Lie theory for asymptotic symmetries in general relativity: The BMS Group

We study the Lie group structure of asymptotic symmetry groups in General Relativity from the viewpoint of infinite-dimensional geometry. To this end, we review the geometric definition of asymptotic simplicity and emptiness due to Penrose and the coordinate-wise definition of asymptotic flatness due to Bondi et al. Then we construct the Lie group structure of the Bondi--Metzner--Sachs (BMS) group and discuss its Lie theoretic properties. We find that the BMS group is regular in the sense of Milnor, but not real analytic. This motivates us to conjecture that it is not locally exponential. Finally, we verify the Trotter property as well as the commutator property. As an outlook, we comment on the situation of related asymptotic symmetry groups. In particular, the much more involved situation of the Newman--Unti group is highlighted, which will be studied in future work.

Энергетический спектр и спектр оптического поглощения эндоэдральных фуллеренов Lu-=SUB=-3-=/SUB=-N@C-=SUB=-80-=/SUB=- и Y-=SUB=-3-=/SUB=-N@C-=SUB=-80-=/SUB=- в модели Хаббарда

Anticommutator Green’s functions and energy spectra of fullerene C80, endohedral fullerenes Lu3N@С80 and Y3N@С80 with the Ih symmetry groups have been obtained in an analytical form within the Hubbard model and static fluctuation approximation. The energy states have been classified using the methods of group theory, and the allowed transitions in the energy spectra of molecules C80, Lu3N@С80 and Y3N@С80have been determined. On the basis of these spectra, an interpretation of experimentally observed optical absorption bands endohedral fullerenes Lu3N@С80 and Y3N@С80.

Computing Classification of Interacting Fermionic Symmetry-Protected Topological Phases Using Topological Invariants*

In recent years, great success has been achieved on the classification of symmetry-protected topological (SPT) phases for interacting fermion systems by using generalized cohomology theory. However, the explicit calculation of generalized cohomology theory is extremely hard due to the difficulty of computing obstruction functions. Based on the physical picture of topological invariants and mathematical techniques in homotopy algebra, we develop an algorithm to resolve this hard problem. It is well known that cochains in the cohomology of the symmetry group, which are used to enumerate the SPT phases, can be expressed equivalently in different linear bases, known as the resolutions. By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies, the computational cost is drastically reduced. In particular, it reduces the computational cost for infinite discrete symmetry groups, like the wallpaper groups and space groups, from infinity to finity. As examples, we compute the classification of two-dimensional interacting fermionic SPT phases, for all 17 wallpaper symmetry groups.

Gauge Interactions

The principle of gauge symmetry is introduced as a consequence of the invariance of the equations of motion under local transformations. We apply it to Abelian, as well as non-Abelian, internal symmetry groups. We derive in this way the Lagrangian of quantum electrodynamics and that of Yang–Mills theories. We quantise the latter using the path integral method and show the need for unphysical Faddeev–Popov ghost fields. We exhibit the geometric properties of the theory by formulating it on a discrete space-time lattice. We show that matter fields live on lattice sites and gauge fields on oriented lattice links. The Yang–Mills field strength is related to the curvature in field space.

Dynamical symmetry indicators for Floquet crystals

Various exotic topological phases of Floquet systems have been shown to arise from crystalline symmetries. Yet, a general theory for Floquet topology that is applicable to all crystalline symmetry groups is still in need. In this work, we propose such a theory for (effectively) non-interacting Floquet crystals. We first introduce quotient winding data to classify the dynamics of the Floquet crystals with equivalent symmetry data, and then construct dynamical symmetry indicators (DSIs) to sufficiently indicate the inherently dynamical Floquet crystals. The DSI and quotient winding data, as well as the symmetry data, are all computationally efficient since they only involve a small number of Bloch momenta. We demonstrate the high efficiency by computing all elementary DSI sets for all spinless and spinful plane groups using the mathematical theory of monoid, and find a large number of different nontrivial classifications, which contain both first-order and higher-order 2+1D anomalous Floquet topological phases. Using the framework, we further find a new 3+1D anomalous Floquet second-order topological insulator (AFSOTI) phase with anomalous chiral hinge modes.

HEAT TRANSFER IN GRADIENT LAMINAR FLOWS

The development of new areas of research in the field of theoretical thermophysics requires reliable analytical solutions that could take into account the main aspects of physical parameters in the studied objects. One such analytical technique is symmetry groups. On the basis of symmetry groups the problem of heat transfer in gradient laminar flows is solved in the paper. For the first time, the symmetries of the energy equation for the boundary layer at an arbitrary changing velocity at marching direction are obtained. Examples of the use of group analysis methods for the study of heat transfer in the boundary layer of an incompressible fluid are demonstrated. The problems of heat transfer in the boundary layer on a heat-conducting wall with a constant temperature and on a heat-insulated wall are considered. Analytical relations for temperature and heat transfer coefficients distribution are obtained.

Exploring multi-Higgs models with softly broken large discrete symmetry groups

We develop methods to study the scalar sector of multi-Higgs models with large discrete symmetry groups that are softly broken. While in the exact symmetry limit, the model has very few parameters and can be studied analytically, proliferation of quadratic couplings in the most general softly broken case makes the analysis cumbersome. We identify two sets of soft breaking terms which play different roles: those which preserve the symmetric vacuum expectation value alignment, and the remaining terms which shift it. Focusing on alignment preserving terms, we check which structural features of the symmetric parent model are conserved and which are modified. We find remarkable examples of structural features which are inherited from the parent symmetric model and which persist even when no exact symmetry is left. The general procedure is illustrated with the example of the three-Higgs-doublet model with the softly broken symmetry group $$\Sigma (36)$$ Σ ( 36 ) .