Compactification of Calabi-Yau manifolds

Question

Hey JO. Calabi-Yau manifolds get compactified to give a four dimensional theory (with extra dimensions). But who says that they should be compactified to give four dimensions? Shouldn't that be a consequence of the theory itself with no artificial techniques applied "from the outside" ? Of course it makes sense to make it four dimensional from a human perspective and older physics, but is that really the most sensible argument ? Because it doesn't fall out of the equations I guess. /edit: This resembles a criticism of stringtheory from Richard Feynman.



It's around 1:00:24

Comments

We don't compactify the CY manifolds themselves, you compactify theories on these manifolds.

---

This question looks very confused to me.

But what is true is that energy is needed to keep the extra dimensions rolled up, so they naturally would tend to be decompactified. Maybe a better related question would be to ask why exactly four dimensions are decompactified in our universe, and what is known about the mechanisms that can lead to decompactification?

---

@dimension10 Just like always, nothing to do with the question, your comment should be hidden. But thx@Dilaton, that was a reasonable comment.

---

@WolfnSheepSkin

If you are not satisfied with the answers given, you should probably edit the question to clarify what exactly you are asking (if you are asking about the decompactification issue I mentioned for example?)

Also, the comments about string-theory criticism included in the current version make the question look rather rhetorical or philosophical and therefore not a good fit for PhysicsOverflow. Voting to (temporary?) close.

---

WolfnSheepSkin is trolling us. This is question must be closed.

---

@Outlander for closing we need an additional closvote (=upvote) here, there are already 2 of the 3 closevotes needed ...

---

I pressed the link but I got the message "The question has been hidden". I would up-vote it anyway.

---

@Outlander oh yes my bad, the community moderation threads where one can vote to close etc need 500 rep to be accessed.

---

No I'm not trolling. That's a reasonable criticism of string theory expressed by Richard Feynman. Morons.

---

Ok HERE IS MY QUESTION: WHAT IS THE REASON THAT EXTRADIMENSIONS WANT TO COMPACTIFY?

---

@WolfInSheepskin Then ask a different question.

---

 @WolfInSheepSkin This is a subject that is studied in string why and how 4 dimensions, and is a rather involved subject. The picture seems to be that string theory has several low energy vacua. One of them could be our universe.

If you really want to understand these things you have to learn the subject.

see this,

http://arxiv.org/abs/hep-th/0509003 http://arxiv.org/abs/hep-th/0610102

Answer 1

The Calabi-Yau manifolds do not get compactified to anything. Strings get compactified on Calabi-Yau manifolds. We can choose any type of manifolds to compactify string theory on, for example a circle, a torus, the Klein-bottle and other similar topological spaces. Witten, Strominger, Candellas et al. trying to extract various properties of string theory that should resemble to properties of the four dimensional space-time we are used to along with mathematical consistency and physical properties (such as the number of supersymmetries they preserve) lead them to consider compactifications on CY. Strings by no means "need" to be compactified in four dimensions. CYs provides us a very possible way that the extra dimensions of string theory are "curled up". That's all.

Comments

So that's no answer at all. Thx for nothing. Everything is just artificially assumed and nothing follows from the equations. That's silly.

---

No, it is you that fail to understand the context here. We ask ourselves "what properties does a 4d theory need to satisfy"? Then once we know what we want we ask ourselves "what manifolds satisfy these properties"? This is how it works. If the answer satisfies these properties, making the theory the way it is supposed to be by physical requirements and mathematical consistencies there is no "ad hoc" procedure to worry about unless you want to discuss philosophical booboo.

---

This resembles criticism of stringtheory from Richard Feynman, not from me. But still this is not an answer. My question is, does it follow from the equations that you need compactification to get a four dimensional theory? Nope it doesn't, and your answer is silly. I updated my question.

---

No, your understanding is incomplete. Does it follow from the equations that the theory should be chiral or non-chiral? The  "need" for compactification makes sense if you want to understand things from a 4d perspective, the one we are familiar with. We are trying to find a way to obtain the data of the four dimensions we live on using this mechanism. Your question in the above comment is the silly one. Do the EOMs of the SM Lagrangian require a SSB? No, but we know particles have mass. Your logic is not valid.

---

Just look at my question again, you comment has nothing to do with it and you don't understand my question.

---

@conformal_gk I think wolfin is asking why a particular CY manifold should be chosen for our universe, and whether there is a theory that explains why this choice has to be what it is. This is to do with the Cosmic Landscape, right?

Answer 2

Calabi-Yau manifolds come in a complex number of dimensions of 1, 2, or 3. A manifold with \(n\) complex dimensions can be written with  \(2n\) real dimensions. We compactify 10-dimensional theories on 3-dimensional (remember that 3 is the number of complex dimensions) Calabi-Yau manifolds to obtain \(10-6=4\) dimensions. 

Comments

Not sure what this has to do with the question and your answer is off topic, so it should be deleted, but OK you never answered questions in a unique way anyway lol. And an answer with no Lubos reference?? That's weird for you dimension10.

---

wolfin is asking what motivates nature to do it like this? Is it inevitable or are there other possible alternatives?