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I'm a grad student in math, but I don't know as much about physics as I should. I've read a handful of pop expositions of string theory, and they often refer to "curled up dimensions" and I've always wondered what this means. Is this referring to any kind of standard mathematical object, e.g. a bundle of Calabi-Yaus? (Actually learning some physics is one of my long-term self-improvement projects, but for now I'd be happy to demystify this word!)

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I believe that's correct, that the curled up dimensions are a Calabi-Yau manifold, which is the fiber in the bundle. –  Grumpy Parsnip Jan 17 '12 at 17:09
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Perhaps it would be more likely to find authoritative answers at physics.SE? –  Henning Makholm Jan 17 '12 at 17:23
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I think it's better suited here. I've heard tons of physicists describe "extra dimensions", and it's all this wishy washy stuff. I'd like to hear a formal, mathematical description of what a "curled up extra dimension" is. Are they talking about $\mathbb{R}^4 \times (S^{1})^{n}$ with a certain topology given to it? $\mathbb{R}^{4} \times G$, where $G$ is a Lie group? –  Jackson Walters Jan 17 '12 at 17:45
    
@Jackson Walters: Yes, they're talking about a direct product of $\mathbb{R}^4$ with some compact n-dimensional manifold. In addition to the topological stuff, there is going to be some metric structure, and the term has the connotation that any length scale(s) $\ell$ associated with the compact part of the space are all microscopic; if $\ell$ was one meter or a billion light-years, then we would have already detected the extra dimensions. –  Ben Crowell Jan 17 '12 at 18:18
    
@JacksonWalters: FYI we can (and love to) do mathematics at Physics.SE. The only reason you hear wishy-washy stuff is because most of the people physicists have to explain these things to are complete laymen. But those pop-science explanations are not the purpose of Physics.SE. This type of question would be a fantastic fit there. (And yes, I know it's been 3-4 months since this question saw any activity... I'm just posting for future reference :-P) –  David Z May 8 '12 at 16:47

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up vote 9 down vote accepted

The basic idea doesn't have to be that complicated, and it isn't specific to string theory. We seem to live in a spacetime with four dimensions (three spacelike and one timelike). If there was a fifth dimension, say a spacelike one, you'd think we'd have noticed it. However, suppose that that extra dimension has the topology of a circle, i.e., spacetime is topologically a direct product of $\mathbb{R}^4$ with a circle. That circle has some circumference, which sets some length scale $\ell$. If $\ell$ was very small, we might not be able to detect the existence of the extra dimension.

If you like, you can add more than one extra dimension. (String theory requires 6 or 7.) Those extra dimensions can then have intrinsic curvature, which is another way of setting a distance scale $\ell$.

To a physicist, the next obvious question is what sets $\ell$. General relativity is completely invariant under any smooth change of coordinates, so the theory has absolutely no scale that is privileged over any other scale. For example, the spacetime at the surface of the earth has curvature that sets a certain length scale (which is how GR describes the gravitational field), but that scale isn't hard-coded into GR, it's due to the mass of the earth, which is an accident of the formation of the solar system.

The typical answer is that $\ell$ should probably be the Planck length, which is the scale at which you can't describe spacetime without using some theory of quantum gravity, which we don't yet possess. In most theories, the Planck length is $10^{-35}$ m. In some theories, known as theories with large extra dimensions, http://en.wikipedia.org/wiki/Large_extra_dimensions , it's much larger, and therefore possibly accessible by experiment. In these theories, the $10^{-35}$ m value is only the false, apparent value of the Planck length when you do experiments at scales large enough so that only four dimensions seem to exist.

If you have more than one extra dimension, you can have nontrivial topologies. The Calabi-Yau stuff is one possibility, which is purely motivated by string theory. Since string theory is probably wrong (e.g., we don't see evidence for supersymmetry at the LHC), there is no really good reason to focus on Calabi-Yau manifolds. Furthermore, results from the LHC seem to show that large extra dimensions don't exist: http://arxiv.org/abs/1012.3375

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What do you mean when you say that "in most theories" the Planck length is $10^{-35}$m? I thought that it was defined as the unique way to make a quantity with the dimensions of length from the dimensionful quantities $\hbar$, $G$ and $c$ (reduced Planck constant, Newton's constant, and speed of light) which are all experimentally measurable and not subject to your choice of theory. –  Chris Taylor Jan 17 '12 at 17:37
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@Chris Taylor: In theories with large extra dimensions, the microscopic value of G is different from the value of G measured in macroscopic experiments. The whole motivation for these theories is to make the microscopic value of G be such that it produces a Planck length that is the same as the electroweak scale. –  Ben Crowell Jan 17 '12 at 18:08
    
Re large additional dimensions, here is a paper that explains the physical motivation better than the WP article does: arxiv.org/abs/hep-ph/0402168 Their $R$ is the same as my $\ell$. –  Ben Crowell Jan 17 '12 at 18:23
    
Thanks, Ben -- this is exactly what I was looking for. Hopefully someday I'll have a better understanding of all of this. –  Mark G Jan 17 '12 at 18:41

I think they generally refer to the dimensions of the Fiber in a Fiber Bundle.

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