My impression in that most theories assume three spatial dimensions and one temporal dimension, though could in principle be formulated in others numbers of dimensions without inconsistencies. I know, however, that string theory makes specific predictions for the number of dimensions. It cannot be formulated in an arbitrary number of dimensions.

Is string theory unique in this regard? Are there other theories that predict (or even restrict) the numbers of dimensions? I'm interested in viable theories as well as toy models and historical ideas that turned out to be wrong.

I know for example that spontaneous symmetry breaking cannot occur in $d\le2$ spatial dimensions (Coleman/Mermin-Wagner theorem). Whilst it might be reasonable to expect that SSB is required in a theory of nature, I'm not sure whether I consider this to be a prediction for $d$ in QFT. I don't want to consider anthropic "predictions" a la Hawking.


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I just want to point out that the dimensionality of spacetime is a bit of a fluid concept in string theory. Superstring theory can only be formulated in 10 dimensions, but it can be shown to be dual to an 11-dimensional theory called M-theory. It has also been conjectured to be dual to a 12-dimensional theory known as F-theory, although whether or not this theory is actually 12-dimensional is a bit disputed at this point (for example one of the originators of F-theory for example told me that he no longer considers it to be truly 12-dimensoinal).

Furthermore, just because string theory is 10 (or more)-dimensional, it can describe a variety of smaller-dimensional systems if the extra dimensions are curled up. This is why string theory is a candidate theory of everything for us 4-dimensional beings.

Then there are holographic dualities. The most well-understood is the equivalence between string theory on a certain 10 dimensional background and a CFT in 4 dimensions.

All these examples indicate that the question of "how many dimensions does your theory have" doesn't always have a clear answer. Before AdS/CFT everyone would have thought that N=4 supersymmetric Yang-Mills was unambiguously 4-dimensional, now we see that we can just as well describe it as a 10-dimensional semi-classical gravity theory. In fact if the theory is strongly coupled, this is perhaps a more fundamental description.


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