Combinatorial proof of a non-renormalization theorem

A bstract 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 $$ {s}_e={a}_e^{-\frac{1}{2}}\left({x}_e^{+}-{x}_e^{-}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>s</mml:mi> <mml:mi>e</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>a</mml:mi> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:msubsup> <mml:mfenced> <mml:mrow> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mi>e</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mo>−</mml:mo> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mi>e</mml:mi> <mml:mo>−</mml:mo> </mml:msubsup> </mml:mrow> </mml:mfenced> </mml:math> , where $$ {x}_e^{\pm } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>x</mml:mi> <mml:mi>e</mml:mi> <mml:mo>±</mml:mo> </mml:msubsup> </mml:math> are the positions of the two end vertices of e , and a e is a Schwinger parameter. The “topological propagator” $$ {P}_e={e}^{-{s}_e^2}{\textrm{d}s}_e $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>P</mml:mi> <mml:mi>e</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:msubsup> <mml:mi>s</mml:mi> <mml:mi>e</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:msup> <mml:msub> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> </mml:mrow> <mml:mi>e</mml:mi> </mml:msub> </mml:math> includes a part proportional to d x v and a part proportional to 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 ∧ α Γ = 0 for all graphs except for trees.