Combinatorial proof of a Non-Renormalization Theorem
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908402493423616 |
|---|---|
| author | Balduf, Paul-Hermann Gaiotto, Davide |
| author_facet | Balduf, Paul-Hermann Gaiotto, Davide |
| contents | We provide a direct combinatorial proof of a Feynman graph identity which implies a wide generalization of a formality theorem by Kontsevich. For a Feynman graph $Γ$, we associate to each vertex a position $x_v \in \mathbb R$ and to each edge $e$ the combination $s_e = a_e^{-\frac 12} \left( x^+_e - x^-_e \right)$, where $x^\pm_e$ are the positions of the two end vertices of $e$, and $a_e$ is a Schwinger parameter. The "topological propagator" $P_e = e^{-s_e^2}\text d s_e$ includes a part proportional to $\text d x_v$ and a part proportional to $\text d a_e$. Integrating the product of all $P_e$ over positions produces a differential form $α_Γ$ in the variables $a_e$. We derive an explicit combinatorial formula for $α_Γ$, and we prove that $α_Γ\wedge α_Γ=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03192 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Combinatorial proof of a Non-Renormalization Theorem Balduf, Paul-Hermann Gaiotto, Davide Mathematical Physics High Energy Physics - Theory Combinatorics 81T18, 81Q30, 05C31, 18G85 We provide a direct combinatorial proof of a Feynman graph identity which implies a wide generalization of a formality theorem by Kontsevich. For a Feynman graph $Γ$, we associate to each vertex a position $x_v \in \mathbb R$ and to each edge $e$ the combination $s_e = a_e^{-\frac 12} \left( x^+_e - x^-_e \right)$, where $x^\pm_e$ are the positions of the two end vertices of $e$, and $a_e$ is a Schwinger parameter. The "topological propagator" $P_e = e^{-s_e^2}\text d s_e$ includes a part proportional to $\text d x_v$ and a part proportional to $\text d a_e$. Integrating the product of all $P_e$ over positions produces a differential form $α_Γ$ in the variables $a_e$. We derive an explicit combinatorial formula for $α_Γ$, and we prove that $α_Γ\wedge α_Γ=0$. |
| title | Combinatorial proof of a Non-Renormalization Theorem |
| topic | Mathematical Physics High Energy Physics - Theory Combinatorics 81T18, 81Q30, 05C31, 18G85 |
| url | https://arxiv.org/abs/2408.03192 |