Flow Polynomials as Feynman Amplitudes and their $\alpha$-Representation

Abstract

Let $G$ be a connected graph; denote by $\tau(G)$ the set of its spanning  trees. Let $\mathbb F_q$ be a finite field, $s(\alpha,G)=\sum_{T\in\tau(G)}  \prod_{e \in E(T)} \alpha_e$, where $\alpha_e\in \mathbb F_q$. Kontsevich  conjectured in 1997 that the number of nonzero values of $s(\alpha, G)$ is a polynomial in $q$ for all graphs. This conjecture was disproved by  Brosnan and Belkale. In this paper, using the standard technique of the Fourier transformation of Feynman amplitudes, we express the flow polynomial $F_G(q)$ in terms of the "correct" Kontsevich formula. Our formula represents $F_G(q)$ as a linear combination of Legendre symbols of $s(\alpha, H)$ with coefficients $\pm 1/q^{(|V(H)|-1)/2}$, where $H$  is a contracted graph of $G$ depending on $\alpha\in \left(\mathbb F^*_q \right)^{E(G)}$, and $|V(H)|$ is odd.