New role of topology in physics: momentum-space topology of quantum vacua. Lecture 1

Topology in momentum space is becoming the leading mathematical tool in modern physics. It determines the main properties of ground states of condensed matter systems and quantum vacua of relativistic quantum fields in particle physics. The quantum vacuum of Standard Model of particle physics, both in its massless and massive phases, belong to the broad class of topological media, which also include topological superfluids and superconductors, topological insulators, topological semi-metals, etc. Momentum space topology protects the massless (gapless) fermions in bulk, or on the surface of matter, or inside the core of topological defects. The gaplessness of fermions is not sensitive to details of the microscopic physics (atomic or trans-Planckian): due to robustness of topology, irrespective of the deformation of the parameters of the microscopic theory, the energy spectrum of these fermions remains strictly gapless, and they survive in the low temperature limit. This solves the main hierarchy problem in particle physics: for fermionic vacua with topologically protected Fermi points (Dirac, Weyl or Majorana points) the masses of elementary particles are naturally small as compared to the Planck scale value.
There is a number of topological invariants in momentum space of different dimensions, which characterize different topological substances. These topological invariants determine universality classes of the topological matter; the type of the effective theory, which emerges at low energy; quantization of physical parameters in emergent topological quantum field theories. In many cases they also give rise to emergent symmetries, including the effective Lorentz invariance, and emergent phenomena such as effective gauge and gravitational fields. Some invariants are relevant for vacua of quantum chromodynamics describing topologically different vacua of lattice models with Wilson fermions.
Topological invariants in extended momentum + coordinate space determine the so-called bulk-surface and bulk-vortex correspondence. They connect the momentum space topology in bulk with the traditional topology in real space, which describes topological defects. These invariants determine the gapless fermions living on the surface of a system, at the interface between vacua with different topological charges, and in the core of topological defects (vortices, strings, domain walls, solitons, monopoles, etc.). A particular example of potential application of momentum space topology is formation of dispersionless spectrum on the surface of topological material — the flat band. Due to singular density of states in the flat band, one may expect realization of room-temperature superconductivity in topological matter.
The momentum space topology gives some lessons for quantum gravity. In effective gravity emerging at low energy, the collective variables are tetrad fields and spin connections, while the metric is only the secondary composite object of tetrad fields. This suggests that the Einstein-Cartan-Sciama-Kibble theory with torsion field is more relevant for description of gravity than general relativity. There are also several scenarios of Lorentz invariance violation governed by topology, including splitting of Fermi point in neutrino sector and development of the Dirac points with quadratic and cubic spectrum. The former may serve as possible explanation of the debated OPERA experiments, which suggest superluminal neutrino; the latter leads to the natural emergence of the Horava-Lifshitz anisotropic gravity.