TITLE: Does string theory really associate 6 dimensions to electromagnetism & the nuclear forces?
QUESTION [0 upvotes]: 1) I understand string (superstring) theory often ends up with 10 dimensions, 9 space-like and 1 timelike. Typically I read that these are all associated to space-time.
2) So, I was interested when I came across a statement (without reference to source) that string theory also associates four dimensions to space-time (three space-like, one time-like), one to electromagnetism and five to the nuclear forces (these last six also being space-like, and compacted).
Question: is (2) just bad pop-science writing/wrong? If it is right, can someone give a clear & concise summary how this is supposed to work (e.g. why only 1D for electromagnetism etc).

REPLY [1 votes]: The quote in question reads:

It turns out that in order to encompass both of these two forces, we have to add another five dimensions to our mathematical description. There’s no a priori reason it should be five; and, again, none of these additional dimensions relates directly to our sensory experience. They are just there in the mathematics. So this gets us to the 10 dimensions of string theory. Here there are the four large-scale dimensions of spacetime (described by general relativity), plus an extra six ‘compact’ dimensions (one for electromagnetism and five for the nuclear forces), all curled up in some fiendishly complex, scrunched-up, geometric structure.

As written, this is misleading, and venturing on totally wrong. Superstring theory requires 10 dimensions to realize the symmetries of special (and general) relativity on a theory of a quantum string. The six extra dimensions of space are required for these 'spacetime' symmetries to be respected, and are certainly not there, divided into 1 + 5, for the fundamental forces and particles that we observe.
However, there is a technical sense in which 'electromagnetism' is a "1d" force. Quantum electrodynamics follows from 'gauging' (making local) a global phase symmetry $\psi \to e^{iq\phi}\psi$ of the wavefunction (a quantum field). The group of phase rotations is called $U(1)$, and is a one-dimensional group. More technically, a theory defined in $d$-dimensional spacetime with electromagnetism can be considered to have a $U(1)$ fibration that describes the EM gauge potential. In string theory, this can come from any number of more complicated geometric internal structures. It is very unlikely that this is what was being referenced, especially since from this point of view you would need 3 + 8 = 11 more 'dimensions' ($\dim(SU(2)) + \dim(SU(3)) = 11$) to account for the Standard Model's description of the nuclear forces.
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