Fractional quantum Hall effect on compact surfaces

The thesis is organized as follows. At first, in the second chapter we will give an elementary introduction to the QHE. We review the theories for the IQHE and FQHE. In particular, We emphasis the role of topology in the QHE and show that the Hall conductance is related to the topological invariant. It is also shown that the quasiparticles in the FQHE are anyons, which satisfy fractional statistics. In chapter 3, we first review the standard hierarchical theory of the FQHE. We consider the FQHE on the sphere and use the projective coordinates on the sphere. The guiding principle for constructing the state of the FQHE on the sphere is the rotational invariance of the wave function. Spin-nonpolarized FQHE will be also discussed in this chapter. When the magnetic field is not strong enough, the electron spin may not be polarized. Halperin had proposed a class of state with half spins reversed which are spin-singlet states. The hierarchical wave function for the spin-non-polarized case, for example, the spin-singlet FQHE, is not still fully understood. So it remains an interesting problem. We will construct the hierarchical wave function based on the Halperin spin-singlet state. We will also calculate the spin of the quasiparticle in the FQHE on the sphere. However, some questions remain unanswered. Is the spin of the quasiparticle topological-independent or not? This problem will be answered in chapter 6. In chapter 4, the hierarchical wave function on the torus will be analyzed in details. We construct the wave function of the quasiparticles and discuss the fractional statistics on the torus. Modular invariance is also discussed. The degeneracy of the hierarchical state is obtained explicitly by constructing the wave function. In chapter 5, we discuss the quantum mechanics and quantum Hall effect on Riemann surfaces. In particular, we construct the Laughlin wave function on Riemann surfaces of high genus. The degeneracy of the Laughlin states is obtained explicitly and our result about the degeneracy is different from the one in the literature. In chapter 6, we will calculate the spin of the quasiparticle on Riemann surfaces by using the braid group on Riemann surfaces. We find that the spin of the quasiparticle is topological-independent.