Flow polynomials as Feynman amplitudes and their α-representation

Abstract

© 2017, Australian National University. All Rights Reserved.Let G be a connected graph; denote by τ(G) the set of its spanning trees. Let Fq be a finite field, (formula presented), where αe ∈ Fq. Kontsevich conjectured in 1997 that the number of nonzero values of s(α, 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 FG(q) in terms of the “correct” Kontsevich formula. Our formula represents FG(q) as a linear combination of Legendre symbols of s(α, H) with coefficients (formula presented), where H is a contracted graph of G depending on (formula presented), and |V (H)| is odd.