# Calabi Yau compactification and M theory

Ok, I really know nothing about this subject. Yeah, I know the definition, a Kahler manifold with a vanishing first chern class and a su(n) holonomy. I do think it's a cool mathematical concept that you could compact non-compact dimensions and in doing such you can generate the coupling constants in physics for the compact parts, su(3)Xsu(2)Xu(1), but how do you compact a non compact dimension? I can kind of see it like taking and infinitely long piece of string and mapping it onto a circle over and over, but how is this done mathematically, or is the Calabi yau manifold already considered to have six compact dimensions? Now M theory is considered to be in eleven dimensions so wouldn't you need seven compact dimensions not six to leave you with $$\mathbb{R}^4$$, but the only seven dimensional irreducible Riemann manifolds I see are so(7) and $$G_2$$. I don't think $$G_2$$ is large enough to squeeze the standard model into so that just leaves you with so(7). So to my extremely naive understanding of things it would seem to me that your looking for a 11 dimensional non-compact Einstein manifold whose maximal compact subgroup is so(7).
Something like $$dS_4 \otimes so(4)$$
I take it no such thing exists.

## Answers and Replies

ONow M theory is considered to be in eleven dimensions so wouldn't you need seven compact dimensions not six
I understand very little of string theory, but this part I think I can explain. There is some sense in which the eleventh dimension in M theory isn't "real". From wikipedia:

The coupling constant of string theory, which determines the probability of strings to split and reconnect, can be described by a field called dilaton. This in turn can be described as the size of an extra (eleventh) dimension which is compact. In this way string theory in ten dimensions (more precisely, type IIA string theory) can be described as the compactification of M-theory in eleven dimensions. Furthermore, different versions of string theory are related by different compactifications in a procedure known as T-duality.

So, M-theory is technically in eleven dimensions, but before you can consider an actual string theory based on M-theory the dilaton eats the eleventh dimension, kaluza-klein style, and you're left with ten. I think?

What I'm not sure about is whether the branes (which correspond to the cailbi-yau manifolds, right?) exist in the 10-dimensional perspective on the theory or just the 11-dimensional perspective.

blechman
The 11th dimension of M-theory is not part of the CY. You are typically interested in backgrounds that look like: $M_4\times CY_6\times(S^1/Z_2)$ - so you have Minkowski space x some Calabi-Yau manifold, and the whole thing lives on an orbifold.

Superstring theory itself does not make sense in 11 dimensions - it has fatal anomalies. So the string theory must live (propogate) on a 10D manifold, and you can think of the 11th dimension of M-theory as the "string thickness" - but the string doesn't "move" in that direction.

I'm not sure what Coin is saying about branes. Branes are not CY spaces! Rather, they are surfaces where open strings are allowed to end. They exist in string theory as well as M-theory and don't see the 11th dimension. At least, this is how D-branes work. I admit I know nothing about M-branes.

I hope that makes sense.

The 11th dimension of M-theory is not part of the CY. You are typically interested in backgrounds that look like: $M_4\times CY_6\times(S^1/Z_2)$ - so you have Minkowski space x some Calabi-Yau manifold, and the whole thing lives on an orbifold.

Superstring theory itself does not make sense in 11 dimensions - it has fatal anomalies. So the string theory must live (propogate) on a 10D manifold, and you can think of the 11th dimension of M-theory as the "string thickness" - but the string doesn't "move" in that direction.

I'm not sure what Coin is saying about branes. Branes are not CY spaces! Rather, they are surfaces where open strings are allowed to end. They exist in string theory as well as M-theory and don't see the 11th dimension. At least, this is how D-branes work. I admit I know nothing about M-branes.

I hope that makes sense.

Ok, so the Calabi Yau manifolds are six dimensional compact manifolds where the ends of the D branes can attach. From this picture, where does the Minkowski space and the orbifold enter, and when they use the term compactification what are they referring to? Sorry, I really know nothing about this subject. Also, I've heard Witten say you can not define quantum gravity in de sitter space, or an S matrix for de Sitter space. Why is this true?

blechman
Ok, so the Calabi Yau manifolds are six dimensional compact manifolds where the ends of the D branes can attach.

No, D-branes are surfaces where the ends of the strings live. The background spacetime is Minkowski space x 6 curled up dimensions. You need 6 extra dimensions to cancel quantum anomalies. There are scenarios where there a less than 6, but you can never have more. At least in SUPERstring theory.

M-theory roughly says that there is now an additional dimension. It's not that things propogate in that direction, but rather you can think of the strings as having a "width" into that dimension. Think of the string as a strip, where the width goes into the 11th dimension and fills it completely, so there's no more room to move in that direction. That's a bit of a weak analogy, I admit, but at least it might get you thinking in the right direction.

From this picture, where does the Minkowski space and the orbifold enter, and when they use the term compactification what are they referring to? Sorry, I really know nothing about this subject.

*Any* dimension that is "curled up" (not infinite in extent) is a "compactified dimension". So the 6 dimensions that make the CY, as well as the 11th dimension of M-theory are all compactified dimensions. The 4 dimensions that we see are not compact (they are the usual Minkowski space).

Also, I've heard Witten say you can not define quantum gravity in de sitter space, or an S matrix for de Sitter space. Why is this true?

I'm not exactly sure. This is not my area of expertise. I will take a guess, but I would advise you not to take me at my word! You might not be able to define an S-matrix because of issues with "asymptotic states" that you need for such a construction. Recall that for an S-matrix to make sense, you have to have states that, when far enough away from the scattering potential, act as free states. This does not happen on a deSitter background. That's my guess. It could be wrong, so if anyone has a better explanation, I'd be happy to hear it.

Haelfix
Alternatively you can compactify Mtheory/SUGRA on manifolds with G2 holonomy (so called Joyce manifolds) that give rise to effective 4d N=1 Supersymmetric systems.

These have a host of nice properties for phenomenology, but have issues retrieving the chiral spectrum unless you consider rather singular metrics.

MTd2
Gold Member
Superstring theory itself does not make sense in 11 dimensions - it has fatal anomalies. So the string theory must live (propogate) on a 10D manifold, and you can think of the 11th dimension of M-theory as the "string thickness" - but the string doesn't "move" in that direction.

What kind of object exists in 11 dimensions that are actualy used to infer a complete theory, that is, M-Theory?

blechman
What kind of object exists in 11 dimensions that are actualy used to infer a complete theory, that is, M-Theory?

Again, let me emphasize that this is NOT my area of expertise, but it is my understanding that in M-theory, the 11th dimension is somewhat special - nothing "moves around" in it. Rather, it represents the "thickness" of the string, as I was saying before.

People talk about "M-branes" but I do not know what these are. If they are allowed to propogate in the extra dimension or not. It is my (rather weak) understanding that nobody really knows how M-theory is supposed to work ("M" for "Mysterious!").

As to Haelfix's G2 holonomy - I have heard of this in seminars but I confess to being totally ignorant. As you say, I know that the problem with all these things is the lack of a chiral spectrum. E8 has the same problem, but you can fix it by using orbifolds or CY's (it is my understanding that this is the motivation behind CYs in string theory to begin with!).

Ok, let me see if I got it.

Let me see if I got this straight
There's these branes floating around in 10d space, a 4D minkowski space and six compact dimensions with holonomy SU(3). Now strings can attach to these branes, and depending on the thickness of the string maybe determines what type of string theory we're talking about. I want to consider $$E_8 \times E_8$$. So $$E_8 \times E_8$$ attaches to this brane. Then how do you get $$E_6 \times E_8$$ where $$E_6$$ is your GUT and the other $$E_8$$ is in some hidden sector?

As far as not being able to describe an S matrix for de Sitter space,

You might not be able to define an S-matrix because of issues with "asymptotic states" that you need for such a construction. Recall that for an S-matrix to make sense, you have to have states that, when far enough away from the scattering potential, act as free states. This does not happen on a deSitter background. That's my guess. It could be wrong, so if anyone has a better explanation, I'd be happy to hear it.

this was my thought too, but it seems more like a theoretical problem than a practical one, to me, since a point at infinity for all intensive purposes is not really that far away.

As far as $$G_2$$ being the compact part of the background space, then you would only be left with $$\mathbb{R}^3$$ for the non compact part, which doesn't seem very interesting to me. That is unless you go up to eleven dimensions then you could have $$G_2 \times \mathbb{R}^4$$ where $$\mathbb{R}^4$$ is minkowski space, but now wouldn't you have anamolies?

I also think I heard Witten say it may not be possible for quantum gravity to even exist.
I do not know the context in which he made this statement, maybe someone could clarify it for me. Is he saying there are no quantum gravitational effects? I don't find that likely. Or, is he saying that mathematics can't describe quantum gravity? Either way, it sounds like a crazy statement, to me.

I still have no idea how the orbifold enters the picture, and am really not clear how any of this works.

blechman
Let me see if I got this straight
There's these branes floating around in 10d space, a 4D minkowski space and six compact dimensions with holonomy SU(3). Now strings can attach to these branes, and depending on the thickness of the string maybe determines what type of string theory we're talking about. I want to consider $$E_8 \times E_8$$. So $$E_8 \times E_8$$ attaches to this brane. Then how do you get $$E_6 \times E_8$$ where $$E_6$$ is your GUT and the other $$E_8$$ is in some hidden sector?

this just comes from the boundary conditions inside the compact dimensions - they can break the symmetry down to a smaller group.

As far as not being able to describe an S matrix for de Sitter space[...]
this was my thought too, but it seems more like a theoretical problem than a practical one, to me, since a point at infinity for all intensive purposes is not really that far away.

I'm not sure what you mean. If you cannot define asymptotic states, then you cannot define an S-matrix. What does "theoretical" and "practical" mean in this context?

As far as $$G_2$$ being the compact part of the background space, then you would only be left with $$\mathbb{R}^3$$ for the non compact part, which doesn't seem very interesting to me. That is unless you go up to eleven dimensions then you could have $$G_2 \times \mathbb{R}^4$$ where $$\mathbb{R}^4$$ is minkowski space, but now wouldn't you have anamolies?

again, someone else should probably expain this better. But for the G_2 case, I think you actually do compactify the 7 dimensions of M-theory. You don't have anomalies as long as the string doesn't propagate in one of the 7 dimensions that make up the G_2.

I also think I heard Witten say it may not be possible for quantum gravity to even exist.
I do not know the context in which he made this statement, maybe someone could clarify it for me. Is he saying there are no quantum gravitational effects? I don't find that likely. Or, is he saying that mathematics can't describe quantum gravity? Either way, it sounds like a crazy statement, to me.

Well, although I don't consider myself part of the "Witten-cult", he is a smart guy and I'm sure whatever he said, he meant. But someone else should clarify this point, since I"m not sure what it's referring to.

I still have no idea how the orbifold enters the picture, and am really not clear how any of this works.

The orbifold is there in Horava-Witten theory where the 11th dimension is just a line segment. That's all that is.

The 11th dimension of M-theory is not part of the CY. You
I'm not sure what Coin is saying about branes. Branes are not CY spaces! Rather, they are surfaces where open strings are allowed to end. They exist in string theory as well as M-theory and don't see the 11th dimension. At least, this is how D-branes work. I admit I know nothing about M-branes.

Whoops! Okay, I guess I really messed that up. I probably shouldn't have included that last paragraph.

Do you mind if I poke at this a little more? Basically I think what had me confused is, sometimes I see writings where something that happens to a brane is discussed as if it were the same as a change to the background geometry of the string theory related to that brane. For example, here's a paper (discussed here earlier in the "time stopping" topic), where they discuss a hypothetical universe whose "braneworld is about to change from Lorentzian to Euclidean signature". When this happens they *appear* to be saying both that a brane (the embedded in 11d space) is changing its geometry, and also that the geometry of some universe is changing.

When I first saw that, I thought that the idea was that there was a universe made up of strings stuck to a certain brane, and when the geometry of the brane changed then the geometry of the universe changed-- and I took this to mean that the surface of the brane was actually the spacetime manifold of the universe. Now that I think about it I think I can see reasons why this must be wrong.

How, if at all, am I supposed to interpret this kind of thing, like from the link above, when I see it?

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blechman
The idea of "braneworlds" is that we mere mortals "live" on hypersurfaces of the 11D universe. So, for instance, we might live on D3-brane (four-dimensional surface) that itself is embedded in the larger 10 or 11 dimensions. Our brane has curvature of its own, even if the 11dimensions are flat. That's the idea.

In these scenarios, for example, what we perceive as "quarks" are actually the ENDS of the strings that live on our brane. And mesons are, for example, a full string (one end is a quark, the other end is an antiquark, and the fact that it's a string is why you can't split the quark-antiquark apart - you would just break the string and make 2 mesons). This idea actually goes WAY back to the founding of string theory, whose original purpose was to describe the strong nuclear force. Recently, these ideas have resurfaced, with branes being the place where we "observe" the mesons as q-qbar pairs.

Hope that helps!

That does help, thanks. So in the "braneworld" scenario, is there NO calabi-yau manifold? The D3-brane is just flat with respect to the other six dimensions?

Or is the curved-braneworld approach in some way dual to the approach where the "extra" six dimensions are curled up as a calabi-yau manifold?

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M-theory roughly says that there is now an additional dimension. It's not that things propogate in that direction, but rather you can think of the strings as having a "width" into that dimension. Think of the string as a strip, where the width goes into the 11th dimension and fills it completely, so there's no more room to move in that direction. That's a bit of a weak analogy, I admit, but at least it might get you thinking in the right direction.

Do you have a reference for your statement about the 11th dimension? Particle physicists are hoping to see gravitional effects from this 11th dimension who size could be on the order of mm.

blechman
Do you have a reference for your statement about the 11th dimension? Particle physicists are hoping to see gravitional effects from this 11th dimension who size could be on the order of mm.

Check out Brian Greene's book.

I don't know about particle physics searching for the "11th dimension of M-theory" - we are searching for AN extra dimension, but not necessarely this one coming from M-theory. Also, mm is a little too optimistic from the studies I've read. Check out papers by Adelberger, et al, for example.

nrqed
Homework Helper
Gold Member
The idea of "braneworlds" is that we mere mortals "live" on hypersurfaces of the 11D universe. So, for instance, we might live on D3-brane (four-dimensional surface) that itself is embedded in the larger 10 or 11 dimensions. Our brane has curvature of its own, even if the 11dimensions are flat. That's the idea.

In these scenarios, for example, what we perceive as "quarks" are actually the ENDS of the strings that live on our brane. And mesons are, for example, a full string (one end is a quark, the other end is an antiquark, and the fact that it's a string is why you can't split the quark-antiquark apart - you would just break the string and make 2 mesons). This idea actually goes WAY back to the founding of string theory, whose original purpose was to describe the strong nuclear force. Recently, these ideas have resurfaced, with branes being the place where we "observe" the mesons as q-qbar pairs.

Hope that helps!

A few stupid questions...

Something that always puzzled me in those scenarios is this: if we are on the brane, then the brane is filling all space for us. How do we perceive this? I mean, particles are actually the ends of open strings attached to the brane but what about the brane itself. It has an energy density (it has a tension). How does that appear to us? Th eobvious guess might be dark energy or even dark matter but I have never seen this stated explicitly.

Second question: if an open string is attached to a brane then the brane is not a manifold, no? I mean, if I picture an open string attached to a brane, at the point of contact (which has a certain width corresponding to the width of the string) we don't have a manifold (for the same reason that a two dimensional plane with a one-dimensional line piercing it is not a manifold). Maybe that's not a problem or maybe I misunderstand "strings attached to a brane".

Third dumb question: I don't quite understand why closed strings may leave the surface of the brane but not open strings. If I try to picture a closed string emitted from the surface of a brane, it seems impossible without tearing. I probably miss something obvious.

arivero
Gold Member
And mesons are, for example, a full string (one end is a quark, the other end is an antiquark, and the fact that it's a string is why you can't split the quark-antiquark apart - you would just break the string and make 2 mesons). This idea actually goes WAY back to the founding of string theory, whose original purpose was to describe the strong nuclear force. Recently, these ideas have resurfaced, with branes being the place where we "observe" the mesons as q-qbar pairs.

Really? Is there some recent paper telling of branes being <<the place where we "observe" the mesons as q-qbar pairs>>?

arivero
Gold Member
Hmm and with N=3, of course.

I have read Greene's book. My basic problem with attempting to find a higher dimension is that all the arguements [I have seen] for finding a higher dimension are based on analogies with 'seeing' the effects of the 3rd dimesion in lower dimensions. Is there any mathematical studies of what can be seen of the 3rd dimension in 2D space?

blechman

A few stupid questions...

I am *WAY* outside my area of expertise here (I was Adam Falk's grad student, after all!) But I can try to give a few "stupid answers" to your NOT SO stupid questions. I make no promises that they are completely accurate...

Something that always puzzled me in those scenarios is this: if we are on the brane, then the brane is filling all space for us. How do we perceive this? I mean, particles are actually the ends of open strings attached to the brane but what about the brane itself. It has an energy density (it has a tension). How does that appear to us? Th eobvious guess might be dark energy or even dark matter but I have never seen this stated explicitly.

There has been research done in this direction (brane tension as dark energy), but as I know nothing about it, I am hesitant to say anything. Branes are topological defects, and therefore can support various (form) fields in the KK-decomposition. This is another way branes can make themselves known to us.

Second question: if an open string is attached to a brane then the brane is not a manifold, no? I mean, if I picture an open string attached to a brane, at the point of contact (which has a certain width corresponding to the width of the string) we don't have a manifold (for the same reason that a two dimensional plane with a one-dimensional line piercing it is not a manifold). Maybe that's not a problem or maybe I misunderstand "strings attached to a brane".

You are right: branes are not (Riemann) manifolds. They are "topological defects". The idea is this: if you take a look at the open string action and you do the usual thing to find the equations of motion you end up with a total divergence which becomes a surface term (this is just the usual classical Hamilton principle story). This term has to vanish, but there are several ways this can happen. One way is to make the derivative of the string coordinate fields vanish at the boundaries (Neuman BC). This is what you naively do to get an open string. The other thing that will work is if the string is periodic - this will give you a closed string. HOWEVER, it turns out you can also fix the value of the coordinates themselves (Dirichlet BC), and this is something quite different. When you fix the endpoints of the string, you generate a coordinate manifold of all the points where the (open) strings can end. THIS object is what is called a "D-brane" (D=Dirichlet). So it is a topological manifold.

Third dumb question: I don't quite understand why closed strings may leave the surface of the brane but not open strings. If I try to picture a closed string emitted from the surface of a brane, it seems impossible without tearing. I probably miss something obvious.

see my previous comments. think about how open strings can form closed strings by endpoints coming together. When that happens the string changes its boundary conditions (Dirichlet -> Periodic) and branes are no longer required. Remember that implicitly whenever there is a brane, it means that there are a bunch of open strings coming off of that brane. A brane without open strings doesn't make sense!

Really? Is there some recent paper telling of branes being <<the place where we "observe" the mesons as q-qbar pairs>>?

it is the definition of a brane (see above, Polchinski's text, Zwieback's text).

Hmm and with N=3, of course.

Well, of course we don't see N=3 (!!!) Usually this works in the Large-N limit. A lot of these ideas were presented old-school in the classic text by Green-Schwarz-Witten, and in a more recent incarnation in hep-th/9905111. Of course, all these sources are quite technical.

Anyway, hope that helps. And remember that this is beyond my area of expertise (I've studied it on the side, but never done actual research in it).

nrqed
Homework Helper
Gold Member
I am *WAY* outside my area of expertise here (I was Adam Falk's grad student, after all!) But I can try to give a few "stupid answers" to your NOT SO stupid questions. I make no promises that they are completely accurate...
Ah! Nice to "meet" you. I am a former student of Peter Lepage..

There has been research done in this direction (brane tension as dark energy), but as I know nothing about it, I am hesitant to say anything. Branes are topological defects, and therefore can support various (form) fields in the KK-decomposition. This is another way branes can make themselves known to us.

You are right: branes are not (Riemann) manifolds. They are "topological defects". The idea is this: if you take a look at the open string action and you do the usual thing to find the equations of motion you end up with a total divergence which becomes a surface term (this is just the usual classical Hamilton principle story). This term has to vanish, but there are several ways this can happen. One way is to make the derivative of the string coordinate fields vanish at the boundaries (Neuman BC). This is what you naively do to get an open string. The other thing that will work is if the string is periodic - this will give you a closed string. HOWEVER, it turns out you can also fix the value of the coordinates themselves (Dirichlet BC), and this is something quite different. When you fix the endpoints of the string, you generate a coordinate manifold of all the points where the (open) strings can end. THIS object is what is called a "D-brane" (D=Dirichlet). So it is a topological manifold.

see my previous comments. think about how open strings can form closed strings by endpoints coming together. When that happens the string changes its boundary conditions (Dirichlet -> Periodic) and branes are no longer required. Remember that implicitly whenever there is a brane, it means that there are a bunch of open strings coming off of that brane. A brane without open strings doesn't make sense!

Thanks for the summary. I know the basi ideas but part of me is puzzled by the branes having different dimensions than branes. Since branes are supposedly states of a more fundamental theory, it means that the fundamental thoery allows states of different dimensionalities to interact. So the wolrdvolume of the whole "thing" (brane with the open strings attached) is not a manifold. But that's the way it is, I guess...I don't know why it bothers me

Thanks again for your comments in this thread. your comments about the 11th dimension (as not being an extra degree of freedom but as being "filled" by the strings) were quite instructive to me.