Combinatorial proof of a Non-Renormalization Theorem

TL;DR

This paper presents a direct combinatorial proof of a Feynman graph identity that leads to a broad generalization of Kontsevich's formality theorem, connecting graph integrals with algebraic structures.

Contribution

It introduces a new combinatorial method to prove a Feynman graph identity, extending the applicability of the formality theorem in mathematical physics.

Findings

Derived an explicit combinatorial formula for the differential form mma_

Proved that mma_      0, establishing a key algebraic property

Connected graph integrals with algebraic formality structures

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 $\Gamma$, 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 $\alpha_\Gamma$ in the variables $a_e$. We derive an explicit combinatorial formula for $\alpha_\Gamma$, and we prove that $\alpha_\Gamma \wedge \alpha_\Gamma=0$.