КФУ. Карточка публикации. Flow polynomials as Feynman amplitudes and their $\alpha$-representation

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

Форма представления

Статьи в зарубежных журналах и сборниках

Год публикации

2017

Язык

английский

Авторы, сотрудники КФУ

Лернер Эдуард Юльевич, автор

Мухамеджанова Софья Альфисовна, автор

Другие авторы

Купцов Андрей Павлович, автор

Библиографическое описание на языке оригинала

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.

Аннотация

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.

Ключевые слова

Flow polynomial, Kontsevich's conjecture, Laplacian matrix, Feynman amplitudes, Legendre symbol, Tutte 5-flow conjecture

Название журнала

The Electronic Journal of Combinatorics

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https://repository.kpfu.ru/?p_id=234426

Полная запись метаданных

Поле DC	Значение	Язык
dc.contributor.author	Лернер Эдуард Юльевич	ru_RU
dc.contributor.author	Мухамеджанова Софья Альфисовна	ru_RU
dc.contributor.author	Купцов Андрей Павлович	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/?p_id=234426	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	Статьи в зарубежных журналах и сборниках	ru_RU