KFU. Card publication. Flow polynomials as Feynman amplitudes and their $\alpha$-representation

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FEDERAL UNIVERSITY

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FLOW POLYNOMIALS AS FEYNMAN AMPLITUDES AND THEIR $\ALPHA$-REPRESENTATION

Form of presentation

Articles in international journals and collections

Year of publication

2017

Язык

английский

Authors, employees of KFU

Lerner Eduard Yulevich, author

Mukhamedzhanova Sofya Alfisovna, author

Other authors

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

Full metadata record

Field DC	Value	Language
dc.contributor.author	Lerner Eduard Yulevich	ru_RU
dc.contributor.author	Mukhamedzhanova Sofya Alfisovna	ru_RU
dc.contributor.author	Kupcov Andrey Pavlovich	ru_RU
dc.date.accessioned	2017-01-01T00:00:00Z	ru_RU
dc.date.available	2017-01-01T00:00:00Z	ru_RU
dc.date.issued	2017	ru_RU
dc.identifier.citation	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.	ru_RU
dc.identifier.uri	https://repository.kpfu.ru/eng/?p_id=234426&p_lang=2	ru_RU
dc.description.abstract	The Electronic Journal of Combinatorics	ru_RU
dc.description.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.	ru_RU
dc.language.iso	ru	ru_RU
dc.subject	Flow polynomial	ru_RU
dc.subject	Kontsevich's conjecture	ru_RU
dc.subject	Laplacian matrix	ru_RU
dc.subject	Feynman amplitudes	ru_RU
dc.subject	Legendre symbol	ru_RU
dc.subject	Tutte 5-flow conjecture	ru_RU
dc.title	Flow polynomials as Feynman amplitudes and their $\alpha$-representation	ru_RU
dc.type	Articles in international journals and collections	ru_RU