FORMATION OF MODERN MATHEMATICAL APPROACH TO SOLVING PROBLEMS OF PHYSICS
Formulation of the problem. Precision studies of the Higgs boson, supersymmetric particles, the magnetic moment of the muon, electric dipole moment of the electron, flavor anomalies demonstrate the deviation beyond Standard Model. They are connected with a new understanding of quantum field theory t...
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| Format: | Article |
| Language: | English |
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Makarenko Sumy State Pedagogical University
2022-04-01
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| Series: | Фізико-математична освіта |
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| Online Access: | https://fmo-journal.org/index.php/fmo/article/view/143 |
| _version_ | 1819022911885803520 |
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| author | Тетяна Обіход |
| author_facet | Тетяна Обіход |
| author_sort | Тетяна Обіход |
| collection | DOAJ |
| description | Formulation of the problem. Precision studies of the Higgs boson, supersymmetric particles, the magnetic moment of the muon, electric dipole moment of the electron, flavor anomalies demonstrate the deviation beyond Standard Model. They are connected with a new understanding of quantum field theory through the unification of gravity with particle physics in the framework of string theory - the powerful instrument, which has changed the theory picture. The article is devoted to the study of new physics through these two components. First, we considered particle physics in terms of the latest experimental data and then moved on to the mathematical apparatus of string theory.
Materials and methods. The N = 2 Yang-Mills theory is the heterotic string analog determined in ten-dimensional space: four usual space-time coordinates and six extra dimensions, known as Calabi-Yau manifold in weighted projective space. We studied the Calabi-Yau manifold in terms of both differential forms and reflexive polyhedrа to extract the elementary particle information. For further work with Calabi-Yau manifolds, differential forms for calculation of cohomology groups and reflexive polyhedrа for calculation of Hodge numbers were used. We used two definitions of general properties of toric varieties: hypersurfaces in terms of differential forms and projective space in terms of reflexive polyhedral. Then we investigated lattice polyhedra ∆ which gives rise to families of Calabi-Yau hypersurfaces in weighted projective space, P∆. Such polyhedra admit a combinatorial characterization and are called reflexive polyhedra.
Results. The comparison of two approaches to the description of Calabi-Yau manifold as a complex manifold and as weighted projective space led us to the conclusion about the equivalence of these two treatments in the context of calculation of the Euler characteristic. As Euler’s characteristic for elementary particle physics is the number of generations of quarks and leptons, the selection of Calabi-Yau manifolds with appropriate topological properties is one of the urgent problems of modern physics. It is necessary to stress that the important result of our paper is the coincidence of the value of the Euler characteristic, found in terms of Dolbeault cohomology and terms of reflexive polyhedral. The obtained information about topological invariants is necessary for predicting the number of generations in particle physics.
Conclusions. Although a unified theory of all interactions has not yet been found, however, certain aspects related to the interpretation of the unified theory of all interactions in terms of modern mathematics give their significant results. Therefore, the use and development of the apparatus of algebraic geometry for finding topological invariants that have the value of observables in physics is an urgent task. |
| first_indexed | 2024-12-21T04:30:32Z |
| format | Article |
| id | doaj.art-f4e122abb01f40f99e53d15c765e97f5 |
| institution | Directory Open Access Journal |
| issn | 2413-1571 2413-158X |
| language | English |
| last_indexed | 2024-12-21T04:30:32Z |
| publishDate | 2022-04-01 |
| publisher | Makarenko Sumy State Pedagogical University |
| record_format | Article |
| series | Фізико-математична освіта |
| spelling | doaj.art-f4e122abb01f40f99e53d15c765e97f52022-12-21T19:15:57ZengMakarenko Sumy State Pedagogical UniversityФізико-математична освіта2413-15712413-158X2022-04-01331262910.31110/2413-1571-2022-033-1-004144FORMATION OF MODERN MATHEMATICAL APPROACH TO SOLVING PROBLEMS OF PHYSICSТетяна Обіход0https://orcid.org/0000-0003-1103-4006Institute for Nuclear Research NAS of Ukraine, UkraineFormulation of the problem. Precision studies of the Higgs boson, supersymmetric particles, the magnetic moment of the muon, electric dipole moment of the electron, flavor anomalies demonstrate the deviation beyond Standard Model. They are connected with a new understanding of quantum field theory through the unification of gravity with particle physics in the framework of string theory - the powerful instrument, which has changed the theory picture. The article is devoted to the study of new physics through these two components. First, we considered particle physics in terms of the latest experimental data and then moved on to the mathematical apparatus of string theory. Materials and methods. The N = 2 Yang-Mills theory is the heterotic string analog determined in ten-dimensional space: four usual space-time coordinates and six extra dimensions, known as Calabi-Yau manifold in weighted projective space. We studied the Calabi-Yau manifold in terms of both differential forms and reflexive polyhedrа to extract the elementary particle information. For further work with Calabi-Yau manifolds, differential forms for calculation of cohomology groups and reflexive polyhedrа for calculation of Hodge numbers were used. We used two definitions of general properties of toric varieties: hypersurfaces in terms of differential forms and projective space in terms of reflexive polyhedral. Then we investigated lattice polyhedra ∆ which gives rise to families of Calabi-Yau hypersurfaces in weighted projective space, P∆. Such polyhedra admit a combinatorial characterization and are called reflexive polyhedra. Results. The comparison of two approaches to the description of Calabi-Yau manifold as a complex manifold and as weighted projective space led us to the conclusion about the equivalence of these two treatments in the context of calculation of the Euler characteristic. As Euler’s characteristic for elementary particle physics is the number of generations of quarks and leptons, the selection of Calabi-Yau manifolds with appropriate topological properties is one of the urgent problems of modern physics. It is necessary to stress that the important result of our paper is the coincidence of the value of the Euler characteristic, found in terms of Dolbeault cohomology and terms of reflexive polyhedral. The obtained information about topological invariants is necessary for predicting the number of generations in particle physics. Conclusions. Although a unified theory of all interactions has not yet been found, however, certain aspects related to the interpretation of the unified theory of all interactions in terms of modern mathematics give their significant results. Therefore, the use and development of the apparatus of algebraic geometry for finding topological invariants that have the value of observables in physics is an urgent task.https://fmo-journal.org/index.php/fmo/article/view/143standard modelcalabi-yau manifoldtoric varietiesdifferential formweighted projective spacereflexive polyhedraldolbeault cohomologyeuler characteristic |
| spellingShingle | Тетяна Обіход FORMATION OF MODERN MATHEMATICAL APPROACH TO SOLVING PROBLEMS OF PHYSICS Фізико-математична освіта standard model calabi-yau manifold toric varieties differential form weighted projective space reflexive polyhedral dolbeault cohomology euler characteristic |
| title | FORMATION OF MODERN MATHEMATICAL APPROACH TO SOLVING PROBLEMS OF PHYSICS |
| title_full | FORMATION OF MODERN MATHEMATICAL APPROACH TO SOLVING PROBLEMS OF PHYSICS |
| title_fullStr | FORMATION OF MODERN MATHEMATICAL APPROACH TO SOLVING PROBLEMS OF PHYSICS |
| title_full_unstemmed | FORMATION OF MODERN MATHEMATICAL APPROACH TO SOLVING PROBLEMS OF PHYSICS |
| title_short | FORMATION OF MODERN MATHEMATICAL APPROACH TO SOLVING PROBLEMS OF PHYSICS |
| title_sort | formation of modern mathematical approach to solving problems of physics |
| topic | standard model calabi-yau manifold toric varieties differential form weighted projective space reflexive polyhedral dolbeault cohomology euler characteristic |
| url | https://fmo-journal.org/index.php/fmo/article/view/143 |
| work_keys_str_mv | AT tetânaobíhod formationofmodernmathematicalapproachtosolvingproblemsofphysics |