Why 10 or 11?

Why does string theory require 10 or 11 spacetime dimensions? The answer at a technical level is well-known, but it’s hard to bring it down to earth. By reading economics blogs by people who check out political theory blogs, I stumbled across an attempt at making it clear — by frequent CV commenter Moshe Rozali, writing in Scientific American. After explaining a bit about supersymmetry, Moshe concludes:

A guide in this pursuit is a theorem devised/put forth by physicists Steven Weinberg and Edward Witten, which proves that theories containing particles with spin higher than 2 are trivial. Remember each supersymmetry changes the spin by one half. If we want the spin to be between -2 and 2, we cannot have more than eight supersymmetries. The resulting theory contains a spin -2 boson, which is just what is needed to convey the force of gravitation and thereby unite all physical interactions in a single theory. This theory–called N=8 supergravity–is the maximally symmetric theory possible in four dimensions and it has been a subject of intense research since the 1980s.

Another type of symmetry occurs when an object remains the same despite being rotated in space. Because there is no preferred direction in empty space, rotations in three dimensions are symmetric. Suppose the universe had a few extra dimensions. That would lead to extra symmetries because there would be more ways to rotate an object in this extended space than in our three-dimensional space. Two objects that look different from our vantage point in the three visible dimensions might actually be the same object, rotated to different degrees in the higher-dimensional space. Therefore all properties of these seemingly different objects will be related to each other; once again, simplicity would underlie the complexity of our world.

These two types of symmetry look very different but modern theories treat them as two sides of the same coin. Rotations in a higher-dimensional space can turn one supersymmetry into another. So the limit on the number of supersymmetries puts a limit on the number of extra dimensions. The limit turns out to be 6 or 7 dimensions in addition to the four dimensions of length, width, height and time, both possibilities giving rise to exactly eight supersymmetries (M-theory is a proposal to further unify both cases). Any more dimensions would result in too much supersymmetry and a theoretical structure too simple to explain the complexity of the natural world.

This is reminiscent of Joe Polchinski’s argument (somewhat tongue-in-cheek, somewhat serious) that all attempts to quantize gravity should eventually lead to string theory. According to Joe, whenever you sit around trying to quantize gravity, you will eventually realize that your task is made easier by supersymmetry, which helps cancel divergences. Once you add supersymmetry to your theory, you’ll try to add as much as possible, which leads you to N=8 in four dimensions. Then you’ll figure out that this theory has a natural interpretation as a compactification of maximal supersymmetry in eleven dimensions. Gradually it will dawn on you that 11-dimensional supergravity contains not only fields, but two-dimensional membranes. And then you will ask what happens if you compactify one of those dimensions on a circle, and you’ll see that the membranes become superstrings. Voila!

This entry was posted in Science. Bookmark the permalink.

67 Responses to Why 10 or 11?

  1. Moshe says:

    My god you guys are quick…in fact this came out as elaboration on a comment I made here to George Musser, in the post you wrote. I had no idea at the time he was associated with SciAm, but he asked a concrete question and did not seem to be inclined to tell me precisely what is wrong with me…your post at the time BTW was really nice.

    I did not know about Joe’s route to string theory, but now I am convinced..

  2. Paul Valletta says:

    Surely a one-dimensional “string” embedded into a “two-dimensional” anything?..will always have an energy factor, that is exponentionally sufficient to break/snap, the two-dimnensional ‘brane’, to such an extent that Branes cannot exist?

    Compactification of any “matter”, from any higher dimension, has a default INCREASE in the energy value?

    The only way around this “string-energy” paradox is if the Brane is “one-dimensional” and exists within a “string”..and not be external?

  3. Sean says:

    Here is the question from George that Moshe was nice enough to try to answer. See, it pays to make useful comments on blogs, you could become famous.

  4. Moshe says:

    Sean, you must have meant rich and famous…

    One clarification, despite starting out as question about string theory, the piece is not very specific to string theory. Basically if you want to combine the ideas of supersymmetry and extra dimensions you put an upper bound on the number of those extra dimensions. This by itself does not tell you there is a consistent theory, or of course if it relevant to nature.

  5. fh says:

    This is very nice, and I can almost follow what’s going on…
    So basically the argument is that since we expect the higher dimensions to show up as symmetries at the 4dimensional viewpoint, and to much symmetry makes for vacuous theories, we can at most go to 10/11D if the additional symmetries show up as supersymmetry on a (flat? Wasn’t that an assumption of Weinberg Witten?) background, right?
    I assume there is no nice and easy way to see why we would expect the extra dimensions to manifest as supersymmetry?

  6. sisyphus says:

    Sean: Nice string – er, thread. Just how little I grasp of this will be made obvious by the following question: Is it possible to have 6 or 7 extra dimensions locally – that is epiphenomnal to our 4D spacetime – while the whole package exists within a global 5D context? I know that this is probably mathematically paradoxical if not downright silly, but I have to ask.

    I hope the answer is ‘no’; if it’s ‘yes’, I’ll never understand the explanation.


  7. J says:

    fh: Actually, the explanation isn’t hard. A supersymmetry has a spinor charges. In every dimension, there is a minimal size for a spinor. When we say “N=x” supersymmetry, we mean that you have x spinors of minimal size. In d=4, the minimal size of a spinor is 4 real dimensional, i.e. in 4 dimensions you always have 4x real supercharges that get mixed by rotations. This number grows very rapidly, more or less exponentially in the dimension.

    In 10 dimensions, the smallest spinor you can make has is 16 real dimensional. However, type II superstrings have N=2, so they have 32 supercharges, which from a 4 dimensional point of view looks like N=8 – 4×8=32. Rotations in 10 dimensions mix them together, but rotations in 4 dimensions mix each 4 dimensional spinor seperately.

    In 11 dimensions, you have to have at least 32 real supercharges for N=1, so you’re already at the bound. N>1 in d=11 or N>0 in d>11 requires so many supercharges that you you have to have spin >2 particles.

  8. Maynard Handley says:

    Ignore the stringiness aspect of all this.
    The argument is predicated on supersymmetry being real.
    What is the current experimental status of this?
    Do we ignore ICE CUBE data, or has it just not been running long enough — there was a brief flurry of activity surrounding this about ten days ago which seemed to degenerate into some people saying it proved something or other, others saying no it did not and I lost track of what was actually going on.

    Beyond ICE CUBE, is this something that will actually be resolved to most people’s satisfaction in 1 yr, 5 yrs, 15 yrs? Or is it basically so slippery a concept that people can continue claiming it’s true (or at least possibly true) for the rest of my life.

  9. Count Iblis says:

    I don’t understand why one expects a quantum theory of gravity to be renormalizable. In principle, one can think of the standard model as a low energy effective theory that is to be obtained from the (unknown) theory of everything by integrating out the unknown high energy physics.

    Adding non renormalizable operators to the fundamental theory has almost no effect on the low energy physics, so I don’t see how one can work ”backward” and theoretically ”derive” the theory of everything from its low energy remnant.

  10. fh says:

    J, thanks. At least on this level this makes sense.

    Count Iblis,
    the point is that if you have a renormalizable theory you can to some degree expect to be able to define it perturbatively, since order by order nothing undefined happens.
    So if you find a renormalizable perturbation expansion of Quantum Gravity you can say you have found a theory. If it’s nonrenormalizable you don’t have a theory.

  11. Joe Polchinski says:

    Hi Sean, thanks for repeating this argument. I’d like to mention that it is just one example. Another starting point (trying to make a minimum length by introducing a position-position uncertainty principle) is followed to its inevitable conclusion in hep-th/9812104 and hep-th/0209105. Yet another (trying to make the graviton as a bound state of two gauge bosons) leads to the same endpoint — this is in a paper with Gary Horowitz that will appear in the next week or so.

  12. Pyracantha says:

    When people are mystified by my behavior, I tell them I’m from the “seventh dimension.” That usually ends the conversation.

    Once again, a large question from the ancient high school physics student:
    what is “supersymmetry?” Is it impossible to explain this on just a blog post? Are any books on it up-to-date?

    Yours, Pyracantha

  13. Sean says:

    Joe, I’m glad to hear (at least implicity) that I got the argument right. And the bound-state idea sounds interesting, I’m looking forward to the paper.

    Pyracantha, supersymmetry is a proposed symmetry that relates the two basic kinds of particles, bosons and fermions. Fermions are “matter” particles that take up space (electrons, quarks, neutrinos), while bosons are “force” particles that happily pile on top of each other (photons, gluons, etc). If supersymmetry is true, every kind of boson/fermion we know of has a partner that is a fermion/boson (respectively), but that is so heavy we haven’t seen it yet. Supersymmetry is a key feature of string theory. There’s a good popular-level book on it by Gordon Kane, called simply “Supersymmetry.”

  14. anon says:

    Count Iblis: you raise an interesting point that’s been bugging me lately. It’s true that a nonrenormalizable theory like a naive approach to quantizing gravity is in some sense not a theory at all, as it never makes predictions about high energies (one must measure an infinite number of constants to fix the theory). On the other hand, we don’t really know that the real world doesn’t work that way. It would be depressing, since it would entail a fundamental limitation to the effectiveness of science. But I can’t think of any reason to think it might not be the case. Can anyone else?

    Joe or Moshe: as an interested observer of string theory, I’ve been a little puzzled by the usual story of the critical dimension and compactification. Do we really have any compelling reason to think that our world is decribed by a compactified 10D string instead of a noncritical string with all sorts of nontrivial fields (dilaton, etc….) turned on? I suppose one could try to think of all such things as “compactifications,” where the extra dimensions are not simply a manifold but some more complicated geometric (or noncommutative-geometric or whatnot) object, but it seems like generically this wouldn’t be a useful viewpoint. It seems particularly worth asking since I get the impression that (at least some large number of) people tend to expect a lot of the usual vacua to have SUSY broken at a high scale, so the motivation for wanting to start with 4D space times a Calabi-Yau seems to me to be lost. Do I misunderstand the situation?

  15. Thomas Larsson says:

    The main problem with SUSY is not whether it predicts 11D, but that God doesn’t seem to care.

    The natural prediction of supersymmetry is that all particles have a superpartner of equal mass. Since this is obviously completely wrong, one must assume that SUSY is broken. The first line of defense had to be abandoned immedately.

    But even if SUSY is broken in a natural way, it is still completely wrong, e.g. because it predicts proton decay a million times faster than experimental limits. Thus one had to posit some ugly ad hoc mechanism which suppresses dimension 4 operators, like R-parity conservation. The second line of defense also had to be abandoned.

    But even broken SUSY with R-parity conservation has severe problems with experiments. One would naturally expect to see a light Higgs, proton decay, muon g-2 deviations from the standard model, permanent electric dipole moments, WIMPS, and perhaps even sparticles, at presently available experiments. This kind of SUSY is not yet completely ruled out (it might be at the LHC), but we know that it is not natural – this is expressed by the buzzwords “SUSY requires fine-tuning at the percent level”. The third line of defense is in trouble.

    In view of these problems, some people have proposed so-called split SUSY. This forth line of defense is finally permanently safe from confrontation with nasty experiments, since it makes no predictions at all which can be tested in any experiment that will ever be feasible. (Probably; it is unclear to me if PEDM data could be used to rule out also split SUSY).

  16. Clifford says:

    anon, the main reason people have studied phenomenological scenarios using critical strings as a starting point is because those were are the best understood…. unless I’ve forgotten something. (Progress in this field often follows the path of least resistance….which means that we sometimes take a while to get to important points we were close to a long time before..) Supersymmetry and higher dimensional Lorentz invariance (two key things that put you there in the first place) both get thrown out the window in a short while anyway, “so why were they so important in the first place?” you might ask….. I’ve been on a lonely campaign to remind people that we need to widen the scope a lot. There may be all sorts of riches living in non-critical string, including a real stab at describing Nature. It is time we did the hard work and made a real assault on that area, with our eyes a bit sharper this time around since we know many of the interesting things that show up beyond perturbation theory…..like branes…..

    See my “News from the front” posts on this blog for more on that.



  17. hackticus says:

    I’m not sure you’ve really succeeded in “bringing it down to earth” when you have to take as your starting point a theorem by Witten about higher spin particles. “Oh, I see, it’s just a straightforward consequence of the Witten-Weinberg theorem” the average man on the street is saying…

  18. PLato says:

    From a layman perspective I like things simplified, and finding model associations that would speak any positon held by the contributors here would help understand this process. But there is something more that needs to be done.

    Would Clifford see diffeently then us, if such a abstract developement would have taken him directly to D-brane analysis? How would he now see, encapsulating such discussion about N dimensional spaces and such. I mean he would had to have been able to see in different ways that most of us wouldn’t he?:)

    Or even a Lubos motl, who having understood these physics processes woud have married thinking of “two sides of a coin” down to the understanding of the dual nature of the blackhole, that such D brane analysis would have made some kind of sense in the physics world.

    So holding Moshe thought and comments on the supersymmetry(Kravstov cosmological computors models) issue this too needs some association, just to help make it easier to understand. Having people like Dvali(D brane effects spreading across surface) throw tidbits in for consideration of conceptual developement is always nice. Fluid flows, and laval nozzles, help greatly, taking vision down to a certain level.

    But the central theme now is where we had gone to such lengths and raising Cerenkov radiation or lagrangian perspective, I thought would be great starting points. along side of He4 or superfluid recognitions?

    As a layman developing concepts, are these starting points wrong having assumed D brane thinking?

  19. Moshe says:

    Clifford, the word “string” is not mentioned in my little piece, it is not necessary…but to your point which IS a point about string theory, Joe’s argument seems relevant: you do want to control UV behavior and for the theory to have at least 4dim, I am not aware of any way of doing that without SUSY, but of course I am one of the students educated after the first stab at the non-critical strings…

  20. Clifford says:


    Fair point. Don’t get me wrong…I like those arguments too…. Nevertheless, (at least) approximately four dimensions and broken susy are experimental facts. I’m just saying that I’m not sure that we understand non-critical strings well enough to know that after all the susy-breaking and dimension changing and strong coupling scenarios that we do once we start with critical strings, we don’t end up some place we could have got to by starting with non-critical strings in the first place…… But given that I don’t have the answer either, this could well be wishful thinking.



  21. Moshe says:

    Thanks Clifford, I think I see what you mean now, that is an interesting point.

  22. Eva Silverstein says:

    Hi Sean et al,

    I think the reason one has trouble providing an argument that 10 or 11 dimensions is required by string theory is because there is no such argument, as mentioned in the comments by anon and Clifford.

    If you impose low energy SUSY, or if you require *exactly* flat spacetime in the prescribed dimension, then you obtain the critical dimension. These criteria are not required (yet) by either phenomenology/cosmology or top-down consistency.

    In particular, the relevant consistency condition in perturbative string theory is simply that the total Weyl anomaly cancels, which a priori can be arranged in a wide variety of ways. For example, one of the simplest ways around the old “no go” arguments for de Sitter space is to take into account the positive term in the moduli potential that arises in supercritical dimensionality (combined with other ingredients such as orientifolds and RR fluxes).

    I would not particularly advocate the noncritical models for phenomenology in the absence of other motivations, but the question of principle (what is allowed vs what is excluded based on first principles) is an important one and I know of no non-circular top down argument for the critical dimension. Conversely, if someone came up with such a proof it would be new and useful. Of course low energy SUSY has a number of phenomenological virtues and it will be extremely important to see whether or not it is there at the LHC.

    Obtaining nearly flat space of course requires fine tuning (this is the cc problem), but it is not known whether the tuning required is actually better or worse in the case of high vs low scale SUSY breaking. There has been some preliminary discussion of this in the literature but no firm conclusion.

    Best regards,

  23. Moshe says:

    Eva, thanks for your comment. To the extent it relates to anything I have written, I have been careful to state the assumptions and what follows from them. Any theory at all that attempts to use SUSY and extra dimensions, in the regime when the notion of dimension makes sense, is bound to find itself in 11 dimensions or less. In other words the numbers 10/11 are not pulled out of a hat, which may surprise a few people…It is an interesting and independent question whether in string theory these are absolutely necessary, but once again there is no mention of strings in my little piece.

    As for that question, the absence of Weyl anomaly is one way of deriving the critical dimension, in the Green-Schwarz formalism the derivation has much more transparent relation to spacetime supersymmetry, perhaps that is a circular argument…

    In addition, as far as I understand there is no one self-consistent framework that incorporates all the ingredients needed for the super-critical scenario. That does not mean there is none, only that critical strings are under much better control, especially when considered non-perturbatively.

  24. Eva Silverstein says:

    Hi Moshe,

    I was not writing as a criticism of your piece (which if it states an assumption of low energy SUSY is perfectly reasonable). People sometimes claim that string theory essentially predicts both 10d and low energy SUSY; it is this of which I am not convinced.

    I worry that this could be a much more sophisticated version of the following. We all learn mechanics first with spherical symmetry, which makes calculations easier. In that context, it would be ridiculous to elevate the symmetry to a principle of nature. Of course SUSY is a much deeper structure, but it is still not obvious to me whether it is ultimately a deep requirement or closer to a theoretical crutch.

    Regarding supercritical constructions: the statement is that enough independent forces to fix the moduli appear in these theories (self-consistenly in the same model). Of course it is true that this arena has not been much studied, and there could be subtleties. On the non-perturbative formulation: recall that there is no known non-perturbative formulation of string theory on a Calabi-Yau. The backgrounds for which a non-perturbative definition presently exist, while very interesting, are few and far between, so I would not use this as a selection mechanism given our present knowledge.

    Best regards,

  25. Lee Smolin says:

    Hi, There is another very elegent way to understand why 10 and 11 are special, which was developed a long time ago by Feza Gursey and others. This has to do with the division algebras: A= R (real), C (complex), Q (quaternions) and O (octonions). There are the only algebras which extend the real numbers in having addition, multiplication and division. It turns out that these are related deeply to supersymmetry. One way to see this is that a supersymmetric extension of Yang-Mills theory must have two component spinors, valued in one of these algebras. The four possibilities give theories in 3,4 6 and 10 dimensions.

    Another way to see this is that the supersymmetry charge is fermionic and this is generated by two component spinors that transform under a representation of SL(2,A). SL(2,R) ~ SO(1,2), SL(2,C) ~ SO(3,1), and SL(2,Q) ~ SO(1,5). There is no SL(2,O), because the octonions are non-associative. But there are close relationships between the representation theory of SO(8)-the transverse directions in 9+1 dim spacetime and the octonions that allow a supersymmetric theory to be defined. There are three 8 dimensional representations-the vector, spinor and conjugate spinor, related by a symmetry called triality that can be seen to underlie the structure of the supersymmetry algebras between 8 and 11 dimensions.

    If I may, one last beautiful fact: These three representations organize themselves into components of a very beautiful algebra-which is the algebra of 3 by 3 hermitian matrices of octonions. Under anticommutators they form a unique structure called the exceptional Jordan algebra. This is a 27 dimensional algebra, and it has 3 SO(8) scalars-so it naturally represents physics in 11=8+3 dimensions