Kazan (Volga region) Federal University, KFU
KAZAN
FEDERAL UNIVERSITY
 
FLOW POLYNOMIALS AS FEYNMAN AMPLITUDES AND THEIR $\ALPHA$-REPRESENTATION
Form of presentationArticles in international journals and collections
Year of publication2017
Языканглийский
  • Lerner Eduard Yulevich, author
  • Mukhamedzhanova Sofya Alfisovna, author
  • Kupcov Andrey Pavlovich, author
  • Bibliographic description in the original language A. P. Kuptsov, E. Yu. Lerner, S. A. Mukhamedjanova, Flow polynomials as Feynman amplitudes and their $\alpha$-representation, Electron. J. Combin., 24 (2017), no. 1, paper 11, 19 pp.
    Annotation 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.
    Keywords Flow polynomial, Kontsevich's conjecture, Laplacian matrix, Feynman amplitudes, Legendre symbol, Tutte 5-flow conjecture
    The name of the journal The Electronic Journal of Combinatorics
    Please use this ID to quote from or refer to the card https://repository.kpfu.ru/eng/?p_id=234426&p_lang=2

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