>>10148157The Kaluza-Klein monopole in string theory is made from the Hopf fibration. String theory is not my expertise though so I'll give you another example.
Suppose you have Helium-3 in its chiral superfluid-A1 phase, then its BdG Hamiltonian is parameterized by the Bloch vectors such that , where is the dispersion. By tuning the chemical potential , you can tune the magnitude of the 3-dimensional -vector without interfering with its topological order, hence He3-A1 is characterized by homotopy classes of maps into the -sphere . Now suppose we add a vortex (i.e. string defect) through the superconducting order parameter along - via a magnetic field threading the superfluid He3, for instance - then the symmetry is broken down to for which is parameterized by
1. the orbital coordinate , and
2. the "unbroken" Bloch vectors , i.e. these crystal momenta stay good quantum numbers even in the presence of the vortex, compactified such that and is defined at ,
then we see that . Hence the Pontrjagyn invariant , or equivalently the Pontrjagyn index , characterizes the topological orders of the superfluid He3-A1 in the presence of a vortex.
