# D8-Branes What do they bring?

1. Apr 14, 2008

### DarKonion

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.

2. Apr 14, 2008

### Golfer

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.

3. Apr 14, 2008

### DarKonion

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.

Last edited: Apr 14, 2008
4. Apr 15, 2008

### kneemo

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.

5. Apr 15, 2008

### DarKonion

You are correct... I am referencing Kaluza-Klein D5 Electromagnetism.

Question: Why only 2 and 5 branes? (I'm confused still on how you come to that conclusion)

Thank you for clearing that up.

I actually understood that ^^

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

Other than that thank you for your help. ^^

6. Apr 16, 2008

### kneemo

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.

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.

7. Apr 16, 2008

### arivero

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, anyone, perhaps this http://www.slac.stanford.edu/spires/find/hep/www?j=CQGRD,5,189. And also this one from Evans http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B298,92

Last edited: Apr 16, 2008
8. Apr 16, 2008

### gorgos

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

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

9. Apr 16, 2008

### kneemo

Close. It's because a 9-brane is the maximal allowed in 10 dimensions.

10. Apr 16, 2008

### kneemo

You're right. The open branes ending on 9-branes constructions (so far) arise from the four Hopf maps ((2.8) of hep-th/0606216), 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.

11. Apr 16, 2008

### nrqed

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?

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.

12. Apr 16, 2008

### DarKonion

Huh?

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 ^^

13. Apr 17, 2008

### 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.

Last edited: Apr 17, 2008
14. Apr 17, 2008

### arivero

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

15. Apr 17, 2008

### nrqed

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!

16. Apr 17, 2008

### nrqed

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?)

17. Apr 17, 2008

### DarKonion

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)

18. Apr 18, 2008

### arivero

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".

Last edited: Apr 18, 2008
19. Apr 18, 2008

### 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.

Last edited: Apr 18, 2008
20. Apr 18, 2008

### nrqed

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.

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!!