The fractional quantum Hall effect on a sphere and the Atiyah-Singer index theorem

Most analyses of the ground state wavefunctions of the QHE ignore electron spin and simply use anti-commuting scalars: either because strong magnetic fields split the spins and the ground state is completely spin polarised or because the ground state is effectively spin degenerate so the only effect of spin is to double the degeneracy of the ground state. A useful approach is that of the Haldane sphere where the system is put on a 2-dimensional sphere and a normal magnetic field is generated by placing a magnetic monopole at the centre of the sphere. The analysis of the fractional quantum Hall effect in this approach is somewhat subtle, as the ground state wavefunction of the unperturbed Hamiltonian is not unique in Haldane's approach. When the fermionic nature of the charged particles is fully incorporate into this picture the analysis is modified and we show that, incorporating Jain's composite fermion picture into the Haldane sphere leads to a unique unperturbed ground state for the fractional quantum Hall effect on a sphere. A mass gap then ensures stability under perturbations. An important tool in the analysis is the Atiyah-Singer index theorem.