# Mathematically rather than physically speaking, is there something “special” about 10 (or 11) dimensions?

As I understand it, string theory (incorporating bosons and fermions) "works" in $9+1=10$ spacetime dimensions. In the context of dual resonance theory, I've read descriptions of why that is "physically", i.e., how in the process of formulating a quantum theory of relativistic interacting (closed and open) strings, the only way one can do things like get rid of ghosts, for example, is to have ten dimensions.

I'm looking for a way to appreciate more intuitively this restriction on dimensionality. Is there something special about ten-dimensional spaces purely from the point of view of mathematics? Without reference to the particular detailed exigencies of building a physical theory based on strings, does ten-dimensional topology have some unique features that are lost when one dimension is added or subtracted?

Another way to pose the question: For purposes of this discussion, let's stipulate by fiat that the universe has $8+1=9$ dimensions. Would we humans simply be back to the drawing board in terms of coming up with a unified theory of the fundamental forces? Would anything in string theory be salvagable?

I realize that M-theory adds a dimension. So if someone wants to answer the question by telling me what's unique or special about eleven dimensions, that's fine.

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Sure, 1+2+3+4=10. (P.S. I'm serious. Gravity is the curvature of a 4d manifold (spacetime), em is for 1d (U(1) bundles), weak is for 2d (SU(2) bundles) and strong is for 3d (SU(3) bundles. KK theory unifies gravity and em, so it has 4+1 = 5 dimensions, strings unify all of them, so they've 1+2+3+4 dimensions, but m-theory has an (irritating) extra 1 dimension.) –  Dimensio1n0 Jul 30 '13 at 15:43
Perfect username for the comment! –  Relative0 Aug 9 '13 at 18:59

Dear Andrew, the derivation of the critical dimension - which can be done in many "physical" ways - is ultimately a reflection of mathematics and a rather deep one. All of string theory is studied at the level of "mathematics" (as opposed to "experiments", for example). Maybe you need some "substantially simpler mathematics" or "one that doesn't depend on quantum mechanics"? Well, if the maths is too simple, it works in any dimension and there's nothing special in $D=10$.

There are other, in principle non-stringy properties of 10 dimensions. Some of them are clearly not independent of string theory - they refer to a subset of properties of string theory. For example, 10 dimensions are the dimensionality in which the minimally supersymmetric gauge theory with a Majorana Weyl (real chiral) gaugino exists. This gauge theory is embedded in the $N=1$ $D=10$ vacua of string theory as the low-energy limit.

There are also other properties of 10 dimensions which, at least at this moment, look totally independent of string theory. For example, the allowed topologies of black hole solutions undergo some transformation as the dimension changes and 10 is an interesting turning point. Unfortunately, I have forgotten the details - I've heard only one talk about it. The result doesn't seem to imply that higher-dimensional gravity would be "inconsistent", so that's why I say that it doesn't seem to be a strictly stringy result, but there's still something special about $D=10$ here, too.

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Thanks, Lubos. In the context of supersymmetry, does the number of particle generations affect the dimensionality of the theory? In other words, if there were four particle generations instead of three, would that point to some dimensionality of the universe other than 9 + 1? –  Andrew Wallace May 20 '11 at 13:19
great, sounds interesting that comment about topologies of BH solutions being special at 10D. If you happen to remember the details or any references let us know –  lurscher May 20 '11 at 16:06
What about 11D? –  Gulshan Oct 28 '12 at 14:56
@Gulshan: 11=10+1. –  Dimensio1n0 Jul 30 '13 at 15:43

In the representation theory of Virasoro algebra there is a mathematically strict theorem: http://en.wikipedia.org/wiki/Goddard%E2%80%93Thorn_theorem for bosonic string theory, and similarly for superstring theory. Intuitively the critical dimensions come from zeta function regularization.

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This is not a very useful answer for someone who doesn't already know it. –  ACuriousMind Mar 9 at 12:53
@ACuriousMind Hum, I agree. But then what kind of answer would be useful. I think neither the requirement of the positivity of Hilbert space nor the approach of cancellation of Weyl anomaly is "intuitive". Since the questioner asks for mathematical aspects of critical dimension, I think it's necessary of putting no ghost theorem here. –  Vese Akitsuki Mar 9 at 13:06
Yeah, I agree that there isn't an "intuitive" answer, but usually, answers that take that stance elaborate a bit on why there isn't any intuition, or at least summarize the theorem they refer to. Currently, this is close to being a link-only answer, which are generally not well-liked. –  ACuriousMind Mar 9 at 13:08
Thx! I'll be careful next time. –  Vese Akitsuki Mar 9 at 13:17