Topology in momentum space is the main characteristics of the ground states of a system at zero temperature, the quantum vacua. The gaplessness of fermions in bulk, on the surface or inside the vortex core is protected by topology. Irrespective of the deformation of the parameters of the microscopic theory, the energy spectrum of these fermions remains strictly gapless. This solves the main hierarchy problem in particle physics. The quantum vacuum of Standard Model is one of the representatives of topological matter alongside with topological superfluids and superconductors, topological insulators and semi-metals, etc. There is a number of of topological invariants in momentum space of different dimensions. They determine universality classes of the topological matter and the type of the effective theory which emerges at low energy, give rise to emergent symmetries, including the effective Lorentz invariance, and emergent gauge and gravitational fields. The topological invariants in extended momentum and coordinate space determine the bulk-surface and bulk-vortex correspondence, connecting the topology in bulk with the real space. The momentum space topology gives some lessons for quantum gravity. In effective gravity emerging at low energy, the collective variables are the tetrad field and spin connections, while the metric is the composite object of tetrad field. This suggests that the Einstein-Cartan-Sciama-Kibble theory with torsion field is more relevant. There are also several scenarios of Lorentz invariance violation governed by topology, including splitting of Fermi point and development of the Dirac points with quadratic and cubic spectrum. The latter leads to the natural emergence of the Horava-Lifshitz gravity.
Breaking de Sitter Symmetry with Cosmic Topology: Cosmological Role of Infrared Vacuum Freedom
Case Western Reserve University（凯斯西储大学）；Instituto de Física Teórica (IFT) UAM-CSIC（马德里自治大学-西班牙国家研究委员会理论物理研究所）；Imperial College London（帝国理工学院）；INFN, Sezione di Padova（意大利国家核物理研究院帕多瓦分部）；Università degli Studi di Padova（帕多瓦大学）；Laboratoire Univers et Théories, Observatoire de Paris, Université PSL, Université Paris Cité, CNRS（巴黎天文台宇宙与理论实验室（巴黎文理研究大学、巴黎西岱大学、法国国家科学研究中心））
