Co-Homology of Differential Forms and Feynman Diagrams
In the present review we provide an extensive analysis of the intertwinement between Feynman integrals and cohomology theories in light of recent developments. Feynman integrals enter in several perturbative methods for solving non-linear PDE, starting from Quantum Field Theories and including Gener...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-09-01
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| Series: | Universe |
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| Online Access: | https://www.mdpi.com/2218-1997/7/9/328 |
| _version_ | 1827681119705759744 |
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| author | Sergio Luigi Cacciatori Maria Conti Simone Trevisan |
| author_facet | Sergio Luigi Cacciatori Maria Conti Simone Trevisan |
| author_sort | Sergio Luigi Cacciatori |
| collection | DOAJ |
| description | In the present review we provide an extensive analysis of the intertwinement between Feynman integrals and cohomology theories in light of recent developments. Feynman integrals enter in several perturbative methods for solving non-linear PDE, starting from Quantum Field Theories and including General Relativity and Condensed Matter Physics. Precision calculations involve several loop integrals and an onec strategy to address, which is to bring them back in terms of linear combinations of a complete set of integrals (the master integrals). In this sense Feynman integrals can be thought as defining a sort of vector space to be decomposed in term of a basis. Such a task may be simpler if the vector space is endowed with a scalar product. Recently, it has been discovered that, if these spaces are interpreted in terms of twisted cohomology, the role of a scalar product is played by intersection products. The present review is meant to provide the mathematical tools, usually familiar to mathematicians but often not in the standard baggage of physicists, such as singular, simplicial and intersection (co)homologies, and hodge structures, that are apt to restate this strategy on precise mathematical grounds. It is intended to be both an introduction for beginners interested in the topic, as well as a general reference providing helpful tools for tackling the several still-open problems. |
| first_indexed | 2024-03-10T07:09:50Z |
| format | Article |
| id | doaj.art-9f8889e2931a416581160d037b8e608c |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-10T07:09:50Z |
| publishDate | 2021-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Universe |
| spelling | doaj.art-9f8889e2931a416581160d037b8e608c2023-11-22T15:32:56ZengMDPI AGUniverse2218-19972021-09-017932810.3390/universe7090328Co-Homology of Differential Forms and Feynman DiagramsSergio Luigi Cacciatori0Maria Conti1Simone Trevisan2Dipartimento di Scienza ed Alta Tecnologia, Università degli Studi dell’Insubria, 22100 Como, ItalyDipartimento di Scienza ed Alta Tecnologia, Università degli Studi dell’Insubria, 22100 Como, ItalyDipartimento di Scienza ed Alta Tecnologia, Università degli Studi dell’Insubria, 22100 Como, ItalyIn the present review we provide an extensive analysis of the intertwinement between Feynman integrals and cohomology theories in light of recent developments. Feynman integrals enter in several perturbative methods for solving non-linear PDE, starting from Quantum Field Theories and including General Relativity and Condensed Matter Physics. Precision calculations involve several loop integrals and an onec strategy to address, which is to bring them back in terms of linear combinations of a complete set of integrals (the master integrals). In this sense Feynman integrals can be thought as defining a sort of vector space to be decomposed in term of a basis. Such a task may be simpler if the vector space is endowed with a scalar product. Recently, it has been discovered that, if these spaces are interpreted in terms of twisted cohomology, the role of a scalar product is played by intersection products. The present review is meant to provide the mathematical tools, usually familiar to mathematicians but often not in the standard baggage of physicists, such as singular, simplicial and intersection (co)homologies, and hodge structures, that are apt to restate this strategy on precise mathematical grounds. It is intended to be both an introduction for beginners interested in the topic, as well as a general reference providing helpful tools for tackling the several still-open problems.https://www.mdpi.com/2218-1997/7/9/328Feynman integralstwisted cohomologyintersection theory |
| spellingShingle | Sergio Luigi Cacciatori Maria Conti Simone Trevisan Co-Homology of Differential Forms and Feynman Diagrams Universe Feynman integrals twisted cohomology intersection theory |
| title | Co-Homology of Differential Forms and Feynman Diagrams |
| title_full | Co-Homology of Differential Forms and Feynman Diagrams |
| title_fullStr | Co-Homology of Differential Forms and Feynman Diagrams |
| title_full_unstemmed | Co-Homology of Differential Forms and Feynman Diagrams |
| title_short | Co-Homology of Differential Forms and Feynman Diagrams |
| title_sort | co homology of differential forms and feynman diagrams |
| topic | Feynman integrals twisted cohomology intersection theory |
| url | https://www.mdpi.com/2218-1997/7/9/328 |
| work_keys_str_mv | AT sergioluigicacciatori cohomologyofdifferentialformsandfeynmandiagrams AT mariaconti cohomologyofdifferentialformsandfeynmandiagrams AT simonetrevisan cohomologyofdifferentialformsandfeynmandiagrams |