# D8-Branes What do they bring?

## Main Question or Discussion Point

In the theory of superstrings (super due to the addition of supergravity) D-branes are compactified on a circle. We know, theoretically, that D5 is compactified to give us electromagnetism. Then there is the M-theory that gives us D11 as supergravity. Now the only question I have, which I have searched for and cannot find, is what does D8 give us?

Now as a warning, please ignore my basic terminology. I am not an expert in this field and will never claim to be. I just am curious to see if anyone knows.

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D9 not D8

I think you might mean D9. D9 would give you 'strings' and thus all the particles of the universe. D9 would consist of D3 which is the normal spatial dimensions and a D6 which are compacted and needed for the C-Y manifolds. Time would make it a D10.

I understand that strings are D9 existing in 10 dimensional space (9 dimensions + 1 time) but I don't understand the significance of D8.

I do hope that I am explaining my question correctly... I just want to know what the compactification of D8 does to spacetime.

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In the theory of superstrings (super due to the addition of supergravity) D-branes are compactified on a circle. We know, theoretically, that D5 is compactified to give us electromagnetism. Then there is the M-theory that gives us D11 as supergravity. Now the only question I have, which I have searched for and cannot find, is what does D8 give us?
Hello DarKonion

In your question, it appears you are using 'D' for mathematical dimension, as in D=5 Kaluza-Klein theory and D=11 supergravity. This is different than asserting that, for instance, D5 branes give rise to electromagnetism and D11 branes yield supergravity. I'll explain why below.

When superstring theory is studied nonperturbatively, one indeed finds the theory admits objects with p spatial dimensions, called p-branes. For the p=1 case, we recover fundamental string, which can be regarded as a 1-brane. In D=11 supergravity and M-theory, one only finds the 2-brane and 5-brane.

Superstring theories contain p-branes of even and odd dimension, ranging from p=-1 to p=9, called Dp-branes. Open fundamental strings can have endpoints that are free to move about (Neumann boundary conditions) or fixed to some p-dimensional object (Dirichlet boundary conditions) which we call Dp-branes or just D-branes. The 'D' in D-brane is short for Dirichlet.

It turns out that Yang-Mills quantum field theories reside on the worldvolumes of D-branes. Given N (coincident) D-branes, the gauge symmetry of the Yang-Mills worldvolume theory is the freedom the string has in deciding which of the N branes to end on. The symmetry group arising from this freedom, in the case of N (coincident) D-branes with oriented open strings, is the unitary group U(N), e.g., N D8-branes which coincide corresponds to an unbroken U(N) gauge group.

So for N=1, just one D-brane, the string has U(1) freedom in deciding where to end on the single D-brane. The worldvolume U(1) Yang-Mills theory of this single D-brane is of the type used to describe electromagnetism.

It is also possible for D-branes to end on other branes. In Type I string theory (which lives in 10 dimensions), one can have D1, D5, D7 and D8 branes end on 9-branes. In all these cases, such configurations give rise to tachyons for open 1, 5, 7, and 8-branes.

Hello DarKonion

In your question, it appears you are using 'D' for mathematical dimension, as in D=5 Kaluza-Klein theory and D=11 supergravity. This is different than asserting that, for instance, D5 branes give rise to electromagnetism and D11 branes yield supergravity. I'll explain why below.
You are correct... I am referencing Kaluza-Klein D5 Electromagnetism.

When superstring theory is studied nonperturbatively, one indeed finds the theory admits objects with p spatial dimensions, called p-branes. For the p=1 case, we recover fundamental string, which can be regarded as a 1-brane. In D=11 supergravity and M-theory, one only finds the 2-brane and 5-brane.
Question: Why only 2 and 5 branes? (I'm confused still on how you come to that conclusion)

Superstring theories contain p-branes of even and odd dimension, ranging from p=-1 to p=9, called Dp-branes. Open fundamental strings can have endpoints that are free to move about (Neumann boundary conditions) or fixed to some p-dimensional object (Dirichlet boundary conditions) which we call Dp-branes or just D-branes. The 'D' in D-brane is short for Dirichlet.
Thank you for clearing that up.

It turns out that Yang-Mills quantum field theories reside on the worldvolumes of D-branes. Given N (coincident) D-branes, the gauge symmetry of the Yang-Mills worldvolume theory is the freedom the string has in deciding which of the N branes to end on. The symmetry group arising from this freedom, in the case of N (coincident) D-branes with oriented open strings, is the unitary group U(N), e.g., N D8-branes which coincide corresponds to an unbroken U(N) gauge group.

So for N=1, just one D-brane, the string has U(1) freedom in deciding where to end on the single D-brane. The worldvolume U(1) Yang-Mills theory of this single D-brane is of the type used to describe electromagnetism.
I actually understood that ^^

It is also possible for D-branes to end on other branes. In Type I string theory (which lives in 10 dimensions), one can have D1, D5, D7 and D8 branes end on 9-branes. In all these cases, such configurations give rise to tachyons for open 1, 5, 7, and 8-branes.
So my question is: Why do they end on D9 branes?

Other than that thank you for your help. ^^

Question: Why only 2 and 5 branes? (I'm confused still on how you come to that conclusion)
One way to see this is by studying p-branes in 11-dimensional supergravity (the long wavelength limit of M-theory), where one recovers only the 2-brane and 5-brane as solitons.

So my question is: Why do they end on D9 branes?
D-branes ending on D9 branes is just an example. We can have other configurations as well. I just wanted you to know that it's possible for objects other than strings to end on branes.

arivero
Gold Member
Guess that at the end, it is 1,2,4,8 = 0,1,3,7 = 8-0, 8-1, 8-3, 8-7 = 8,7,5,1. Call them spheres with continous comb, call them division algebras... Check Duff, http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+A+DUFF&FORMAT=www&SEQUENCE=citecount%28d%29 [Broken]

When superstring theory is studied nonperturbatively, one indeed finds the theory admits objects with p spatial dimensions, called p-branes. For the p=1 case, we recover fundamental string, which can be regarded as a 1-brane. In D=11 supergravity and M-theory, one only finds the 2-brane and 5-brane.

....
It is also possible for D-branes to end on other branes. In Type I string theory (which lives in 10 dimensions), one can have D1, D5, D7 and D8 branes end on 9-branes. In all these cases, such configurations give rise to tachyons for open 1, 5, 7, and 8-branes.
....
Question: Why only 2 and 5 branes? (I'm confused still on how you come to that conclusion)
....
So my question is: Why do they end on D9 branes?

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So my question is: Why do they end on D9 branes?

To my opinion because number 9 maximal one-digit number.

So my question is: Why do they end on D9 branes?

To my opinion because number 9 maximal one-digit number.
Close. It's because a 9-brane is the maximal allowed in 10 dimensions.

Guess that at the end, it is 1,2,4,8 = 0,1,3,7 = 8-0, 8-1, 8-3, 8-7 = 8,7,5,1. Call them spheres with continous comb, call them division algebras...
You're right. The open branes ending on 9-branes constructions (so far) arise from the four Hopf maps ((2.8) of http://arxiv.org/abs/hep-th/0606216" [Broken]), which map to the spheres $$S^1\cong\mathbb{RP}^1$$, $$S^2\cong\mathbb{CP}^1$$, $$S^4\cong\mathbb{HP}^1$$ and $$S^8\cong\mathbb{OP}^1$$, projective lines over the four division algebras.

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nrqed
Homework Helper
Gold Member
Hello DarKonion

In your question, it appears you are using 'D' for mathematical dimension, as in D=5 Kaluza-Klein theory and D=11 supergravity. This is different than asserting that, for instance, D5 branes give rise to electromagnetism and D11 branes yield supergravity. I'll explain why below.

When superstring theory is studied nonperturbatively, one indeed finds the theory admits objects with p spatial dimensions, called p-branes. For the p=1 case, we recover fundamental string, which can be regarded as a 1-brane. In D=11 supergravity and M-theory, one only finds the 2-brane and 5-brane.
Thanks a lot for this very informative post. I will take advanatge of having ane expert around to ask simple questions.

I guess you partially answer this question when talking about solitons in another of your posts but let me ask anyway. As far as I understand, supergravity is a qft of point particles so I was wondering how branes arise in that context? There are no extended objects as fundamental states so what is a brane? You mentioned solitons but they are solitonic excitation of what field?? Solitonic excitation of the quantum fields of the bosons, the fermions?

Superstring theories contain p-branes of even and odd dimension, ranging from p=-1 to p=9, called Dp-branes. Open fundamental strings can have endpoints that are free to move about (Neumann boundary conditions) or fixed to some p-dimensional object (Dirichlet boundary conditions) which we call Dp-branes or just D-branes. The 'D' in D-brane is short for Dirichlet.

It turns out that Yang-Mills quantum field theories reside on the worldvolumes of D-branes. Given N (coincident) D-branes, the gauge symmetry of the Yang-Mills worldvolume theory is the freedom the string has in deciding which of the N branes to end on. The symmetry group arising from this freedom, in the case of N (coincident) D-branes with oriented open strings, is the unitary group U(N), e.g., N D8-branes which coincide corresponds to an unbroken U(N) gauge group.

So for N=1, just one D-brane, the string has U(1) freedom in deciding where to end on the single D-brane. The worldvolume U(1) Yang-Mills theory of this single D-brane is of the type used to describe electromagnetism.

It is also possible for D-branes to end on other branes. In Type I string theory (which lives in 10 dimensions), one can have D1, D5, D7 and D8 branes end on 9-branes. In all these cases, such configurations give rise to tachyons for open 1, 5, 7, and 8-branes.
What explains why all possible branes can not end on 9-branes? Why not D2, D3, etc?

And what do you mean by an "open 1 brane" or an "open 5 brane"?

Thanks a lot for your expertise.

Huh?

You're right. The open branes ending on 9-branes constructions (so far) arise from the four Hopf maps ((2.8) of (((((Unable to post URLs yet))))), which map to the spheres $$S^1\cong\mathbb{RP}^1$$, $$S^2\cong\mathbb{CP}^1$$, $$S^4\cong\mathbb{HP}^1$$ and $$S^8\cong\mathbb{OP}^1$$, projective lines over the four division algebras.
Ok, you lost me. Can you explain a bit farther? I though I understood at
Arivero said:
Guess that at the end, it is 1,2,4,8 = 0,1,3,7 = 8-0, 8-1, 8-3, 8-7 = 8,7,5,1. Call them spheres with continous comb, call them division algebras...
but now I seem to be lost again. I think it was your sphere-mapping equation... Try starting there ^^

Haelfix
Nrqed, I can point you to a reference for why the 2 and 5 branes (and further families) arise in the nonperturbative sector of supergravity (the former is an electric solution, the latter a solitonic or magnetic solution). They are quite general objects, and arise in both bosonic theories as well as the full sector.

http://arxiv.org/abs/hep-th/9701088

The relevant material is motivated and flushed out in the first 10 pages (albeit a little adhoc since it requires an ansatz, but you will see that it is at least self consistent)

The example on page 9 illustrates the proof of concept fairly simply. I do not know of any other simpler way to demonstrate this.

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arivero
Gold Member
Ok, you lost me. Can you explain a bit farther? I though I understood at but now I seem to be lost again. I think it was your sphere-mapping equation... Try starting there ^^
Perhaps my fault. The "comb-able spheres" I named come indeed from the same origin, but they are S0, S1, S3 and S7. Furthermore, it is not unusual that some of these results are found in math for "euclidean" (as opposite to minkowski) spaces, and then one must guide oneself by the signature of the space (positive minus negative eigenvalues of the metric). So in a lot of results minkowsky space of N dimensions look as euclidean of (N-1)-(1) dimensions

nrqed
Homework Helper
Gold Member
Nrqed, I can point you to a reference for why the 2 and 5 branes (and further families) arise in the nonperturbative sector of supergravity (the former is an electric solution, the latter a solitonic or magnetic solution). They are quite general objects, and arise in both bosonic theories as well as the full sector.

http://arxiv.org/abs/hep-th/9701088

The relevant material is motivated and flushed out in the first 10 pages (albeit a little adhoc since it requires an ansatz, but you will see that it is at least self consistent)

The example on page 9 illustrates the proof of concept fairly simply. I do not know of any other simpler way to demonstrate this.

Haelfix, thanks for the great reference! It sounds like teh perfect starting point for me to try to understand this material. I glanced through it quickly.
I have read some similar stuff before but what always stumps me is the following question (which maybe will be answered after a careful reading of the paper but you may be able to give a quick answer):

What I see as the solution seems to be a metric plus some field configurations for the field present in the theory (scalar fields or gauge fields in general). What is the p-brane in this? The paper says that it's an extended "object" but is it a fiedl configuration of the scalar field or gauge fields? Or is it something which does not appear at all in the initial lagrangian? Does one find that the scalar/gauge fields propagate in confined hyperplanes and this tells us that there is some underlying "object" ?

I guess I am trying to grasp the big picture before working out the details.

Thanks!

nrqed
Homework Helper
Gold Member
Haelfix, thanks for the great reference! It sounds like teh perfect starting point for me to try to understand this material. I glanced through it quickly.
I have read some similar stuff before but what always stumps me is the following question (which maybe will be answered after a careful reading of the paper but you may be able to give a quick answer):

What I see as the solution seems to be a metric plus some field configurations for the field present in the theory (scalar fields or gauge fields in general). What is the p-brane in this? The paper says that it's an extended "object" but is it a fiedl configuration of the scalar field or gauge fields? Or is it something which does not appear at all in the initial lagrangian? Does one find that the scalar/gauge fields propagate in confined hyperplanes and this tells us that there is some underlying "object" ?

I guess I am trying to grasp the big picture before working out the details.

Thanks!
I guess that what I am asking is the following: if you look at the solution provided in th epaper on page 9, what would tell someone that there is p-brane involved? (it might not be directly obvious, I understand, but then what calculation would show this?)

Perhaps my fault. The "comb-able spheres" I named come indeed from the same origin, but they are S0, S1, S3 and S7. Furthermore, it is not unusual that some of these results are found in math for "euclidean" (as opposite to minkowski) spaces, and then one must guide oneself by the signature of the space (positive minus negative eigenvalues of the metric). So in a lot of results minkowsky space of N dimensions look as euclidean of (N-1)-(1) dimensions
Ok... I feel dumb asking this... but is Minkowski's spacetime the 4D we know of (3+1)? and if that's so, then is Euclidean spacetime = 1/(3+1) *reciprocal*? or is it just negative? (never understood Euclidean spacetime, probably the root to my total confusion)

arivero
Gold Member
Ok... I feel dumb asking this... but is Minkowski's spacetime the 4D we know of (3+1)? and if that's so, then is Euclidean spacetime = 1/(3+1) *reciprocal*? or is it just negative? (never understood Euclidean spacetime, probably the root to my total confusion)
Euclidean is a bad denomination, due to physicists, meaning only that the metric has all the four eigenvalues positive. Mathematicians prefer pseudoriemannian and riemaniann, I believe. Worse, a Minkowsky has some work on fractal dimensions, so "Minkowski dimension" sometimes really refer, in math books, to a non integer dimension. Obviously, this meaning is not intended here, and it is almost completely unrelated.

To be clearer, or tu put more mud: Minkowski spacetime is 3+1, thus signature 3-1 = 2. Euclidean is 4+0, thus signature 4-0 = 4. Superstrings live in Minkowsky 9+1, thus signature 9-1=8. Bosonic strings live in Minkowsky 25+1, thus signature 25-1=24. A lot of mathematical results depend on signature modulo 8 and similar periodicities (there is other known periodicity in lattices, and it is mod 24).

As for the spheres, the collection 1,2,4,8 also maps to underline the peculiarities of spacetime of dimension 3,4,6,10. This is the origin of the so-called "Brane scan".

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Haelfix
Hi Nrqed.

Unfortunately it is not obvious at all that there is such objects in the theory. We are after all, looking at quantum nonperturbative *things* and showing their existance at the classical level. The way the paper proceeds is to write an effective action (that at first glance seems hard to reconcile with the full SUGRA action, but they justify this later), and then solve for the eoms.

They then make an ansatz for the solution by looking for Pbrane solutions (read flat euclidean spatial hypersurfaces embedded in the ambient spacetime). They then solve for this and indeed single them out. The example then is a special case where they are trivially apparent and by consistency are required to have the 2 and 5 dimensions as expected.

They then go on to further justify the consistency of the ansatz (preservation of SUSY etc), as well as show how they get the initial lagrangian (dimensional reduction) from 11D SUGRA.

As far as I know, string theory sort of stumbled on the solutions in a round about way as well, the details of that I do not pretend to understand.

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nrqed
Homework Helper
Gold Member
Hi Nrqed.

Unfortunately it is not obvious at all that there is such objects in the theory. We are after all, looking at quantum nonperturbative *things* and showing their existance at the classical level. The way the paper proceeds is to write an effective action (that at first glance seems hard to reconcile with the full SUGRA action, but they justify this later), and then solve for the eoms.
Hi. Thanks for your input. I would really like to understand this concept (p-branes from th eeffective actions) because I rememeber reading about this years ago and hitting a wall in my comprehension.

They then make an ansatz for the solution by looking for Pbrane solutions (read flat euclidean spatial hypersurfaces embedded in the ambient spacetime). They then solve for this and indeed single them out. The example then is a special case where they are trivially apparent and by consistency are required to have the 2 and 5 dimensions as expected.
This is where I get stuck. I understand that they separate the metric into a part in p dimensions and D-p dimensions. But one could do that for any metric.

My problem is that I don't see what is the telling sign that there is a p-brane in the theory. If we look at the final solution (which I am willing to accept....for now my question is not about the derivation or the justification of the ansatz but about the interpretation of the result), what shows that there is a brane? I guess there is some "discontinuity" in the spacetime metric on the hyperplane? (maybe discontinuity is not the best word here)

But is this discontinuity simply due to an energy-momentum distribution of the fields present in the theory? Or is it something else entirely? If it was simply due to the energy-momentum distribution of the fields in the theory then there would be nothing really special, so I guess that the fields are not the source of this spacetime discontinuity.

Do you see what I mean? I am just trying to see exactly how one can tell there is a p-brane by analyzing the solution.

Thanks!!

Haelfix
Hi Nrqed,

I guess i'd respond by saying the solutions must of course satisfy the various criteria for the definition of a PBrane.

At this classical level, you can think of them merely as foliating spacetime with leaves that are isometric to (p+1) dimensional spacetime. So they must satisfy the various axioms and symmetries you would want of such an embedded object (poincare invariance, preservation of some SuSY etc). Further they have a well defined mass and charge, so they will couple to (p+1) gauge potential forms (much like a point particle does to a lower dimensional entity). They also are extremal (another discussion) and satisfy some sort of energy bound (I get confused on this part). Really in a sense, you are characterizing the type of objects that can exist and act independantly in the theory.

The analogy to keep in the back of your mind, is that what monopoles/dyons are to yang mills theory, they are to supergravity. Another reason that they are interesting, is b/c at the quantum level they act as a sort of boundary condition for the string theory completion of supergravity.

As for the singularity structure of the metric at their positions... Well, that takes a bit of discussion b/c care needs to be utilized in finding the natural frame (sort of like how you need to find the right coordinates for blackhole solutions in GR). Often, people distinguish between various types of pbranes (fundamental vs dyonic) based on the existance of time like noncoordinate singularities in the appropriate frame, but I digress....

So im not sure if this helps answer your questions, maybe a good review article from somewhere else might help (anyone know of a good one?) as im getting dangerously close to the limits of my heuristic knowledge of the subject.

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nrqed
Homework Helper
Gold Member
Hi Nrqed,

I guess i'd respond by saying the solutions must of course satisfy the various criteria for the definition of a PBrane.

At this classical level, you can think of them merely as foliating spacetime with leaves that are isometric to (p+1) dimensional spacetime. So they must satisfy the various axioms and symmetries you would want of such an embedded object (poincare invariance, preservation of some SuSY etc). Further they have a well defined mass and charge, so they will couple to (p+1) gauge potential forms (much like a point particle does to a lower dimensional entity). They also are extremal (another discussion) and satisfy some sort of energy bound (I get confused on this part). Really in a sense, you are characterizing the type of objects that can exist and act independantly in the theory.

The analogy to keep in the back of your mind, is that what monopoles/dyons are to yang mills theory, they are to supergravity. Another reason that they are interesting, is b/c at the quantum level they act as a sort of boundary condition for the string theory completion of supergravity.

As for the singularity structure of the metric at their positions... Well, that takes a bit of discussion b/c care needs to be utilized in finding the natural frame (sort of like how you need to find the right coordinates for blackhole solutions in GR). Often, people distinguish between various types of pbranes (fundamental vs dyonic) based on the existance of time like noncoordinate singularities in the appropriate frame, but I digress....

So im not sure if this helps answer your questions, maybe a good review article from somewhere else might help (anyone know of a good one?) as im getting dangerously close to the limits of my heuristic knowledge of the subject.
Hi Haelfix,

I think that I overcame my mental block. My problem was with the following: I did not know if the p-brane was just a field configuration of the field presnt in the theory or some independent "object". And if it is an independent object, it seemed to me that there should be a term in the Lagrangian corresponding to this "object" acting as a source of the field and ccreating the curvature in spacetime. But I did not see any such term so i was confused.

I read the article you linked to and a few other papers and now it's more clear to me.

It became clear when I read about an analogy with an extremal charged black hole, M=Q. Then one could uncover the Reissner-Nordstrom black hole solution before even introducing a point charged mass at the origin. One could build the solution (using for example the harmonicity of the extremal solutions) without introducing the point mass at the origin. After finding the solution for the spacetime and the gauge field, one can then realize that there is a point charged mass at the origin and then introduce after the fact a source term for the gravitational and electromagnetic fields at the origin.

The same logic is used to uncover pbrane solutions. And then, after the fact, one introduces source terms for the electric and magnetic fields.

This is what I was missing: I was looking for the source terms associated to the pbrane and since I did not see any in the starting lagrangian, I was confused about what the pbrane was, exactly. I was missing the fact that the solutions could be found (using symmetry arguments alone) even without introducing source terms, and I was missing the fact that after doing that one had to go back and then *add* the source terms.

Anyway, I am sure this is all obvious to you...I was completely confused about the basic idea. Thank you for helping clarify things for me.

Patrick