Combinatorial proof of a non-renormalization theorem

Internet publication

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 xv∈R and to each edge e the combination se=a−12e(x+e−x−e), where x±e are the positions of the two end vertices of e, and ae is a Schwinger parameter. The "topological propagator" Pe=e−s2edse includes a part proportional to dxv and a part proportional to dae. Integrating the product of all Pe over positions produces a differential form αΓ in the variables ae. We derive an explicit combinatorial formula for αΓ, and we prove that αΓ∧αΓ=0.

Authors: Balduf, P-H; Gaiotto, D

• Publication status: Published
• Peer review status: Not peer reviewed
• Host title: arXiv
• Publication date: 2024-08-06
• DOI: 10.48550/arXiv.2408.03192
• Language: English