1. The theory of the two-dimensional (2D) magnetoexcitons was completed taking into account the electron-hole (e-h) Coulomb exchange interaction in addition to the direct one. The exchange e-h Coulomb scattering takes place with the annihilation and the creation of the e-h pairs with the resultant electronic charges equal to zero. They have dipole-dipole interaction, when the interband dipole moments c v r - are different from zero. It happens when the crystals have the dipole active optical quantum transitions. We have considered the semiconductor layers of the type GaAs with s-type conductions band and p-type valence band with magnetoexcitons formed by electrons with spin projections e 1 2 z s = ± and by heavy holes with full angular momentum projections h 3 2. z j = ±

The Lorentz force in the Landau gauge description determines the positions of the Landau quantization oscillations of the electrons and holes and their distances in the frame of the magnetoexcitons. Their relative and center of mass motions are interconnected. In difference on the direct Coulomb e-h interaction, which gives rise to the quadratic dispersion lawh2k2 2M(B)with magnetic mass M(B) depending on the magnetic field strength B, the exchange e-h Coulomb interaction gives rise to linear dispersion law known as Dirac cone g hv k with group velocity g v depending on the interband dipole moment in the way: 2 g 0 v , c vr l B - » » where 0 l is the magnetic length.

2. The thermodynamic properties of the ideal 2D Bose gas with linear dispersion law were discussed in the Ref [1]. The critical temperature of the Bose-Einstein condensation (BEC) of the 2D magnetoexcitons is different from zero even at the infinite homogeneous surface area and following [1] is proportional to the group velocity: cT : g : B v . In the case of the magnetoexcitons it increases with the increasing magnetic field strength B.

3. It was shown that the Chern-Simons (C-S) gauge field created by the quantum point vortices in the conditions of the fractional quantum Hall effects (FQHEs) leads to the formation of the composite electrons and holes with equal integer numbers of the attached to each particle quantum point vortices. The coherent superposition of the velocities of these vortices leads to the formation of the C-S vector potential, which depends on the difference between the density operators ˆe r of the electrons and ˆh r of the holes. The C-S vector potential generates the effective magnetic field acting on the particles in addition to the external magnetic field. In the mean field approximation, when the average densities of electrons and of the holes coincide the effective C-S magnetic and electric fields vanish and the Landau quantization of the composite particles with the bare electron and hole effective masses take place only under the influence of the external magnetic field [2].

 1. Moskalenko, S.A. and Snoke, D.W. Bose-Einstein condensation of excitons and biexcitons and coherent nonlinear optics with excitons (2000), Cambridge University Press, p.189. 2. Moskalenko, S.A. Moskalenko, V.A. Mold. J. Phys. Sci. 17(1-2), (2018).