Invariant Differential Forms on Complexes of Graphs and Feynman Integrals
We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential forms in question generate the stable real cohomology of the...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2021 |
| Автор: | |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2021
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/211424 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Invariant Differential Forms on Complexes of Graphs and Feynman Integrals. Francis Brown. SIGMA 17 (2021), 103, 54 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862534407120748544 |
|---|---|
| author | Brown, Francis |
| author_facet | Brown, Francis |
| citation_txt | Invariant Differential Forms on Complexes of Graphs and Feynman Integrals. Francis Brown. SIGMA 17 (2021), 103, 54 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential forms in question generate the stable real cohomology of the general linear group, as shown by Borel. By integrating such invariant forms over the space of metrics on a graph, we define canonical period integrals associated to graphs, which we prove are always finite and take the form of generalised Feynman integrals. Furthermore, canonical integrals can be used to detect the non-vanishing of homology classes in the commutative graph complex. This theory leads to insights about the structure of the cohomology of the commutative graph complex, and new connections between graph complexes, motivic Galois groups, and quantum field theory.
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| first_indexed | 2026-03-12T14:58:26Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211424 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T14:58:26Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Brown, Francis 2026-01-02T08:30:09Z 2021 Invariant Differential Forms on Complexes of Graphs and Feynman Integrals. Francis Brown. SIGMA 17 (2021), 103, 54 pages 1815-0659 2020 Mathematics Subject Classification: 18G85; 11F75; 11M32; 81Q30 arXiv:2101.04419 https://nasplib.isofts.kiev.ua/handle/123456789/211424 https://doi.org/10.3842/SIGMA.2021.103 We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential forms in question generate the stable real cohomology of the general linear group, as shown by Borel. By integrating such invariant forms over the space of metrics on a graph, we define canonical period integrals associated to graphs, which we prove are always finite and take the form of generalised Feynman integrals. Furthermore, canonical integrals can be used to detect the non-vanishing of homology classes in the commutative graph complex. This theory leads to insights about the structure of the cohomology of the commutative graph complex, and new connections between graph complexes, motivic Galois groups, and quantum field theory. This project has received funding from the European Research Council (ERC)under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 724638). Many thanks to M.Chan (who spotted a mistake in an earlier proof of Theorem 7.4), S. Galatius (who pointed out that (4.1) follows from the Amitsur–Levitzki theorem), S. Payne, G. Segal, for discussions, and especially R. Hain and K. Vogtmann, of whom the present project is an offshoot of joint work. I am very grateful to O. Schnetz for computing the above examples of canonical integrals, M. Borinsky for sharing his computations of Euler characteristics, C. Dupont and the referees for many helpful comments and corrections. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Invariant Differential Forms on Complexes of Graphs and Feynman Integrals Article published earlier |
| spellingShingle | Invariant Differential Forms on Complexes of Graphs and Feynman Integrals Brown, Francis |
| title | Invariant Differential Forms on Complexes of Graphs and Feynman Integrals |
| title_full | Invariant Differential Forms on Complexes of Graphs and Feynman Integrals |
| title_fullStr | Invariant Differential Forms on Complexes of Graphs and Feynman Integrals |
| title_full_unstemmed | Invariant Differential Forms on Complexes of Graphs and Feynman Integrals |
| title_short | Invariant Differential Forms on Complexes of Graphs and Feynman Integrals |
| title_sort | invariant differential forms on complexes of graphs and feynman integrals |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211424 |
| work_keys_str_mv | AT brownfrancis invariantdifferentialformsoncomplexesofgraphsandfeynmanintegrals |