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Cyclotron resonance in graphene is studied with focus on many-body corrections to the resonance energies, which evade Kohn's theorem. The genuine many-body corrections turn out to derive from vacuum polarization, specific to graphene, which diverges at short wavelengths. Special emphasis is placed on the need for renormalization, which allows one to determine many-body corrections uniquely from one resonance to another. For bilayer graphene, in particular, both intralayer and interlayer coupling strengths undergo infinite renormalization; as a result, the renormalized velocity and interlayer coupling strength run with the magnetic field. A comparison of theory with the experimental data is made for both monolayer and bilayer graphene.

The electromagnetic response of graphene in a magnetic field is studied, with particular emphasis on the quantum features of its ground state (vacuum). The graphene vacuum, unlike in conventional quantum Hall systems, is a dielectric medium and carries an appreciable amount of electric and magnetic susceptibilities. The dielectric effect grows rapidly with increasing filling factor ν in such a way that reflects the 'relativistic' Landau-level characteristics of graphene as well as its valley and spin degeneracy. A close look into the dielectric function also reveals that the Coulomb interaction is efficiently screened on the scale of the magnetic length, leading to a prominent reduction of the exciton spectra in graphene. In addition, an effective gauge theory of graphene is constructed out of the response. It is pointed out thereby that the electric susceptibility is generally expressed as a ratio of the Hall conductance to the Landau gap.

Bilayer graphene in a magnetic field supports eight zero-energy Landau levels, which, as a tunable band gap develops, split into two nearly-degenerate quartets separated by the band gap. A close look is made into the properties of such an isolated quartet of pseudo-zero-mode levels at half filling in the presence of an in-plane electric field and the Coulomb interaction, with focus on revealing further controllable features in bilayer graphene. The half-filled pseudo-zero-mode levels support, via orbital level mixing, charge carriers with nonzero electric moment, which would lead to fieldinduced level splitting and the current-induced quantum Hall effect. It is shown that the Coulomb interaction enhances the effect of the in-plane field and their interplay leads to rich spectra of collective excitations, pseudospin waves, accessible by microwave experiments; also a duality in the excitation spectra is revealed.

In a magnetic field bilayer graphene supports an octet of zero-energy Landau levels with an extra twofold degeneracy in Landau orbitals n = 0 and n = 1. It is shown that this orbital degeneracy is lifted due to Coulombic quantum fluctuations of the valence band (the Dirac sea); this is a quantum effect analogous to the Lamb shift in the hydrogen atom. A detailed study is made of how these zero-energy levels evolve, with filling, into a variety of pseudo-zero-mode Landau levels in the presence of possible spin and valley breaking and Coulomb interactions, and a comparison is made with experimental results.

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A superfield formulation is presented of the central charge anomaly in quantum corrections to solitons in two-dimensional theories with N = 1 supersymmetry. Extensive use is made of the superfield supercurrent, that places the supercurrent J µ α , energy-momentum tensor Θ µν and topological current ζ µ in a supermultiplet, to study the structure of supersymmetry and related superconformal symmetry in the presence of solitons. It is shown that the supermultiplet structure of (J µ α , Θ µν , ζ µ ) is kept exact while the topological current ζ µ acquires a quantum modification through the superconformal anomaly. In addition, the one-loop superfield effective action is explicitly constructed to verify the BPS saturation of the soliton spectrum as well as the effect of the anomaly.

A superspace variational formulation of supercurrents and their associated anomalies is presented, with emphasis on showing its foundation on superconformal symmetry. Parameters of superconfor-ma1 transformations are embedded in a spinor superfield, in terms of which local-superconformal variations of superfields are defined. The one-loop supercurrent anomalies are calculated for the Wess-Zumino model and supersymmetric QED by the path-integral method.