposition principle? Shouldn't they (or Strings inside them) possibly be able to interact with each other, via interference or something like that?

  2. (variation of the same question) How Strings are moving from 1 CY manifold to another if none of manifolds are continuing outside their Planck volume?

  3. (this one is related, but different from others; possibly concerning a landscape problem) How these manifolds can remain topologically identical for such a long time (since the Big Bang) if they're not connected and not interacting with each other?

  • 2
    $\begingroup$ Why do you think there is "one CY manifold curled up in every 3D Planck volume of our universe"? $\endgroup$ Jul 6, 2021 at 11:23
  • $\begingroup$ @NiharKarve Ultimately, I grasped (from various popular sources) that extra dimensions are (must be?) finite. Therefore, CY manifold is a solution; and if different sizes of these manifolds are possible (which answers the question #2) - then another question is, what is the physics behind their boundary conditions? Why this exact $n = 7$ size and shape? And why they're all topologically identical, despite different sizes (question #3 extend) - ? What are these "blobs" if not manifestations of the tiniest volume of spacetime itself?.. $\endgroup$ Jul 6, 2021 at 11:34
  • 4
    $\begingroup$ You have a couple of misconceptions about compactification. "Compactifying M-theory/string theory" does not mean that a physical process triggers 7/6 dimensions to spontaneously become small; rather, we posit that the spacetime is a product manifold of 3+1d flat spacetime and an "internal" manifold (e.g. Calabi-Yau) whose size is small. Correspondingly, there isn't a different a Calabi-Yau manifold at every point in spacetime. $\endgroup$ Jul 6, 2021 at 11:51
  • $\begingroup$ @NiharKarve imagine then a spaghetti pot - that's our observable universe. There're 7 (just 7) very long spaghetti inside that pot, twisted and curled up with each other something like CY manifold - that's our 7 extra dimensions. So, if we'll look at this pot as a whole - we won't see CY manifold as it's usually presented, but rather a $10^{180ish}$ times repeated pattern: that's how many Planck volumes inside our pot. And if we'll look at just 1 Planck volume of our pot - we would see just 1 CY manifold as it's usually presented (and a half for a half Planck volume, and so on). $\endgroup$ Jul 6, 2021 at 12:19
  • 1
    $\begingroup$ There is not one CY per Planck volume. If you want the small dimensions to be continuous, the large dimensions should surely also be continuous, so we can't have a discrete bunch of CY manifolds. Rather, like Nihar Karve said, we assume spacetime to be the product of 3+1d spacetime and a CY, in the same way that a ribbon is the product of a line segment with the real line. With the ribbon, we can see how it looks like a line segment times the real line, but we can't count how many line intervals it contains, since these segments are attached in a continuous way. $\endgroup$
    – Stijn B.
    Aug 16, 2021 at 6:48


Your Answer

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