Various types of topological defects in graphene are considered in the framework of the continuum model for long-wavelength electronic excitations, which is based on the Dirac-Weyl equation. The condition for the electronic wave function is specified, and we show that a topological defect can be presented as a pseudomagnetic vortex at the apex of a graphitic nanocone; the flux of the vortex is related to the deficit angle of the cone. The cases of all possible types of pentagonal defects, as well as several types of heptagonal defects (with the numbers of heptagons up to three, and six), are analyzed. The density of states and the ground state charge are determined.

Vacuum polarization effects in the cosmic string background are considered. We find that a current is induced in the vacuum of the quantized massive scalar field and that the current circulates around the string which is generalized to a (d − 2)-brane in locally flat (d + 1)-dimensional space-time. As a consequence of the Maxwell equation, a magnetic field strength is also induced in the vacuum and is directed along the cosmic string. The dependence of the current and the field strength on the string flux and tension is comprehensively analyzed. Both the current and the field strength are holomorphic functions of the space dimension, decreasing exponentially with the distance from the string. In the case of d = 3 we show that, due to the vacuum polarization, the cosmic string is enclosed in a tube of the magnetic flux lines if the mass of the quantized field is less than the inverse of the transverse size of the string core.

We use the low-energy effective field theory for holes coupled to the staggered magnetization in order to investigate the localization of holes on a Skyrmion in a square lattice antiferromagnet. When two holes get localized on the same Skyrmion, they form a bound state. The quantum numbers of the bound state are determined by the quantization of the collective modes of the Skyrmion. Remarkably, for p-wave states the quantum numbers are the same as those of a hole-pair bound by one-magnon exchange. Two holes localized on a Skyrmion with winding number n = 1 or 2 may have s-or d-wave symmetry as well. Possible relations with preformed Cooper pairs of high-temperature superconductors are discussed.

Electronic excitations in a graphitic monolayer (graphene) in the longwavelength approximation are characterized by the linear dispersion law, representing a unique example of the really two-dimensional "ultrarelativistic" fermionic system which in the presence of topological defects possesses rather unusual properties. A disclination that rolls up a graphitic sheet into a nanocone is described by a pointlike pseudomagnetic vortex at the apex of the cone, and the flux of the vortex is related to the deficit angle of the conical surface. A general theory of planar relativistic fermionic systems in the singular vortex background is employed, and we derive the analytical expression for the charge which is induced at finite temperature on some graphitic nanocones.