Interplay of real space and momentum space topologies in strongly correlated fermionic systems

We discuss the momentum-space topology of 3+1 and 2+1 strongly correlated fermionic systems. For the 3+1 systems the important universality class is determined by the topologically stable Fermi points in momentum space. In the extreme limit of low energy the condensed matter system of this universality class acquires all the symmetries, which we know today in high energy physics: Lorentz invariance, gauge invariance, general covariance, etc. The chiral fermions as well as gauge bosons and gravity field arise as fermionic and bosonic collective modes of the system. This introduces the conceptual similarity between such condensed matter and the quantum vacuum of Standard Model, which also contains the Fermi point. For the 2+1 fermionic systems the momentum space topology gives rise to the quantization of different physical parameters, such as Hall conducivity and spin-Hall conductivity, and to quantum phase transitions with the abrupt change of the momentum-space topology. The combined p-space and r-space topology determines in particular the topology of the energy spectrum of fermion zero modes living within the topological defects; fermionic charges of the topologically nontrivial extended objects; axial anomaly; etc.