Abstract:
We show that the notion of mutual statistics arises naturally from the representation theory of the braid group over the multi-sheeted surface. A Hamiltonian which describes particles moving on the double-sheeted surface is proposed as a model for the bilayered fractional quantum Hall effect (FQHE) discovered recently. We explicitly show that the quasi-holes of the bilayered Hall fluid display fractional mutual statistics. A model for 3-dimensional FQHE using the multi-layered sample is suggested.

Abstract:
The path integral approach to representing braid group is generalized for particles with spin. Introducing the notion of {\em charged} winding number in the super-plane, we represent the braid group generators as homotopically constrained Feynman kernels. In this framework, super Knizhnik-Zamolodchikov operators appear naturally in the Hamiltonian, suggesting the possibility of {\em spinning nonabelian} anyons. We then apply our formulation to the study of fractional quantum Hall effect (FQHE). A systematic discussion of the ground states and their quasi-hole excitations is given. We obtain Laughlin, Halperin and Moore-Read states as {\em exact} ground state solutions to the respective Hamiltonians associated to the braid group representations. The energy gap of the quasi-excitation is also obtainable from this approach.

Abstract:
We give a brief introduction to the phenomenon of the Fractional Quantum Hall effect, whose discovery was awarded the Nobel prize in 1998. We also explain the composite fermion picture which describes the fractional quantum Hall effect as the integer quantum Hall effect of composite fermions.

Abstract:
It is shown that the Jain mapping between states of integer and fractional quantum Hall systems can be described dynamically as a perturbative renormalization of an effective Chern-Simons field theory. The effects of mirror duality symmetries of toroidally compactified string theory on this system are studied and it is shown that, when the gauge group is compact, the mirror map has the same effect as the Jain map. The extrinsic ingredients of the Jain construction appear naturally as topologically non-trivial field configurations of the compact gauge theory giving a dynamical origin for the Jain hierarchy of fractional quantum Hall states.

Abstract:
Experimental data for fractional quantum Hall systems can to a large extent be explained by assuming the existence of a modular symmetry group commuting with the renormalization group flow and hence mapping different phases of two-dimensional electron gases into each other. Based on this insight, we construct a phenomenological holographic model which captures many features of the fractional quantum Hall effect. Using an SL(2,Z)-invariant Einstein-Maxwell-axio-dilaton theory capturing the important modular transformation properties of quantum Hall physics, we find dyonic diatonic black hole solutions which are gapped and have a Hall conductivity equal to the filling fraction, as expected for quantum Hall states. We also provide several technical results on the general behavior of the gauge field fluctuations around these dyonic dilatonic black hole solutions: We specify a sufficient criterion for IR normalizability of the fluctuations, demonstrate the preservation of the gap under the SL(2,Z) action, and prove that the singularity of the fluctuation problem in the presence of a magnetic field is an accessory singularity. We finish with a preliminary investigation of the possible IR scaling solutions of our model and some speculations on how they could be important for the observed universality of quantum Hall transitions.

Abstract:
The edge states of a sample displaying the quantum Hall effect (QHE) can be described by a 1+1 dimensional (conformal) field theory of $d$ massless scalar fields taking values on a $d$-dimensional torus. It is known from the work of Naculich, Frohlich et al.\@ and others that the requirement of chirality of currents in this \underline{scalar} field theory implies the Schwinger anomaly in the presence of an electric field, the anomaly coefficient being related in a specific way to Hall conducvivity. The latter can take only certain restricted values with odd denominators if the theory admits fermionic states. We show that the duality symmetry under the $O(d,d;{\bf Z})$ group of the free theory transforms the Hall conductivity in a well-defined way and relates integer and fractional QHE's. This means, in particular, that the edge spectra for dually related Hall conductivities are identical, a prediction which may be experimentally testable. We also show that Haldane's hierarchy as well as certain of Jain's fractions can be reproduced from the Laughlin fractions using the duality transformations. We thus find a framework for a unified description of the QHE's occurring at different fractions. We also give a derivation of the wave functions for fractions in Haldane's hierarchy.

Abstract:
We explore the consequences of introducing a complex conductivity into the quantum Hall effect. This leads naturally to an action of the modular group on the upper-half complex conductivity plane. Assuming that the action of a certain subgroup, compatible with the law of corresponding states, commutes with the renormalisation group flow, we derive many properties of both the integer and fractional quantum Hall effects, including: universality; the selection rule $|p_1q_2 - p_2q_1|=1$ for quantum Hall transitions between filling factors $\nu_1=p_1/q_1$ and $\nu_2=p_2/q_2$; critical values for the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalisation group flow lead to the semi-circle rule for transitions between Hall plateaus.

Abstract:
One kind of hierarchical wave functions of Fractional Quantum Hall Effect (FQHE) on the torus are constructed. The multi-component nature of anyon wave functions and the degeneracy of FQHE on the torus are very clear reflected in this kind of wave functions. We also calculate the braid statistics of the quasiparticles in FQHE on the torus and show they fit to the picture of anyons interacting with magnetic field on the torus obtained from braid group analysis.

Abstract:
A low energy effective Hamiltonian for the fractional quantum Hall effect is obtained by using irreducible representations of the symmetry group. It is found that the model described by the effective Hamiltonian is similar to the Heisenberg spin chain. Symmetries of the effective Hamiltonian are studied in order to decompose the relevant Hilbert space. This decomposition will be useful for further numerical analysis on the gap of the quantum Hall system.

Abstract:
Unlike regular electron spin, the pseudospin degeneracy of Fermi points in graphene does not couple directly to magnetic field. Therefore, graphene provides a natural vehicle to observe the integral and fractional quantum Hall physics in an elusive limit analogous to zero Zeeman splitting in GaAs systems. This limit can exhibit new integral plateaus arising from interactions, large pseudoskyrmions, fractional sequences, even/odd numerator effects, composite-fermion pseudoskyrmions, and a pseudospin-singlet composite-fermion Fermi sea. The Dirac nature of the B=0 spectrum, which induces qualitative changes in the overall spectrum, has no bearing on the fractional quantum Hall effect in the $n=0$ Landau level of graphene. The second Landau level of graphene is predicted to show more robust fractional quantum Hall effect than the second Landau level of GaAs.