# Constants of nature as determined by string theory

Hi all,

I remember in the Elegant Universe (the documentary), at some point the following lines were said:

"So what exactly, in nature, sets the values of these 20 constants so precisely? Well the answer could be the extra dimensions in string theory. That is, the tiny, curled up, six-dimensional shapes predicted by the theory cause one string to vibrate in precisely the right way to produce what we see as a photon and another string to vibrate in a different way producing an electron. So according to string theory, these miniscule extra-dimensional shapes really may determine all the constants of nature, keeping the cosmic symphony of strings in tune."

I would like to know how is it the vibration causes the strings to produce the "effect" of a photon, plus i realize that the above said is in plain language, can anyone clarify on this, and perhaps provide a link to a paper(s) that describe this? I am currently reading some lecture on "String theory on calabi-yau manifolds" but its a long document so I don't know if I'll find it in there. Thanks for the help.

- Vikram

I'll try to answer this question twice, first for the specific case of heterotic strings on Calabi-Yau manifolds, and then for string theory on general backgrounds. But first we need to look at some details which precede string theory and even quantum theory.

http://en.wikipedia.org/wiki/Vibrating_string" [Broken] is a type of wave motion. A complicated oscillatory motion can be understood as the combination of motions in separate modes, distinguished by wavelength or by the number of nodal points. See the picture: There's a fundamental mode, consisting of one big wave, and then faster vibrations consisting of smaller waves.

That's for a "string" with one end fixed at a point. For a closed string moving freely through space (imagine a rubber band floating through the space shuttle), it's all one loop, it's as if the ends are joined together, so the waves in the string can circulate indefinitely. The vibrations of a classical closed string can therefore be understood as a sum of waves moving in opposite directions around the string. In each direction, there will be a fundamental mode and then higher-frequency modes with smaller, sharper waves.

Now we come to quantum theory. In quantum theory we have probabilities. As it turns out, there are two ways to treat oscillatory motion in quantum theory. In the more obvious one, each mode has an infinite series of "energy levels"; at each level, there's a little more energy concentrated into the vibrational mode, and the average size of the oscillations gets bigger. The vibrations of the string in space are like this.

In the other type of oscillation, there are just two energy levels, which you could call "nothing happening" and "something happening". These are "fermionic modes" and they have a variety of mathematical descriptions, none of them very intuitive. The simplest way is to think of them as something that travels inside the string. As with ordinary waves, the fermionic modes can be moving in either direction around the string. The very definition of a superstring is actually, ordinary string plus these fermionic modes. (The ordinary modes are "bosonic".)

The heterotic superstring is a type of superstring which, along with the spatial, bosonic modes of vibration which arise from existing in ten dimensions, has different numbers of fermionic modes going "clockwise" and "anticlockwise" (or right and left) around the string. That's its distinguishing feature. These fermionic modes combine according to a big algebra or symmetry group called E8 x E8 (this is the symmetry group in the type of heterotic string that people think might describe the real world; there's another type with SO(32) symmetry).

The observable particles would all correspond to different lowest-energy modes of the string; the next higher levels would be very massive and unstable. If we were to consider the behavior of the heterotic superstring in ten large flat dimensions (i.e. no CY compactification considered yet), we can divide the excitations into two classes, "supergravity" and "super-Yang-Mills". The "supergravity" excitations include an ordinary, bosonic vibratory state which is the graviton, and a corresponding fermionic vibratory state which is a particle called the gravitino. The "super-Yang-Mills" excitations also include a boson and a fermion. All the observed particles - photon, electron, etc - have to come from these super-Yang-Mills excitations.

Without going into the details right now, maybe the important thing to understand is the various fermionic modes combine to produce a large number of super-Yang-Mills excitations, and then the geometry and topology of the Calabi-Yau affects them differently, just as it also affects the conventional, spatial oscillations of the superstring. Being wrapped around one hole in the CY is physically different from being wrapped around another hole in the CY, and it means that the string will be interacting differently with the geometric moduli defining the size and shape of the CY. In such a model, this is what gives the particles their different masses. But because CY dynamics is so difficult to calculate, people have mostly settled for getting other properties right, like the low-energy symmetries - some part of the E8 x E8 symmetry that still survives even after compactification. http://arxiv.org/abs/hep-th/0501070" [Broken] is an example of such a model.

So, there's the first answer. I promised a second answer. Here I just want to say that the heterotic string is just one branch of string theory. String theory really descends from an 11-dimensional theory in which you have 2-dimensional "supermembranes", and a superstring in 10 dimensions is actually a supermembrane in 11 dimensions, but with one of the supermembrane's internal directions extended along the extra 11th dimension. For example, the heterotic string is actually a supermembrane in the shape of a cylinder stretching through the 11th dimension, and attached to two "domain walls" (http://www.sukidog.com/jpierre/strings/mtheory.htm" [Broken]). So when we talk about a heterotic string in a space which consists of three large space dimensions, one time dimension, and six small space dimensions in a CY shape... really there are two three-dimensional boundary spaces, and a four-dimensional "bulk" between them, and the string we see is just the circle at one end of the cylinder. (There's one "E8" set of fermionic modes at each end of the supermembrane, which is where the combined E8 x E8 comes from.) But the width of the eleventh dimension is small, like the CY.

That may have confused you unnecessarily, but I just want to say that the superstrings of string theory are really supermembranes from the 11-dimensional theory, M-theory, and you can define M-theory on spaces apart from the background I just described. The eleventh dimension can be a loop rather than a line (the line is the gap between the boundary spaces), and in that case, the supermembrane can wrap around the loop to give rise to a different sort of superstring. In these other string theories, things are a little different. The same picture of bosonic and fermionic vibrations still exists, but you can have membranes as well as strings, and you can have "open strings" that are attached to membranes, and so there are extra factors at work. The world might be based on those other types of strings rather than on heterotic strings, and so we have to consider other models too.

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This is the first time I see a description of the actual string vibrations that a layman can easily understand, thanks!

As it turns out, there are two ways to treat oscillatory motion in quantum theory. In the more obvious one, each mode has an infinite series of "energy levels"; at each level, there's a little more energy concentrated into the vibrational mode, and the average size of the oscillations gets bigger. The vibrations of the string in space are like this.

In the other type of oscillation, there are just two energy levels, which you could call "nothing happening" and "something happening". These are "fermionic modes" and they have a variety of mathematical descriptions, none of them very intuitive. The simplest way is to think of them as something that travels inside the string.

Are these two the internal/external vibrations forbidden by the Coleman-Mandula theorem without SUSY?

That's a tough question and almost a matter of perspective.

There are two perspectives on the CM theorem. One is that it says you can't mix space-time symmetries with internal symmetries. The other is that it says you can't mix bosons with fermions. Bosons and fermions differ by spin, which is an angular momentum and so it's a space-time property, and you can reasonably view the supersymmetry algebras as fundamentally being extensions of the Poincare algebra of space-time symmetries. However, spin also looks a little like an internal symmetry, if you just think of it as an index attached to the particle.

There also happen to be two levels at which field theory shows up in string theory: on the world-sheet and in space-time. The space-time coordinates of the string behave just like scalar bosonic fields lying along the length of the string. Then in the superstring we have these fermionic fields along its length as well, which don't have a straightforward space-time interpretation. You can work in "superspace", adding "anticommuting coordinates" to the ordinary space-time coordinates, but this might be regarded as a little artificial.

Then there's the effective field theory we get at the level of space-time outside the string, the "zero-length limit" where string fields get approximated by ordinary quantum fields and the string is approximated by a point. For example, when I said the heterotic string behaves like supergravity coupled to a super-Yang-Mills gauge field.

At the world-sheet level, the fermionic excitations are extended along the string, so from the perspective of a point on the string they aren't "internal". But at the space-time level, the whole string is being approximated by a single point, and so everything becomes "internal". At either level, the theory is supersymmetric, so it's getting around the CM theorem.

You could say that the world-sheet level illustrates the power of supersymmetry to integrate bosons and fermions, while the space-time level illustrates the power of supersymmetry to combine internal and external symmetries, but that would be a bit too cute.

I just don't think there's an objective answer to your question. The mathematical facts are unambiguous, but as I've tried to explain, whether the fermionic modes of the superstring are an example of one of the un-combinable ingredients referred to in the theorem, depends both on the level at which you analyze string theory, and on what you consider the significance of the CM theorem to be.

Not quite sure I understand the answer, but thanks for trying :)