There are two questions here:
One of the confusing points about String Theory is the existence of extra dimensions. These are explained by saying that the these extra dimensions are compactified.
- Does this they are something similar to a principal bundle? For example electromagnetism is described as a circle bundle; is string theory similar, except that instead of a circle, the standard fibre is something else, for example a Calabi-Yau space?
Now, there is another place where extra dimensions are found and thats Quantum Mechanics. Here, not only do we have extra dimensions, we have an infinite number of them - the Hilbert space of states.
- Where are they - or is this the wrong question to ask because they are a mathematical artifact - an artifact of this particular description?