The implementation of modular invariance on the torus at the quantum level is discussed in a group-theoretical framework. Two cases must be considered, depending on the cohomology class of the symplectic form on the torus. If it is of integer cohomology class n, then full modular invariance is achieved at the quantum level only for those wave functions on the torus which are periodic if n is ev...

The Landau problem is discussed in two similar but still different non-commutative frameworks. The “standard” one, where the coupling to the gauge field is achieved using Poisson brackets, yields all Landau levels. The “exotic” approach, where the coupling to the gauge field is achieved using the symplectic structure, only yields lowest-Landau level states as advocated by Peierls, and widely us...

Addition formulas for theta functions of arbitrary order are shown and applied to the theoretical understanding of the fractional quantum Hall effect in a multi-layer two-dimensional many-electron system under periodic conditions.

A two-dimensional harmonic oscillator, when rotated by the oscillator frequency, generates Landau-like levels. A further cranking results in condensates and gaps resembling the fractional quantum Hall effect. For a filling fraction ν = p/q, with q odd, the model predicts that the gap is proportional to √ ne/pq, where ne is the electron density of the sample. This agrees well with recent experim...

The “Laughlin” picture of the Fractional Quantum Hall effect can be derived using the “exotic” model based on the two-fold centrally-extended planar Galilei group. When coupled to a planar magnetic field of critical strength determined by the extension parameters, the system becomes singular, and “Faddeev-Jackiw” reduction yields the “Chern-Simons” mechanics of Dunne, Jackiw, and Trugenberger. ...

We consider two ways of introducing minimal Abelian gauge interactions into the model presented in [1]. These two approaches are different only if the second central charge of the planar Galilei group is nonzero. One way leads to the standard gauge transformations and the other one to a generalized gauge theory with gauge transformations accompanied by time-dependent area-preserving coordinate ...

A major obstacle toward a quantitative verification, by comparison to experiment, of the theory of the excitations of the fractional quantum Hall effect has been the lack of a proper understanding of disorder. We circumvent this problem by studying the neutral magneto-roton excitations, whose energy is expected to be largely insensitive to disorder. The calculated energies of the roton at 1/3, ...

The microscopic approach for studying the half-filled state of the fractional quantum Hall effect is based on the idea of proposing a trial Fermi wave function of the Jastrow-Slater form, which is then fully projected onto the lowest Landau level. A simplified starting point is to drop the projection operator and to consider an unprojected wave function. A recent study claims that such a wave f...