Flow polynomials as Feynman amplitudes and their \(\alpha\)-representation. (English) Zbl 1355.05136
Summary: 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 \(\alpha_e\in \mathbb F_q\). M. Kontsevich [Math. Phys. Stud. 20, 139–156 (1997; Zbl 1149.53325)] conjectured that the number of nonzero values of \(s(\alpha, G)\) is a polynomial in \(q\) for all graphs. This conjecture was disproved by P. Brosnan and P. Belkale [Duke Math. J. 116, No. 1, 147–188 (2003; Zbl 1076.14026)]. 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.
Keywords:
flow polynomial; Kontsevich’s conjecture; Laplacian matrix; Feynman amplitudes; Legendre symbol; Tutte 5-flow conjectureReferences:
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