Combinatorial proof of a non-renormalization theorem

Abstract

Abstract 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 ∈ ℝ and to each edge e the combination s e = a e − 1 2 x e + − x e − $$ {s}_e={a}_e^{-\frac{1}{2}}\left({x}_e^{+}-{x}_e^{-}\right) $$ , where x e ± $$ {x}_e^{\pm } $$ 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 d s e $$ {P}_e={e}^{-{s}_e^2}{\textrm{d}s}_e $$ includes a part proportional to dx v and a part proportional to da 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 ∧ α Γ = 0 for all graphs except for trees.

Keywords

• BRST Quantization
• Topological Field Theories