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A loop is a 1-sphere that can vibrate in increasingly complex ways as it is embedded in higher dimensional spaces.

  1. Does string theory assume that 1-spheres are the only possible vibrating structures, or does it also look at the vibrations of higher spheres, e.g. (2,3,...,8)-spheres? If only 1-spheres are assumed, what is the reason for that constraint?

  2. Whether for 1-spheres or $n$-spheres, are the vibration modes of string theory assumed always to be in 9-space, or are more restricted modes in lower-dimensional subspaces also explored?

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Dilaton, thanks, those are more than just relevant: I think they are the same question in slightly different forms. I looked but did not find them prior to asking (wrong keywords no doubt). @Qmechanic, while Mitchell Porter's answer is great and needs to be kept, shouldn't my question get axed for redundancy? I seem to be the third person who has asked this! Only the second part of my question -- whether subspace subsets of vibration modes are relevant -- seems to have any uniqueness, and I'm not sure about that. Your call, but I stand ready and willing for the ax in this instance. –  Terry Bollinger Aug 11 '13 at 18:31

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The higher-dimensional objects are the "branes", which were first found as extended black holes in supergravity. In M-theory, the fundamental objects are the 2-brane and the 5-brane, and strings arise from compactifying a brane, e.g. a 2-brane is wrapped around the eleventh dimension, and shows up as a string in the 10-dimensional limit.

The classical vibrations of a string can be described e.g. by Fourier analysis. For the quantum vibrations of a string, you treat the Fourier modes as quantum harmonic oscillators, and get the basic string spectrum. If the string is interacting, then it can emit and absorb new strings too.

A similar analysis can be performed for branes, but it is much harder; it has been the work of many years and is still unfinished. The classical vibrations of a brane are straightforward enough, it's just like the sheet of a drum. The quantum vibrations of a brane are described by a "worldvolume theory" whose degrees of freedom include the position of the brane in the external space - entirely analogous to the scalars in the "worldsheet theory" of a string, which correspond to the position vectors of the points along the string.

But just as a string can emit and absorb other strings, a brane can emit and absorb strings and other branes. More precisely, some strings and branes will be attached - the open strings attached to a D-brane, or the "open M2-brane" of M-theory, which is like a cylinder with the circular ends attached to the parent brane - and they may detach and become closed strings/branes, or while attached they may emit and absorb closed strings/branes.

Another phenomenon is the brane stack, where N branes are on top of each other. In the case of open strings, this means that each string stretches from brane "i" to brane "j" for some i,j <= N. The interactions of these strings are described by a gauge theory of rank N, such as U(N).

AdS/CFT at strong coupling is also relevant here, though I have never quite got the details straight. It appears to be like this: the CFT is the worldvolume theory of the black brane stack, and "string theory in AdS space" is what happens behind the event horizon of the stack. Another thing I'm not clear on is the extent to which every string and brane can be viewed as a sort of black hole or black hole remnant.

The bottom line is that the vibrations of higher-dimensional objects are certainly a part of string theory (e.g.), but the basic theory of branes is still being worked out, and the more quantum it gets, the less it looks like a simple "blob that vibrates".

Many people like to say that the ultimate form of string theory will not even feature space and time as fundamental, they will instead be "emergent" from something else. If that happens, what we now call the vibration of a string or a brane will also have to be "something else" fundamentally, though I don't know what. Or perhaps it won't even be part of the simplest formulation - perhaps stringy vibration is a sort of dynamical gauge freedom, a manifestation of a redundancy that is not strictly necessary.

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Thanks for this very nice easily readable and enjoyable text :-), +1 –  Dilaton Aug 11 '13 at 8:15
    
Mitchell Porter, thanks, that is definitely one of the more most straightforward and comprehensible quick overviews I've seen of string theory concepts. It was also very unexpected for me: While I was aware that branes were a related concepts (I almost mentioned them in the question), I had been under the incorrect impression that they were only used as vast, cosmic-scale concepts. With regards to my second question, is the compactification approach compatible with the object being limited to vibrations in subspaces, e.g., the 1-sphere within 2D vs 3D space? Is this also in string theory? –  Terry Bollinger Aug 11 '13 at 18:23
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Hi Mitchell, maybe you are interested in this too? Sorry if not ... –  Dilaton Aug 18 '13 at 0:51
    
@Terry Bollinger... Along with the perspective of strings moving through space, string theory also has the worldsheet perspective, where you have a 1+1 dimensional space which is the interior of the string, and then N bosonic fields on the worldsheet which correspond to the dimensions of the background space in the usual picture, and finally enough fermionic fields to cancel the conformal anomaly on the worldsheet... –  Mitchell Porter Sep 4 '13 at 8:20
    
It is possible for string theory, in this 1+1 dimensional form, to have nongeometric phases or nongeometric compactifications (just google those phrases for details), in which not all the worldsheet bosonic degrees of freedom are interpretable as spatial dimensions. So it seems to be possible to have a nongeometric phase in which less than the usual 9+1 dimensions exist as space, e.g. there might just be 3+1 dimensions, and then the remaining 6 bosons are nongeometric... –  Mitchell Porter Sep 4 '13 at 8:22

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