On to part 2, Duality!
Strings can wrap around a compact dimension, which leads to a number w, the ‘winding number’ of the string. Classically one would expect that to be fixed, but when we get interactions w will be able to change, i.e. be dynamic, and will have a corresponding quantum number.
Huggett goes on to explain T-duality. Winding and momentum can change roles (n <–> w) in the Hamiltonian (and R <–> 1/R) . The dynamics of the spatial wavefunction becomes the dynamics of the winding wavefunction in the dual, and vice-versa… the pattern of observed quantities is preserved. T-duality is like a translation manual between two theories.
This raises Huggett’s 3rd question: How should we understand T-duality? Do duals describe physically distinct situations?
Beyond Spacetime 
In brief, given the strategy described by Carl, both winding and momentum are represented by wavefunctions living in on circles – momentum in physical space, with radius R, and winding in a ‘new’ space of radius 1/R. If you now consider a dual system in which physical space has radius 1/R, it follows that the winding wavefunction lives in a space of radius 1/(1/R) = R. If you also exchange the momentum and winding wavefunctions, then nothing changes – except which space is physical space. But is that any change at all?